---
source_pdf: Density_Field_Dynamics__A_Complete_Unified_Theory__v5_0.pdf
title: "Density Field Dynamics: A Complete Unified Theory"
version: "v5.0"
date: "2026-10-26"
site: https://densityfielddynamics.com/
author: Gary Alcock
framework: Density Field Dynamics (DFD)
doi: "10.5281/zenodo.18066593"
format_note: "Markdown extracted mechanically from the source PDF for clean AI ingestion. No editorial changes; mathematical typography in the PDF is authoritative. v5.0 supersedes v4.0."
---

Density Field Dynamics: A Complete Unified Theory
Gary Alcock1, ∗
1

Independent Researcher, Los Angeles, CA, USA
(Dated: v5.0 — October 2026)
(Foundations: 19 Aug 2025 [3]; Unified v1.0: 25 Dec 2025 [4])
This is release v5.0; the prior public version is v4.0. It adds evidence appendices, the measured
clock coefficient (item 6 below), two further results, and the DLSC structural extension: the earlyclock CMB fit (App. ZS: all six spectrum-shape inputs supplied upstream of the data with per-input
grades printed, three derived or adopted-and-priced, the tilt a stated number and ωb at its calibrated
conventional-point value, with two further calibrations free (the clock and the amplitude As e−2τ ),
H0early = 67.52 ± 0.09); the gravity completion and Kohn null (App. ZU) — Newton recovered as an
exact zero-parameter identity of the quantized ψ action, with an exactly zero self-gravity sideband on
a trapped crystal’s center-of-mass mode that makes the Schrödinger–Newton shift (semiclassically
∆ω/ω of 1.2–6.7 × 10−13 for Os, 1.1–6.2 × 10−15 for Si) a falsifier of the
√ completion, not a DFD
prediction; and the domain-local screen calibration (App. ZT), a0 (x, t) = 2 α c Hscr (x, t), conditional
on the DLSC axiom (2 discrete bits, 0 fitted continuous parameters), with SH0ES (73.49 ± 0.93 in
2025) as the named external falsifier of that early-clock calibration at 5.0–6.4σ; under the two-clock
postulate of item 6, SH0ES’s value is read on the bound-matter clock, 1.5σ above H0loc = 72.09.
Density Field Dynamics (DFD) is a scalar refractive-index theory of gravity defined by the postulate
that spacetime is flat but permeated by a scalar field ψ(x, t) establishing an optical refractive index
n = eψ . Light propagates according to the eikonal of the optical metric ds̃2 = −c2 dt2 /n2 + dx2 ,
while matter responds to the effective potential Φ = −c2 ψ/2. This framework has an optical scalar
sector ψ that governs clock rates, refraction, and quasi-static dynamics, together with a transversetraceless radiative sector hTT
ij for gravitational waves, derived as the spin-2 irreducible component
of the same zero-mode parent tensor on CP 2 × S 3 whose trace yields ψ [1]. It reproduces all classic
tests of general relativity in the weak-field limit (γ = β = 1, all PPN parameters matching GR),
gravitational waves at speed c with two tensor polarizations, and MOND-like phenomenology at
galactic scales through a nonlinear crossover
√ function µ(x) = x/(1 + x), the weak-response form of an
exponential response, and scale a0 = 2 α cH0loc , both derived from S 3 topology (Appendix N;
the form of µ holds given
pthe ground-state principle, with its branch chosen by data, Appendix BS;
equivalently a0 = 2α29 c7 /Gℏ on bound-matter clocks (item 6 below), with the exponent and
prefactor derived, Appendix AP). A dedicated model-independent SPARC shape analysis further
finds nopt = 1.15 ± 0.12 (95% CI [1.00, 1.50]) in the family µn (x) = x/(1 + xn )1/n , with DFD’s n = 1
inside the confidence region and Standard MOND’s n = 2 strongly disfavored.
This paper presents DFD as a unified framework: (1) Fine-structure constant: α−1 =
137.036 from the microsector spectral action on CP 2 × S 3 with Toeplitz truncation at kmax = 60. The
closed form is convention-locked: a forced binary fork between regular-module and fermion-rep
microsectors is resolved by a no-hidden-knobs policy, with the surviving branch giving a closed form
that is exact under two stated choices (the ρ-shifted Casimir on S 3 = SU (2) and α’s own trace with
the heat-kernel regulator; §X C): its integers alone give the measured value to 21 ppm, and with the
bracket coefficient c = 1/13, whose provenance is open, it lies a fractional 4.9 × 10−9 (0.0049 ppm)
from the CODATA 2022 value, 32 times that value’s uncertainty. The trace in the closed form is
the hypercharge trace, and its identification with 1/e2 rather than with the hypercharge coupling is
not derived (§X I). (1b) Weinberg angle: sin2 θW = 3/13 = 0.2308 from gauge partition (3, 2, 1)
exp
and a stated trace normalization (Appendix Z). Agreement with sin2 θW
= 0.23129 [2]: 0.2%
(tree-level vs MS); (1c) Strong coupling: αs (MZ ) √
= 0.1187 from ΛQCD = MP α19/2 = 61.20 MeV
and the selected proper-time→ MS matching factor 4π (Appendix
Z). Agreement with PDG 2024
√
(0.1180 ± 0.0009): 0.8σ; (2) Higgs hierarchy: v = MP × α8 × 2π = 246.09 GeV (observed: 246.22
GeV)—the
exponent is derived, so the 17 orders of magnitude are topological, not fine-tuned, while
√
the prefactor 2π is one fitted input with a no-go on deriving it. The Higgs quartic λH = 1/8
from dimension counting is a conjecture (Appendix Z); it gives tree-level mH = 123 GeV and 124.6
GeV at the top-quark
scale, 0.40% below the measured value; (3) Nine charged fermion masses:
√
mf = Af αnf v/ 2 with sector-dependent exponents fits with a 1.42% mean error at leading order,
with CP2 kernels Kd = J3 , Ku = I4 fixed by symmetry and democratic coupling (Lemma K.2),
QCD factors from b0 = 7, generation operator G = diag(2/3, 1, 1) from the primed microsector trace
(Theorem K.4, Appendix K); the top and charm are derived, and the other seven species rest on
seven data-selected prefactors (App. FM);
(4) CKM pattern: Wolfenstein parameters fit the integer pattern (λ, A, ρ̄, η̄) = (31, 108, 43
, 49)×α,
2
with the CP-even apex ρ̄ = 43
α
=
21.5α
selected
by
the
Euler-projection
postulate
of
the
strengthened
2
DFD–SD branch (Appendices Z and AO). Per-channel agreement against PDG 2024: λ at 0.5%

2
(1.8σ), η̄ at 1.5% (0.7σ), A at 4.6% (2.5σ, sensitive to the open inclusive/exclusive |Vcb | tension), ρ̄
at 1.4% (0.2σ). The magnitude integers are suggestive of cohomological origin, while the apex is
selected at theorem grade inside the strengthened DFD–SD branch by the Euler-projection postulate
(Appendix AO), which, consistent with the no-forcing anti-theorem of that appendix, is a new
structural postulate rather than a consequence of the pre-strengthened axioms. The branch predicts
γ = 66.3093◦ and J = 2.9731 × 10−5 . PMNS trimaximal TM1 (App. X), with sin2 θ13 = 3α a stated
conjecture; (5) Strong CP: the spectral action fixes θYM = 0, given that nothing is added to it
for the gauge fields, and θ̄ = 0 at tree level and loop-bounded below 10−16 , with det(Mu Md ) kept
positive by the det-orthogonal structure on the branch of Postulate Y.10; the CP anomaly vanishes
because the fermion Clifford module on the even-dimensional (8) mapping torus has spectral pairing
and no zero modes, forcing η = h = 0 (Appendices L, AO). No axion required. The minimal
real-kernel flavor sector gives J = 0 (Theorem AO.7); the observed nonzero CKM phase (J ̸= 0) is
supplied by the conjugation-odd, determinant-orthogonal Berry offset of the strengthened DFD–SD
branch (Postulate Y.10, Appendix AO), which preserves θ̄ = 0;
(6) G–H0 invariant (exponent derived; coefficient measured): The dimensionless constraint
GℏH02 /c5 = C α57 has its exponent derived via Gaussian mode integration on the finite-dimensional
microsector (Appendix O): the exponent 57 is forced by primed-determinant scaling on the finite
microsector state space; the per-mode suppression factor α follows from uniform gauge normalization
with exact eigenvalue cancellation (Lemma O.5); the identification with the physical hierarchy uses
the finite dimensionality of the microsector to eliminate all UV ambiguities (Lemma O.6). The
constraint is written GℏH02 /c5 = C α57 : the exponent 57 is a theorem, while the dimensionless
coefficient C is proven not derivable from the printed corpus (eight-route no-go, App. ZS) and rests
on one measured input, the vacuum coefficient R = ΩΛ C = 0.6042 ± 0.0027, with today’s early-clock
value C = 0.87784 ± 0.00266 (BAO+θ∗ ; plik-lite 0.8772), with the cross-channel H0early = 67.52 ± 0.09
km/s/Mpc concordant across five CMB-anchored channels; the Hubble tension is not read as a
ψ-screen bias of Planck (the background ψ mode adds no screen beyond the expansion it describes,
and the optical metric leaves θ∗ exactly invariant, Theorems in App. ZS). Both measurements of the
Hubble rate are robust, and DFD reads them as one universe on two clocks (Postulate ZS.11): matter
keeps a clock offset by one constant λ, set when matter first binds and falls behind the expansion, so
the CMB-anchored channels read H0early = 67.52, and bound-matter clocks and rulers read the C = 1
value H0loc = α28.5 /tP = 72.09 km/s/Mpc, from the measured G with zero parameters (Sec. XIX);
its size λ = H0loc /H0early is the one measured input (Sec. ZS 5). The local evidence agrees with the
bound-matter
the Cepheid-free local mean 71.87 ± 0.96 lies 0.2σ from H0loc , bound systems
√ clock:
loc
read a0 = 2 α cH0 = 1.1966 × 10−10 m/s2 , consistent with SPARC within its systematic error, and
the coefficient booking of the Extended Derivations’ five-presentation Λ theorem (C = 1.027 ± 0.029,
the ladder’s value re-expressed) lies 0.9σ from C = 1; the price is a laboratory-unit age of 12.92 Gyr
against ages of the universe of 13.5–13.8 Gyr that one programme infers from globular clusters with
a ΛCDM formation delay, and the absolute a0 scale (Euclid/DESI lensing RAR) tests the postulate
once baryonic masses are known to about 2%; (7) Solar corona (Sec. XIV): electromagnetic fields
couple to ψ at tree level through their energy and the optical index n = eψ , with no threshold, so
DFD predicts the standard Ly-α/O VI asymmetry ratio; the 2007–2009 SOHO/UVCS data do not
measure that ratio; (8) CMB from baryon loading + derived χ: Peak ratio R = 2.34 from the
analytic baryon-loading estimate, all peak locations set by the clock coefficient measured from BAO
and θ∗ , with no ψ-lensing map (App. ZS), third-peak height
√ from the derived χ-matter dark matter
— a cold, collisionless component of mass 32.00 eV (= 2π 158 M̄P α13 , the flux-period identification)
whose abundance Ωχ h2 = 0.1193 is computed forward from the finite SU (2)60 Chern–Simons vacuum
(−0.6σ from Planck), with no initial-condition angle
√ and no fitted density (App. AV, Thm. AV.13);
Ledger grade: the carrier/operator is derived, the 158 mass multiplicity is one stated count, and
the abundance is forced-amplitude/stated- convention, its measure conditional on the asserted
premise P4′ , rather than a fitted density (App. FK); (9) One dark energy: the α57 vacuum term
(Appendix AU) is DFD’s only dark energy, and the late-time distance excess is not read as an
optical-screen effect (Appendix ZS; Appendix SL);
(10) Clock sector and Majorana scale (Appendix P): the electromagnetic-sector proposal
kα = α2 /(2π) and the scale MR = MP α3 are obtained under the Appendix O protocol, each
conditional on an explicitly stated assumption (for kα , the one-gauge-vertex axiom: a postulated
non-metric operator outside the minimal action, Thm. XI.2). The clock sector is interpreted
in a channel-resolved way: same-ion E3/E2 measurements strongly constrain any pure α-sector
coupling law, while cross-species and nuclear-clock channels discriminate the extended non-metric
channel. The cavity–atom sector likewise has geometric cancellation at tree level and a residual
screened signal, not an order-unity slope; the 2026 Th-229 reproducibility result already excludes
the unscreened strong-sector amplitude and compresses the surviving nuclear-clock window into
the rough range 26 Hz to O(1 kHz); (10b) Neutrino mass spectrum (Appendix X): Closed
at zero continuous parameters; one discrete branch choice (Branch B, among four equally natural

3
fractions) is selected by the oscillation data, with everything downstream α-locked (branch-selection
remark in Appendix X). Branch B exponents k = α−3/11 , r = α−7/20 from microsector integers;
absolute scale m3 = (14/13)πMP α14 from finite-d priming. Predictions:
∆m221 = 7.48 × 10−5 eV2 ,
P
∆m231 = 2.51 × 10−3 eV2 (NuFIT 6.0: χ2 = 0.025, p = 0.99);
mν = 61.46 meV; combined
hierarchy exponent k2 r2 = α−137/110 ; (11) Dust branch under K ′ = µ (Appendix Q): The
temporal kinetic function K(∆) is built on the same S 3 saturation-union composition law that gives
the weak-response µ(x); the identification K ′ (∆) = µ(∆) is adopted, not forced. Key results: (i)
temporal deviation invariance is forced by the composition law; (ii) the unique temporal segment
scalar is ∆ = (c/a0 )|ψ̇ − ψ̇0 |; (iii) with K ′ (∆) = µ(∆), the dust branch emerges with w → 0,
c2s → 0. A no-go lemma proves the naive quadratic identification gives w → 1/2 (not dust). Full
P (k) matching is a program item, not a theorem. (12) Screen-closure identities (Sec. XVI A 4;
the optical-distance dressing they rest on is retired, App. SL): The identities are retired
with that dressing; distance duality, ∆ψdual = 0 by Etherington reciprocity, stands (item E1).
Additional sectors: (13) Antimatter gravity (Sec. XV): Species-dependent sensitivities σA
from non-metric ψ-sector couplings predict matter–antimatter differential acceleration ∆aH H̄ /a ≈
2|σH̄ − σH |. At the metric level, DFD reproduces GR’s universal free fall; C-odd couplings (nB ,
nL ) are bounded by ordinary-matter EP tests to |∆aH H̄ /a| ≲ 10−11 ; (14) EM–ψ coupling
(Appendix R): Parameter λ controls electromagnetic back-reaction on ψ in a parametric-mode
model that needs a dynamical ψ mode, which the local ψ of Axiom WS.1 lacks, and whose absolute
normalization is open. Existing cavity stability provides an accidental bound |λ − 1| ≲ 3 × 10−5 .
An intentional 2ω modulation search could reach |λ − 1| ∼ 3 × 10−9 with upgraded apparatus;
(15) IBVP well-posedness (Sec. III E): Existence, uniqueness, and continuous dependence
hold at theorem grade for the initial-boundary value problem of the scalar’s numerical hyperbolic
relaxation on bounded domains (under Axiom E the scalar itself is an instantaneous constraint);
(16) Late-time observations (Sec. XVI N): the DES Y3 Weyl potential is 2–3σ shallower at
low z (not a DFD prediction); DESI DR2 hints at w(z) ̸= −1 (an open tension: DFD’s w = −1);
wide binaries are contested, with DFD predicting no boost near the Sun; EG is consistent on the
current Planck-like matter fraction, and the dependency-aware S8 ≃ 0.822 result is consistent with
KiDS-Legacy (0.45–0.74σ for the primary/joint anchors) and 2.8σ above DES Year 6 (survey errors
only).
New structural results: (17) The Padé identity and the horizon (Appendix AA):
The lapse-squared scalar L(u) ≡ c2 /|gtt | of isotropic-coordinate Schwarzschild is mathematically
identical to the squared [1, 1] Padé approximant of the exponential exp(2u), the value of DFD’s
lapse-squared eψ at ψ = 2u. Writing Pm,m (u) for the diagonal [m, m] Padé approximant of eu
(so Pm,m (u) ≈ eu ): LGR (u) = [P1,1 (u)]2 = [(1 + u/2)/(1 − u/2)]2 , and the identity extends to a
Padé hierarchy in which [Pm,m (u)]2 = exp(2u) + O(u2m+1 ). Agreement through O(u2 ) reproduces
the post-Newtonian parameter β = 1 in both theories (the spatial-curvature parameter γ = 1 for
DFD follows separately from the physical metric, Sec. IV). Beyond that order DFD’s exterior is
not the exponential: with the scalar-longitudinal piece σ of the parent strain kept (Sec. V A 3), the
Einstein–Hilbert term (App. WS) makes the exterior of a static star the Schwarzschild
metric in


DFD’s coordinates, lapse e−ψ/2 and det gij = e3ψ , with ψ = − ln 1 − 2GM/c2 R at areal radius
R. The pole at u = 2 is therefore a horizon of DFD’s black holes as well: Schwarzschild’s event
horizon at R = 2GM/c2 , with Schwarzschild’s photon sphere and shadow (Secs. VI B and VI C).
A firewall theorem distinguishes DFD from Yilmaz-type exponential metrics, which arise within
GR as wormhole solutions with exotic matter (Theorems AA.2, AA.3); (18) Uniqueness of the
internal manifold (Appendix F 6): The manifold X = CP 2 × S 3 used throughout DFD is not
an ansatz. It is the unique compact manifold, up to diffeomorphism, satisfying seven satisfiable
vacuum axioms: compactness, minimality of dimension, nondegenerate Kähler-times-frame product
structure, a parallelizable frame factor, χtop (Mc ) = 3 (three generations), simple connectedness
of the frame factor, and positive internal Ricci curvature (Theorem AB.1). Derived rather than
assumed: dimension 7, the Spinc structure, π3 = Z, even mapping-torus dimension (Strong-CP),
and kmax = 60 (through the conditional determinant-line count, given the generation count, the
flux quantum and the five multiplet types). The (3, 2, 1) block structure lives on the internal mode
bundle over spacetime, not on T X (no tangent-bundle (3, 2, 1, 1) splitting exists, Lemma AB.13);
the electroweak group is the isotropy group U (2) of CP 2 at the vacuum point under one stated
identification, while colour and the family content are inputs, as in the Standard Model (App. F).
The forced decomposition 7 = 4 + 3 gives Mc = CP 2 uniquely (del Pezzo surface with χ = 3
under positive Ricci) and Mg = S 3 uniquely (Perelman: closed simply connected 3-manifold). The
integers {3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137} are therefore not free parameters: they are fixed by
this manifold together with the stated inputs and identifications of App. FK.
(19) Quantum sector (Sec. XVII, Appendix QM): the single-particle Schrödinger equation is
derived as an algebraic identity of the master equation (the fixed-N interacting form likewise), with
the quantum phase i identified with the CP 2 Kähler structure; the imports are explicitly inventoried

4
— the value of ℏ, the variable-N field-theoretic map, and the Cl(4, 0) enlargement supplying the Dirac
β — with the Born rule conditional and single-outcome selection not derived (a residue shared by
all interpretations). The operator completion of the same action forces the linear side (App. ZU):
the Schrödinger–Newton frequency shift is the falsifier of the completion, not a DFD prediction
(levitated-optomechanics sideband); Distance duality and screen clarification: (E1) Distance
duality (Sec. XVI): Etherington’s reciprocity theorem holds exactly in DFD’s optical metric, with
no e∆ψ factor in the distance duality relation: DL = (1 + z)2 DA exactly. The notation separates
∆ψscreen (distance bias, Estimators A and C) from ∆ψdual = 0 (DDR consistency, Estimator B).
The gauge emergence framework on CP 2 × S 3 yields the electroweak group as the isotropy group
of CP 2 under one stated identification, with colour, the family content and Ngen = 3 entering as
inputs, as in the Standard Model (App. F), and proton stability (if baryon number is violated
only by the electroweak anomaly). The extended derivations — closures and promotions, the
new-territory frontier, the CKM/τ and series IV–VI derivations, and the mathematical-foundation
and quantum-gravity closures — are documented in the companion volume Density Field Dynamics
v5.0: Extended Derivations and Frontier Predictions. The present paper states the theory in its
current form: axioms, theorems with proofs, predictions, and falsifiers.
DFD fits no continuous parameter to the shape of any observable it predicts, except
the baryon density. Every ratio, hierarchy and spectrum shape in the graded prediction set is
delivered by derived or discretely selected structure, and the tilt ns = 1 − 2/60 is a stated number
(the mechanism is open); the individual results that carry an imported normalization instead (the
charged-fermion prefactors of App. FM, the within-multiplet ηq of App. Y) are graded as such at their
own sites and are not reported as parameter-free. The theory’s cosmology carries four continuous
calibration constants: the vacuum coefficient R = ΩΛ C = 0.6042 ± 0.0027 (calibrating the
absolute scale; today’s early-clock value C = 0.87784 ± 0.00266), the amplitude As of the primordial
perturbations (the formula 32πα5 matches it, but the printed theory contains no mechanism for it:
the perturbations enter through one stated starting condition), the coefficient of ωb h2 (the one shape:
its α4 power is derived, and the printed mechanism gives ηB of about (2–3) × 10−9 , three to five times
Planck’s value), and the reionization optical depth τ (an astrophysical quantity shared with ΛCDM).
That is four against ΛCDM’s six free parameters, and on the Planck likelihood itself — where As
and τ collapse into As e−2τ — three against its five effective parameters (App. ZS; full per-input
accounting in App. FK, Rem. FK.8, and the inputs of its graded predictions in Tables CXXVII
and CXXVIII). The discrete topological sector carries no continuous freedom: the isotropy charge of
CP 2 fixes q1 = 3, the determinant-line lift gives O(9) (with Ngen = 3 a discrete input), and the five
chiral multiplet types fix the padding (status remarks: App. F). Within the bundle decomposition
E = O(a) ⊕ O⊕n , minimal-padding uniquely selects (a, n) = (9, 5) with kmax = 60. The discrete
inputs are priced rather than free: a trials-factor audit (App. CE) enumerates the admissible discrete
space (8.0 × 109 addresses) and shows that, given the stated bracket coefficient c = 1/13, landscape
scanning cannot account for the closeness of the α value. One scale measurement (H0 or G),
together with the measured vacuum coefficient R, then determines all dimensionful quantities via
GℏH02 /c5 = C α57 . This paper presents the mathematical formulation and demonstrates that DFD
constitutes a unified framework for gravity and particle physics, falsifiable with current experimental
technology.
Two further results sharpen the flavor and propagation sectors. The CKM mixing magnitudes
fit, at zero continuous free parameters, the selected integer pattern (λ, A, ρ̄, η̄) = (31, 108, 43
, 49) α
2
over kmax = 60 = |A5 | (Appendix AO), and the weak CP phase vanishes in the literal real-kernel
flavor construction (J = 0 at theorem grade), with the observed nonzero phase supplied by a single
conjugation-odd determinant-orthogonal Berry offset whose apex is fixed by the Euler-projection
postulate of the strengthened DFD–SD branch, giving γ = 66.3093◦ and J = 2.9731 × 10−5
(undetermined by the pre-strengthened axioms, by the no-forcing anti-theorem of Appendix AO).
Separately, the light-propagation law c1 = c/n and the matter-acceleration law a = (c2 /2)∇ψ are
unified as the null and timelike sectors of a single optical-metric wave equation, with the exact
identity a = − 14 ∇(c2light ) (Appendix AN).

Scope and status. This manuscript separates DFD
claims into theorem-grade results, candidate extensions,
program-grade numerical infrastructure, external empirical tests, anti-theorems/no-go results, and implementation artifacts. The verification record (Tier-1 closures, a

∗ gary@gtacompanies.com

reconciliation round, and Tier-2 to Tier-5+6 closures; the
five development-record syntheses are not reprinted in this
deposit) carries 91 results entered at theorem grade.
Their review status differs by tier and is stated here rather
than averaged into a single number. Each Tier-1–6 closure
above was checked against its stated proof at the time of
entry; results inherited from releases predating that tiered
programme have not all been re-examined to the same

5
standard, and a ranked, deliberately unadjudicated list
of the inherited statements flagged by a systematic proofpattern scan is printed in §XVIII K. This manuscript
makes no claim of zero internal inconsistencies.
It maintains instead a corrigenda record: corrections
are made at the statements they amend, the principal
ones are tabulated below, and the standing-review list
is printed for the same reason. The main open items
are named rather than hidden, among them the origin
of the primordial perturbations and the horizon problem,
which enter through one stated starting condition (the
amplitude As is a calibrated number), the interacting
quantum dynamics shared with every quantum field theory, and in the dark sector the inner cores and masses
of the χ halos, the mode-by-mode rule of motion of the
momentum-conserving energy, the adopted floor law and
environmental confinement, and the weak lensing of isolated galaxies beyond their bound radius, where the halo
model of ΛCDM also falls below the measured points
(see §XVIII A; the Qχ = 1 χ-clustering channel is closed
Principal corrections to earlier statements.

by the adopted Rest-Mass Channel axiom, App. GR.3a,
which confines the µ-law to bound systems). The χ relic
abundance is conditional-derived (the Finite SU (2)60 CS
Vacuum Relic with the exact radiation-era bridge κrad ,
Ωχ h2 = 0.1193 at −0.6σ; App. AV, Step 5b), its named
open inputs the two stated normalization conventions,
the stated count 158, the flux-period identification of
fχ = M̄P α3 and the asserted premise that the 2π/α winding instanton changes the flux sector (App. AV). Claims
should be evaluated by the tier labels and falsifier IDs in
the master ledger, not as a single undifferentiated claim.
Companion volume. The extended derivations —
closures and promotions, the new-territory frontier,
the CKM/τ and series IV–VI derivations, and the
mathematical-foundation and quantum-gravity closures
— are documented separately in Density Field Dynamics
v5.0: Extended Derivations and Frontier Predictions. The
present paper contains the theory in its current form.

6
Earlier statement

Current status

Superseded by the strengthened DFD–SD branch, ρ̄ = 43
2 α
(Euler-projection postulate; App. AO).
Minimal real DFD gives CKM CP
Minimal real kernel gives J = 0 (theorem); nonzero J requires
the DFD–SD det-orthogonal offset.
Replaced by the nonlinear kinetic crossover; rotation curves
Direct ka a2 /c2 MOND source
from the AQUAL
field equation (App. AP).
p
a0 calibrated from SPARC/RAR
a0 = 2α29 c7 /Gℏ = 1.1966 × 10−10 m/s2 on bound-matter
clocks (Postulate ZS.11; 1.1208×10−10 on the early clock); the
law is derived, and SPARC validates rather than calibrates.
“0.55% CKM mean agreement”
Retracted (computed against PDG 2018–2020); per-channel
agreement reported instead.
Fermion-mass mean error quoted as “0.047%” Corrected to the table-derived 1.42% at leading order; the
refined 0.61% value is archive-claimed (priming theorem),
and the 0.082% value is not a result, since the running factors behind it are not Standard-Model running (Ext. Deriv.,
App. AH).
Matter-wave T 3 protocol (App. W) screened- Harmonized to the kinematic expression of the matter-wave
coupling form
section, which is not a phase (units s2 /m), so it gives no perfacility estimate or falsification threshold. Status: not a DFD
prediction (the matter-wave sector is null); see the status note
after Eq. (337) in Sec. XIII.
Quantum framework unlisted among inputs
Added as an explicit background-input row in the DoF ledger
and as an open program item; QM is not derived in full
(App. QM lists the imports).
“Five independent routes” to α−1
Reconciled: the companion volume’s independence statement
now defers to the categorical engine count — three structurally
distinct engines, five presentations.
Glueball normalization chain (companion vol- Note added: the printed coefficient forms do not close nuume)
merically and the flavor-scheme input is mislabeled; m0++ =
1.69 ± 0.10 GeV retained as target, scaling m ∝ ΛQCD retained.
Uniqueness axioms (V1)–(V9) of X = CP 2 × S 3 Replaced: (V6) asserted a splitting of T X that no CP 2 × N 3
admits, so the list was unsatisfiable; the seven current axioms
are satisfiable (App. AB).
Pairwise-kSZ consistency stated unconditionally The verdict rests on quasi-Newtonian linear growth, which
the Rest-Mass Channel’s bound-system confinement supplies
(Geff = G for unbound perturbations; App. GR); the suppressed amplitude of App. AF belongs to the baryon-only
construction of App. AE.
CKM apex ρ̄ = 19α

CONTENTS

I. Introduction
A. Theories of Gravity
B. Core Idea: Gravity as an Optical
Medium
C. What DFD Claims and What It
Doesn’t
D. Reader’s Guide
E. Assumptions and Degrees of Freedom
Ledger
F. Completion Ledger
II. Mathematical Formalism
A. The Optical Metric and Geodesics
1. Gordon’s Optical Metric
2. Fermat’s Principle
3. Phase and Group Velocities

20
20
21
21
23
23
24
24
25
25
25
25

B. Action Principle
1. Scalar Sector Action
2. Matter Coupling
3. Gravitational Wave Sector
4. Interaction and Complete Action
C. Field Equations
1. General Nonlinear Form
2. The Optical Source Theorem: DFD’s
replacement for Tµν
D. The µ(x) Crossover Function
1. Admissible Families
E. Conserved Quantities and Symmetries
1. Diffeomorphism Invariance
2. Energy Conservation
3. Local Conservation in PPN
Framework
F. 4D-from-3D: Emergent Spacetime
Structure

26
26
26
27
27
27
27
27
28
29
29
29
29
29
29

7
1. The Fundamental Arena
2. The 3D-to-4D Morphism
G. Physical Interpretation: Vacuum
Loading
H. Summary of Section II
III. Mathematical Well-Posedness
A. Static Solutions: Elliptic Theory
1. Assumptions on µ
2. Existence and Uniqueness
3. Regularity
B. Exterior Domains and Boundary
Conditions
C. Dynamic Solutions: Hyperbolic
Theory
1. First-Order Symmetric Hyperbolic
Form
2. Local Well-Posedness
3. Finite Speed of Propagation
D. Stability
1. Energy Minimization
2. Perturbative Stability
3. No Ghosts
E. Initial-Boundary Value Problems
1. Dynamic Structural Assumptions
2. IBVP Formulation
3. Compatibility Conditions
4. Energy Estimates
5. Main IBVP Theorem
6. Finite Speed of Propagation
7. Parabolic Extension
8. Stability Estimates
9. Numerical Implementation
F. Open Mathematical Problems
G. Summary of Section III
IV. Parametrized Post-Newtonian Analysis
A. The PPN Framework
B. DFD Physical Metric in PPN Form
C. Parameter Extraction: γ = β = 1
D. Vector Sector: α1 = α2 = α3 = 0
E. Preferred-Frame Invisibility: Why the
Foliation Decouples at 1PN
F. Conservation Laws:
ζ1 = ζ2 = ζ3 = ζ4 = 0
G. Summary: DFD Equals GR at 1PN
H. Classic Solar System Tests
1. Light Deflection
2. Shapiro Time Delay
3. Perihelion Precession
4. Gravitational Redshift
5. Frame Dragging and Lense-Thirring
Effect
I. Where DFD Differs from GR
V. Gravitational Waves
A. Two Gravitational Sectors on Flat R3
1. The Optical Sector (DFD Core)

29
29
30
30
30
31
31
31
31
31
32
32
32
33
33
33
33
34
34
34
34
34
34
35
35
35
35
35
35
36
36
36
37
37
38
39
41
41
41
41
41
42
42
42
43
43
43
43

2. The Radiative Sector (Tidal
Disturbances)
43
3. Parent Strain Field and Irreducible
Decomposition
44
4. Spectral-Geometry Origin of the
Two-Sector Structure
44
5. Why cT = c (Structural
Requirement)
45
6. Adiabatic Limit and GW Speed in the
Unified Picture
45
7. Falsifiability
45
B. The Minimal Transverse-Traceless
Sector
46
C. Verification: cT = c from No Derivative
Mixing
46
1. The Flat-Background Wave
Equation
46
2. Why No Derivative Mixing is Natural
in DFD
46
D. Wave Equation and Source Coupling
47
E. Quadrupole Formula and Energy Flux
47
F. Post-Newtonian and ppE Framework
47
1. Conservative and Dissipative
Parametrization
47
2. Phase Coefficients
47
G. Comparison with LIGO-Virgo-KAGRA
Observations
48
1. DFD Predictions for Compact
Binaries
48
2. Comparison with LVK O3 Bounds
48
3. Falsifiability and Future Tests
48
H. Binary Pulsar Verification
48
1. The Hulse-Taylor System
48
2. DFD Prediction
48
3. Quantitative Comparison
49
4. Other Binary Pulsars
49
5. Bounds on DFD Parameters
49
I. Numerical Evolution for Compact
Binaries
49
1. Evolution System
50
2. Boundary Conditions
50
3. AMR Strategy
50
4. Validation Tests
50
J. Summary and Implications
50
VI. Strong Fields and Compact Objects
A. Static Spherical Solutions
B. Optical Causal Structure
C. Photon Spheres
D. Black Hole Shadows: EHT
Comparison
1. DFD in the Strong-Field Regime
2. M87* Shadow
3. Sgr A* Shadow
4. Summary Comparison
E. Constrained µ-Function Family
1. The Constrained Family µα,λ (x)
F. Compact Star Structure

51
51
51
51
52
52
52
52
52
53
53
53

8
G. Potential DFD-Specific Signatures

53

VII. Galactic Dynamics
54
A. The Deep-Field Limit
54
B. Galaxy Rotation Curves
55
C. The Baryonic Tully-Fisher Relation
55
D. The Radial Acceleration Relation
55
E. Predicted acceleration scale and empirical
validation
56
F. Quantitative SPARC Validation
57
G. Model-Independent
Interpolation-Function Shape Test
57
H. Wide Binary Stars
58
I. External Field Effect
59
J. Dwarf Spheroidal Galaxies
60
1. Jeans Analysis with EFE
60
2. Two-Regime Model
60
3. Comparison with Data
60
4. Ultra-Faint Dwarfs: Systematic
Effects
60
K. Cluster-Scale Phenomenology
61
1. Cluster Dynamics in DFD
61
2. Cluster Sample Analysis
61
3. Physical Interpretation
61
4. Nonlinear Averaging and the
Correction Stack
62
5. The Bullet Cluster: Quantitative
Analysis
63
6. Global Consistency: One Function,
Galaxies to Clusters
63
L. Summary: Galactic Phenomenology
64
VIII. The α-Relations: Parameter-Free
Predictions
A. The Fundamental Relations
B. Relation I: The Self-Coupling
ka = 3/(8α)
C. Relation II: The Clock Coupling
kα = α × ae
D. Relation III: The MOND Scale a0
(Derived)
E. Consistency and Cross-Checks
F. The Three-Scale Hierarchy
G. Status Summary
IX. Gauge Coupling Variation and High-Energy
Implications
A. Universal Gauge-ψ Coupling
B. Connection to the β-Function
C. Modified Renormalization Group
Equations
D. Asymptotic Freedom and UV Behavior
E. Nuclear Clock Prediction:
Thorium-229
F. Cosmological α(z) Variation
G. Grand Unification
H. Vacuum Energy Feedback
I. Summary of Falsifiable Predictions

64
65
65
65
66
66
66
67
67
67
68
68
68
68
69
70
71
71

X. Convention-Locked α from the Microsector
A. Design Constraint: No Hidden Tuning
Parameters
B. Operator Choice (Locked)
C. Regularization/Truncation Rule
(Locked)
D. Finite-k Truncation and the
(k + 3)/(k + 4) Factor (Locked)
E. The Forced Microsector Fork
1. Branch A: Regular-Module
Microsector (Survives)
2. Branch B: Fermion-Representation
Microsector (Falsified)
F. Branch Selection by Real-Bimodule
Consistency
G. The Closed-Form Chain
H. Sharp Falsifier
I. The Closed-Form Result
J. Summary
XI. Atomic Clock Tests
A. Local Position Invariance Framework
B. Common-Factor Cancellation and
Observable Residuals
C. Screening: Derivation from a Response
Functional
D. The Same-Ion E3/E2 Constraint
E. Cross-Species Atomic Comparisons
1. ROCIT Statistical Detail
F. Nuclear Clocks: the Strong-Sector
Channel
G. Channel-Resolved Prediction Table
H. Empirical Checks and Current Status
I. Experimental Priorities

71
72
72
72
72
73
73
74
74
74
75
75
76
76
76
77
79
79
80
80
81
82
82
82

XII. Cavity-Atom Redshift Tests
A. Formal Constitutive Proof of the
Cancellation
B. What Survives Physically
C. Four Independent Empirical Checks
D. BACON and the Screening Regime
E. Sector-Resolved Parameterization
F. The 4→3 GLS Protocol
G. Experimental Concept and Controls
H. Expected Signal and Sensitivity
I. Current Status and Priority
J. Summary: Cavity–Atom as a Precision
Residual Test

83

XIII. Matter-Wave Interferometry
A. The ψ-Coupled Schrödinger Equation
B. The T 3 Discriminator (Not a DFD
Prediction)
C. Experimental Designs
1. Design A: Vertical Fountain
2. Design B: Horizontal Rotation
3. Design C: Source Mass Modulation
4. Design D: Dual-Species Protocol

86
86

83
84
84
84
85
85
85
85
86
86

86
87
87
87
87
87

9
D. Discriminants and Systematics Control
E. Sensitivity Forecast
F. Existing Analyses and the T 3 Term
G. MAGIS and AION as Null Tests
H. Complementarity with Cavity-Atom
Test
I. Summary: Matter-Wave Test

87
88
88
88
89
89

XIV. Solar Corona Spectral Asymmetry
Analysis
89
A. Motivation: Intensity Changes Without
Velocity Changes
89
B. Electromagnetic Coupling to ψ
90
C. SOHO/UVCS Ly-α Analysis
90
1. Statistical Methodology: Permutation
Tests and FDR Control
90
D. Multi-Species Test: O VI 103.2 nm
90
1. Data and Methods
90
2. Results
90
E. Consistency Check: Intensity Without
Velocity
90
F. Summary
90
G. Quantitative Multi-Wavelength Test: The
Asymmetry Ratio
91
1. Thermal Width Analysis
91
2. The Generalized Prediction
91
3. Comparison with Observations
91
XV. Antimatter Gravity Tests
A. GR Baseline: Matter–Antimatter
Universality
B. DFD Metric-Level Prediction
C. Non-Metric Couplings and
Species-Dependent Sensitivities
1. Bound-State Mass Shifts
2. CPT Considerations
D. Matter–Antimatter Differential
Acceleration
1. Effective Point-Particle Action
2. Free-Fall Acceleration
E. Three Scenarios for σH̄ − σH
F. Experimental Mapping: ALPHA-g and
Beyond
1. ALPHA-g Free-Fall Measurements
2. Spectroscopy Complement
G. Relation to Ordinary-Matter EP Tests
H. DFD Prediction and Falsification
I. Summary
XVI. Cosmological Implications
A. ψ-Tomography (ψ-Screen) Cosmology
Module
1. DFD postulates and sign
conventions
2. Forward model: three optical
relations
3. Two independent screen estimators
and one consistency check

91
91
92
92
92
92
92
92
92
92
93
93
93
93
93
94
94
94
94
94
95

4. Theorem-level internal closure of the
reconstructed screen
95
5. A GR-independent falsifier
96
6. Evolving “constants” as controlled
parameters
98
7. Practical next steps
98
B. The ψ-Universe framework
98
C. CMB observables as ψ-screened
measurements
99
1. Asymmetry Factor Decomposition
99
D. The optical illusion principle
99
E. Intrinsic anisotropy from ψ-gradients
99
F. Line-of-sight distance bias and apparent
acceleration
100
G. Cluster-scale dynamics: Status
100
H. Scope of CMB claims
100
I. ISW Effect: A Falsifiable Prediction
100
J. The late-time distance ratio is not a
screen
101
K. Late-Time Distance Cross-Consistency 101
L. Matter Power Spectrum from
Microsector
101
M. Power Spectrum Multipole
Confrontation
103
1. Method
103
2. Results
103
3. Interpretation: EFE-Screened Growth
Theorem
103
4. Conclusion and Falsifier
104
N. Observational Status (2024–2025)
104
1. Late-Time Potential Shallowing (DES
Y3)
104
2. Dynamical Dark Energy Hints (DESI
DR2)
104
3. Wide Binaries (Active and
Contested)
104
4. Counter-Evidence and Null Tests
104
5. Observational Summary Table
105
O. Hierarchy of Astrophysical Scales from
α
105
P. Summary
105
XVII. Quantum and Gauge Extensions
A. Status and Conditionality
B. Internal Mode Bundle and Berry
Connections
C. The C3 ⊕ C2 ⊕ C partition
D. Yang-Mills Kinetic Terms from Frame
Stiffness
E. Generation Counting
F. CP Structure
G. Higgs and Mass Spectrum
H. The Fine-Structure Constant from
Chern-Simons Theory
1. Chern-Simons Quantization
2. The Maximum Level: Topological
Derivation
3. Lattice Comparison

105
106
106
107
107
107
107
108
108
108
108
108

10
I. The Bridge Lemma: kmax = 60 from
Closed Index
109
1. Statement
109
2. Proof
109
3. Physical Selection
109
J. Nine Charged Fermion Masses
109
1. The Mass Formula
109
2. Sector-Dependent Exponents
109
3. Structural Ratios
109
K. CKM Matrix from CP 2 Geometry
110
1. Wolfenstein Parameterization
110
2. Geometric Derivation
110
3. Predictions
110
L. Electroweak-Scale Relation
110
1. The Relation
110
2. Physical Origin
110
M. Strong CP: Tree-Level Closure,
Loop-Bounded
110
1. Tree Level
110
2. Loop Level
111
N. PMNS Matrix from CP 2 Geometry
111
1. Observed Mixing
111
2. Physical Mechanism
111
3. Tribimaximal Base
112
4. Corrections
112
O. Infrared Scale for Yang-Mills from DFD
Geometry
112
1. Setup: DFD Spatial Geometry
112
2. Weitzenböck Identity
112
3. The DFD-Induced Infrared Bound
112
4. Clarification: What This Does Not
Claim
112
P. Testable Predictions
113
Q. Caveats and Required Verification
113
XVIII. Open Problems and Limitations
114
A. Main Open Items
115
B. Quantum Superpositions and the Penrose
Paradox
117
C. UV Completion: the Topological
Proposal
118
D. Hyperbolicity and Numerical
Evolution
119
E. Cluster-Scale Phenomenology: the χ
Budget
119
F. Cosmological Constant: the Hierarchy
from Topology
120
G. Full Cosmological Treatment
120
H. Null Predictions: Where DFD Says “No
Effect”
121
I. Experimental Verification Timeline
121
J. Closed Routes
121
K. Standing Internal Review of Inherited
Results
121
L. Summary: Resolved and Remaining
Items
123
XIX. A Topological Link Between H0 and MP

125

A. The Dimensionless Invariant
B. Implication for the Cosmological
Constant Problem
C. Testable Consequence: The Hubble
Constant
D. Epoch structure of the G–H0 invariant:
coupling versus state function
E. The homogeneous optical-time
Hamiltonian (background expansion)
F. The Parameter Structure
XX. Conclusions
A. Summary of Density Field Dynamics
B. What DFD Accomplishes
C. The Critical Tests
D. If DFD Is Confirmed
E. If DFD Is Falsified
F. Comparison with Alternatives
G. Outlook
H. Structural Separation: Gravity vs.
Microsector
I. Final Statement

125
126
126
127
130
131
132
132
133
134
135
135
135
136
136
137

A. Notation and Conventions
140
1. Fundamental Fields and Parameters
140
2. Coordinate and Metric Conventions
140
3. Physical Constants
140
4. Post-Newtonian and Gravitational Wave
Parameters
140
5. Clock and LPI Parameters
140
6. Galactic Dynamics Notation
140
7. Unit Conventions
140
8. Abbreviations and Acronyms
140
9. Sign Convention Summary
140
B. Detailed Derivations
1. Second Post-Newtonian Light
Deflection
a. Setup
b. Ray Equation
c. First-Order (1PN) Deflection
d. Second-Order (2PN) Deflection
2. Perihelion Precession
a. Effective Potential
b. Orbit Equation
c. Precession Rate
d. Mercury
3. Baryonic Tully-Fisher from
µ-Crossover
a. Deep-Field Limit
b. Spherical Symmetry
c. Asymptotic Velocity
d. Zero-Point
4. α-Relation Derivations
√
a. Relation I: a0 = 2 α cH0
b. Relation II: ka = 3/(8α)
c. Relation III: kα = α2 /(2π)
d. Consistency Check

141
142
142
142
142
143
143
143
143
143
143
143
143
144
144
144
144
144
144
144
144

11
5. Matter-Wave Phase Shift
a. Phase Evolution
b. Three-Pulse Interferometer
c. Species-Coupling Expression (Not a
DFD Prediction)
6. Gravitational Wave Emission
a. Quadrupole Formula
b. Binary Inspiral

145
145
145
145
145
145
145

C. Interpolating Function Catalog
1. General Requirements
2. Catalog of Functional Forms
3. Simple Interpolating Function
4. Standard Interpolating Function
5. RAR Empirical Function
6. The n-Family
7. Comparison of Properties
8. Calibration Procedure
9. Physical Interpretation

145
145
145
145
146
146
146
146
146
147

D. Experimental Protocols
1. Clock Comparison Procedure
a. Measurement Overview
b. Technical Requirements
c. Recommended Clock Pairs
d. Data Analysis
e. Systematic Error Budget
f. Windowed vs. Global Analysis
Strategies
2. Cavity-Atom Setup Requirements
a. Experiment Concept
b. Key Configuration
c. Technical Specifications
d. Height Comparison Method
e. Observable
f. Discrimination Significance
3. Matter-Wave Interferometer
Specifications
a. Formula Under Test (Not a DFD
Prediction)
b. Interferometer Requirements
c. Dual-Species Configuration
d. T 3 Signature (Not a DFD
Prediction)
e. Systematic Control
4. Galaxy Rotation Curve Analysis
a. Data Requirements
b. Baryonic Mass Model
c. DFD Fitting Procedure
d. Quality Metrics
5. Decision Matrix: Which Experiment to
Prioritize

147
147
147
147
147
147
148

E. Data Tables
1. Post-Newtonian Parameter Bounds
2. Binary Pulsar Timing Data
3. Clock Sensitivity Coefficients
4. SPARC Galaxy Sample Statistics

150
150
150
150
151

148
148
148
148
149
149
149
149
149
149
149
149
149
150
150
150
150
150
150
150

5. Gravitational Wave Constraints
6. Physical Constants Summary
7. DFD Parameter Summary
8. Experimental Timeline
F. Rigorous Foundations for Gauge
Emergence
1. Minimality of the (3, 2, 1) Partition
2. The SU (N ) Selection Lemma
3. The Spinc Flux Quantization
4. The Spinc Dirac Index on CP 2
5. Generation Count and Flux-Product
Rule
6. Uniqueness of Minimal Flux
7. Global Fermionic Consistency: the
Spin-Z4 B−L Structure and the
Necessity of νR
8. The Self-Coupling Coefficient ka
(Model)
9. Frame Stiffness from Ricci Curvature
10. Proton Stability
11. UV Robustness of Topological Results
12. Summary: Rigorous vs. Conjectural
G. Derivation of α-Relations from Gauge
Emergence
1. The Gauge-ψ Lagrangian
2. The Magnetically Dominated Regime
3. Frame Stiffness Structure
4. Derivation of ka = 3/(8α)
5. Strong CP Prediction
6. Derivation of kα = α2 /(2π)
7. Proton Stability Prediction
8. Summary of Results
H. Higgs and Yukawa Sector from Gauge
Emergence
1. Higgs Emergence from the (3, 2, 1)
Structure
2. Zero-Mode Localization on CP 2
3. Yukawa Hierarchy from Overlap
Integrals
4. CKM Mixing from Geometry
5. Neutrino Masses from See-Saw
6. Summary of Mass Sector

151
151
151
151
151
151
153
154
156
156
157

157
158
158
158
159
160
160
160
160
160
160
161
161
162
162
162
162
163
163
164
164
164

I. Full Cluster Sample Analysis
1. Dataset Description
2. Complete Results Table
3. Statistical Summary (the µ-Law
Alone)
4. Alternative µ1/2 Function
5. Systematic Uncertainties
6. Conclusions
7. The Five Correction Factors,
Examined
8. Galaxy Groups

165
165
165

167
167

J. Derivation of the ψ-CMB Solution

168

165
166
166
166

12
1. The ψ-Acoustic Oscillator
168
2. Peak Height Asymmetry
168
a. Baryon Loading Factor fbaryon
168
b. Integrated Sachs-Wolfe Factor fISW 168
c. Visibility Function Factor fvis
168
d. Doppler Factor fDop
169
e. Total Asymmetry
169
3. Peak Ratio Derivation
169
4. Why the 1/µ Enhancement Cancels
169
5. The ψ-lensing peak-location construction
(not a DFD prediction)
169
a. Gradient-Index Optics
169
b. Application to CMB
170
6. Consistency Checks
170
7. Comparison with ΛCDM
170
8. Falsifiable Predictions
170
9. The Cold-Clustering Wall: a no-go
theorem for the third-peak height
172
K. Microsector Physics: Complete Derivations 174
1. Closed Form for α = 1/137 from
Chern-Simons Theory
174
a. Setup: Chern-Simons on S 3
174
b. The Level Sum and Fine-Structure
Constant
174
c. Heat Kernel on S 3
174
d. Determination of kmax : Closed Spinc
Index
174
2. Lattice Comparison for α = 1/137
174
a. Inputs (Independent of α)
174
b. The Prediction
175
c. Lattice Comparison
175
d. Falsifiability: What Would Have
Failed
175
e. Finite-Size Scaling
175
f. L16 Detailed Results
176
g. Wilson Ratio Scan
176
h. β Bracket Test
176
i. Additional Runs
176
j. Summary: Lattice Evidence
177
3. The UV Cutoff Consistency Check:
kmax = 60 Compared with the Lattice 177
a. The Truncation Scan
177
b. Physical Interpretation
177
c. Why This Is Not Fine-Tuning
178
d. Systematic Independence Tests
178
4. The Bridge Lemma
178
a. Statement
178
b. Proof
178
c. Physical Selection
178
d. Consistency Checks
178
5. Charged Fermion Mass Derivation
179
a. The Mass Formula
179
b. Sector-Dependent Exponent
Assignment
179
c. Prefactor Structure
179
d. Complete Mass Table
179
e. Statistical Summary
179

f. Structural Ratios
180
g. Explicit Finite Yukawa Operator
180
h. Derivation of G[1, 1] = 2/3 from
Primed Microsector Trace
181
6. CKM Matrix from CP 2 Geometry
182
a. Wolfenstein Parameterization
182
b. Geometric Origin of λ
182
c. Higher-Order Parameters
182
d. Predictions and Comparison
182
e. Key Prediction:
p
|Vub /Vcb | = λ ρ̄2 + η̄ 2
182
7. Summary: Microsector Consistency
182
8. The Higgs Scale Hierarchy
183
a. Numerical Verification
183
b. Physical Origin of Factors
183
9. Strong CP: Tree Level and Loop Bound 183
a. Tree Level
183
b. Loop Level
183
10. PMNS Matrix Derivation
184
a. Physical Picture
184
b. Tribimaximal Mixing
184
c. Corrections from Charged Lepton
Masses (localization estimate; the
adopted branch is TM1, App. X)
184
d. Why PMNS ̸= CKM
184
11. Summary: DFD Unified Framework
184
L. Strong CP: Tree-Level Closure,
Loop-Bounded
184
1. What must be shown
184
2. Tree-level inputs
185
3. The Dai–Freed anomaly formula
185
4. Theorem: η vanishes automatically in
even dimensions
185
5. Main theorem: tree-level closure,
loop-bounded
186
6. A conditional alternative: quaternionic
structure
186
7. Falsifiable prediction
186
8. Summary: why the S 3 factor does triple
duty
186
M. Scattered and Locally Emitted Lines in a
Static ψ
1. Definitions and Setup
2. Gaussian Detuning Scaling
3. The Value of Γ
4. Observational Constraint on Γ
N. First-Principles Derivation of µ(x) and a0
1. The S 3 Partition Function (Exact
Result)
2. Microsector-to-ψ Map and Level
Response
3. The Key Theorem: µ is Fixed by a
Composition Law
4. The Acceleration Scale a0 : Variational
Derivation

187
187
187
187
187
188
188
188
188
189

13
a. The Unique IR Control Parameter
189
b. Microsector Scaling Charge
189
c. The Spacetime Functional
189
d. Homogeneous-Limit Theorem
190
e. The MOND Scale Theorem
190
5. Summary and Falsifiable Predictions
190
6. Alternative Derivation: Variational
Approach
191
a. Setup: Auxiliary-Field Action
191
b. Asymptotic Constraints
191
c. Closed-Form Solution
191
d. Comparison with S3 Result
191
7. The Complete Picture: MOND from S3
Topology
192
O. The α57 Mode-Count Exponent and the
G–H0 –α Invariant
193
1. Mathematical core: primed-determinant
scaling fixes the exponent
193
2. Gaussian mode-integration realization 193
3. From determinant ratio to physical
hierarchy: derivation
193
4. The derived invariant
194
5. Connection to the Einstein Product
Condition
195
P. Clock Coupling and Majorana Scale
1. Scope and Convention Lock
2. Theorem P.1: Schwinger Coefficient
ae = α/(2π)
3. Theorem P.2: Clock Coupling
kα = α2 /(2π)
a. Observational Test: Fine-Structure
Constant Variation
4. Majorana Scale MR = MP α3
a. Parallel Structure with Appendix O
b. Neutrino Mass Predictions
5. Summary
Q. Temporal Completion: Dust Branch under
K′ = µ
1. Temporal Deviation Invariance from
Saturation-Union
2. Unique Local Temporal Invariant
3. No-Go Lemma: Quadratic Invariant
Gives w → 1/2
4. Dust Branch from Deviation-Invariant
Closure
5. Summary: What is Theorem-Grade vs.
Program
6. Primordial optical engine: the
background screen flow
R. EM–ψ Back-Reaction Coupling
1. Physical Interpretation of λ
2. Mode Equation and Pumping
Channels
a. Single Lab-Mode Reduction

196
196
196
196
196
197
197
197
198
198
198
198
199
199
200
200
203
203
203
203

b. Channel 1: Driven Resonance
(2ω = Ωψ )
203
c. Channel 2: Parametric Amplification
(2ω ≃ 2Ωψ )
203
3. Geometry Transparency
203
a. When the Driven Overlap Cancels
203
b. How to Restore the Overlap
204
c. Parametric Overlap: Area-Ratio
Law
204
4. Constraints on |λ − 1|
204
a. Accidental Constraint from Cavity
Stability
204
b. Intentional Search: Projected
Reach
205
5. Why λ ̸= 1 Has Not Been Detected
205
6. Intentional Detection Protocol
205
7. Relation to Core DFD Framework
206
8. Summary
206
9. Dual-Sector Extension: The κ
Parameter
206
a. Constitutive Split Preserving
vph = c/n
206
b. The Unified Bracket
206
c. Standing-Wave Energy Equality
206
d. Experimental Tests
207
e. Experimental Discrimination
207
S. Standard Model Extension Dictionary
1. SME Framework Overview
2. DFD↔SME Correspondence
3. Translation Table
4. Experimental Constraints
Reinterpreted
5. Cavity-Atom Comparisons in SME
Language
T. Family and Clock-Type Parametrization of
LPI Tests
1. Two-Parameter Model
2. Constraints from Data
3. Predictions for Untested Channels
4. Relation to DFD Microsector
5. Summary

207
207
207
207
208
208
208
208
208
209
209
209

U. Mathematical Well-Posedness of the DFD
Field Equations
210
1. The Static Field Equation: Elliptic
Theory
210
a. Structural Assumptions on µ
210
b. Weak Formulation and Variational
Structure
210
c. Main Existence and Regularity
Theorems
210
d. Exterior Domains and Optical
Boundary Conditions
211
2. The Dynamic Field Equation: Hyperbolic
Theory
211

14
a. Structural Assumptions for Hyperbolic
Theory
211
b. Reduction to First-Order Symmetric
Hyperbolic Form
211
c. Local Well-Posedness for the Cauchy
Problem
212
d. Initial-Boundary Value Problems
212
e. Finite Speed of Propagation
212
3. Parabolic Extension and Long-Time
Behavior
212
4. Stability and Continuous Dependence 213
5. Open Problems
213
6. Summary: Mathematical Status of
DFD
213
V. Extended Phenomenology and Numerical
Methods
214
1. The External Field Effect (EFE)
214
a. Physical Origin
214
b. Quantitative Formulation
214
c. Observational Signatures
214
2. Wide Binaries
214
a. The Crossover Scale
214
b. Predicted Velocity Anomaly
214
c. GAIA DR3 Constraints
215
3. Finite Element Implementation
215
a. Weak Form for FEM
215
b. Newton Iteration for Nonlinearity
215
c. Mesh Refinement Strategy
215
d. Boundary Conditions
215
e. Convergence Verification
216
4. Matter Power Spectrum from ψ-Screen 216
a. Scale-Dependent ψ Perturbations
216
b. Observational Signatures
216
5. Cooper-Pair Mass Anomaly from A5 Pair
Space
216
6. Summary
216
W. Experimental Protocols and Sensitivity
Analyses
217
1. Cavity-Atom LPI Test: Complete
Protocol
217
a. Observable and Predictions
217
b. Experimental Configuration
217
c. Measurement Cycle
217
d. Systematics Budget
217
e. Blinding Protocol
217
f. Pre-Registered Decision Rule
218
g. Sensitivity Reach
218
2. Multi-Species Clock Comparison
Protocol
218
a. Observable
218
b. Species Selection
218
c. Analysis Protocol
218
3. Matter-Wave Interferometry: T 3 Protocol
(Not a DFD Prediction)
218
a. Observable
218
b. Parity Isolation
219

c. Sensitivity Requirements
d. Interpretation of Outcomes
4. Nuclear Clock Protocol: Th-229
a. Prediction
b. Experimental Requirements
c. Timeline
5. Space Mission Protocols
a. ACES (ISS)
b. Dedicated LPI Mission
6. Summary: Experimental Roadmap

219
219
219
219
219
219
219
219
219
220

X. Neutrino Mass Spectrum from DFD
Microsector
220
1. DFD Inputs from the Microsector
220
2. Why S3 Invariance Cannot Split the
Doublet
220
3. TBM Selects a Canonical Residual S2 221
4. Microsector-normalized residual-S2
spurion
221
5. Combined mass pattern
(microsector-normalized)
221
6. Parameter-free oscillation invariant
(discriminator)
221
7. Complete numerical predictions
221
8. Absolute-scale closure for Branch B from
finite-d priming
222
9. The explicit mass matrix (TBM
eigenbasis)
223
10. Falsification criteria
223
11. External global-fit verification
223
12. Summary: DFD-closed neutrino sector
(zero continuous parameters)
224
Y. Finite Yukawa Operator, Chiral Basis, and
the Af Prefactors
225
1. Purpose and Scope
225
2. Finite Hilbert Space and
Normalization
225
3. Block Decomposition for the (3, 2, 1)
Microsector
225
4. Finite Higgs Connector as an Explicit
Matrix
225
5. Chiral Subspaces and Canonical
Link-States
225
6. Yfinite as an Explicit Operator and Its
Matrix Elements
225
7. Explicit Evaluation in the Canonical Link
Basis
226
8. Universality Wall and the Required
Additional Structure
226
9. The Ldet Twist: A 3ngen Ratio Pattern
(Not a Parameter-Free Mass)
226
10. The ηq -Free Double Ratio and the Nc3
Color Clebsch
227
11. A5 Species Projectors: Breaking the
Universality Wall
228
a. Channel Space as Group Algebra
228

15
b. Generators and Universal
Connector
228
c. Higgs Kernel from Derived εH
229
d. Species Projectors from Conjugacy
Classes
229
e. Cayley Geometry and Hierarchy
Mechanism
229
f. Species-Resolved Prefactors
229
g. Class-Amplitude Formula
229
h. Proposed Species Assignment Rule 230
12. Complete Status Summary
230
13. Complete Derivation: Generation
Projectors and Down-Type Selection
230
a. Regular Module Factorization
230
b. Phase Factorization on Isotypic
Blocks
230
c. Canonical Generation Projectors
231
d. Down-Type Selection via
Conjugation
231
e. Numerical Bin Scan
231
f. Diagonal Bin Structure
231
g. Light Fermion Limitation
231
h. Generation Projector Results
231
14. √
Bin–Overlap Lemma and the Structural
20 Scale
232
a. Normalized Class-State Matrix
Elements
232
b. Bin–Overlap Lemma for the Order-3
Class
232
c. Species Projector Closure
232
d. Af Prefactor Structure
233
Z. Complete Parameter Derivation
1. The Weinberg Angle
2. The CKM Matrix
3. The Higgs Sector
4. The PMNS Correction
5. Master Theorem
6. Integer Catalog
7. Strong Coupling Constant
8. Summary

233
233
234
234
235
235
235
235
236

AA. The Padé Identity: GR’s Isotropic Lapse and
the Exponential
237
1. Statement of the Padé Identity
237
2. Order-by-Order Agreement
237
3. The Padé Pole is GR’s Horizon
238
4. Distinction from Yilmaz-Type
Exponential Metrics
238
5. The Padé Hierarchy
239
6. Framing Summary
239
AB. Uniqueness of the Internal Manifold
CP 2 × S 3
1. Vacuum Axioms
2. Uniqueness Theorem
3. Reduction of the Axioms

240
240
241
241

a. Dimension, product structure, and the
split 7 = 4 + 3
241
(4)
b. Identification of Mc
241
(3)
c. Identification of Mg
241
d. Consistency with remaining axioms 242
e. Uniqueness assembly
242
4. Tangent-Bundle Obstruction Lemma
242
5. Integer Cohomology and Standard Model
Parameters
243
6. Falsifiability
243
7. Summary
244
AC. Linear Growth Closure from the
EFE-Screened DFD Operator
1. AC.1 Purpose
2. AC.2 Linearization of the Nonlinear
Spatial Operator
3. AC.3 Specialization to the DFD
Interpolation Function
4. AC.4 Cosmological External-Field
Screening
5. AC.5 First Numerical Growth
Estimates
6. AC.6 Interpretation
7. AC.7 Falsifier
8. AC.8 Summary

244
244
244
245
245
245
245
246
246

AD. Regular-Module Forcing from Finite
Real-Bimodule Consistency
246
1. AD.1 Purpose
246
2. AD.2 The Finite Algebra and Its
Role
246
3. AD.3 Minimal Faithful Real-Bimodule
Theorem
246
4. AD.4 Branch B Exclusion
247
5. AD.5 Finite-d Consequence
247
6. AD.6 Status
247
AE. Optical-Metric σ8 Closure
248
1. AE.1 Statement and Scope
248
2. AE.2 First-Principles Derivation:
aFRW
=0
248
ext
3. AE.3 Setup: spherical top-hat
statistic
249
4. AE.4 The closure equation
249
5. AE.5 The DFD response Q(δb )
249
6. AE.6 Closure number in the spherical
top-hat convention
249
7. AE.7 Physical interpretation
249
8. AE.8 Required CMB-to-z = 0
amplification
250
9. AE.9 Relation to galaxy clustering
250
10. AE.9b RSD compatibility: deep-MOND
velocity divergence
250
11. AE.10 Falsifiable predictions
251
12. AE.11 Open Theorem Obligation: full
convergence kernel
251
13. AE.12 Summary
252

16
GR. Adjudication of the Cosmological
Growth/Velocity Regime
1. GR.1 Purpose and the contested
scalar
2. GR.2 The regime-partition theorem
3. GR.3 The AQUAL Geff is not the
cosmological growth carrier
4. GR.3a The Rest-Mass Channel
Theorem (the corrected DFD
gravitational action)
5. GR.4 The single DFD growth
prediction
6. GR.5 S8 on the one-clock C=1
background
7. GR.5a The carried σ8 = 0.820 passes
Planck CMB lensing
8. GR.6 Resolving the directional f σ8
ambiguity
9. GR.7 Status
10. GR.8 Falsifier
11. GR.9 Reproducibility

253
253
253
253

257
259
259
259
260
260
260
261

A0Z. Epoch Confrontation of a0 (z)
261
1. Scope, and the external measurement it
builds on
261
2. DFD needs no borrowed background
262
3. The zero-parameter local value
262
4. The amplitude tension
262
5. A cosmic ceiling on any external-field
term
263
6. The survey refitted galaxy by galaxy
263
7. Massive discs at 0.6 < z < 2.5
264
8. Dated falsifiers, registrable now
265
AF. Galaxy Bias in DFD: a Frame-Consistent
Derivation from the σ8 Closure
266
1. AF.1 Setup: bias-frame ambiguity in
modified gravity
266
2. AF.2 Galaxy formation operates in the
W-frame
267
3. AF.3 Theorem: DFD galaxy bias from
the closure
267
4. AF.4 Cross-frame consistency: what
each probe measures
268
5. AF.5 Falsifiable predictions
268
6. AF.6 Scale dependence
268
7. AF.7 Open extensions and
limitations
269
8. AF.8 Summary
269
AG. Master Claims Confrontation Matrix
1. Tier Definitions
2. Block 1: Fundamental Constants and
Microsector
3. Block 2: Gravitational Tests (PPN,
Pulsars, GW, Strong-Field)
4. Block 3: Galactic Dynamics
5. Block 4: Cluster Scale

270
270
270
271
273
275

6. Block 5: Cosmology
276
7. Block 5a: Growth–Lensing Consistency
(EG , ACT DR6 × BOSS)
277
8. Block 6: Laboratory Tests (Clocks,
Cavities, Matter Waves, Antimatter)
277
9. Block 7: Strong-Field, Hyperbolicity, and
Open Items
278
10. Summary statistics
279
11. Reading guide for referees
279
12. Use of this matrix
280
AN. The Optical-Metric Master Equation: Unified
Light and Matter Motion
280
1. The isotropic optical metric
280
2. The master wave equation and its eikonal
limit
280
3. Gravity as the gradient of the squared
light speed
281
4. Relation to the action, and the boundary
on matter
281
5. Status and falsifiability
281
AO. The CP Sector: Strong-CP Protection, the
CKM No-Go, and Euler-Projected Closure 282
1. Purpose and scope
282
2. CKM and CP notation
283
3. DFD microsector and cohomological
inputs
283
4. The minimal real-kernel no-go theorem 284
5. Strong-CP protection
284
6. Det-orthogonal geometric CP and
nonzero CKM phase
285
7. Cohomological CKM magnitude
skeleton
286
8. Euler-projected selection of the CP-even
apex
286
a. Fubini–Study overlap motivation for
Postulate E.1
287
9. Final Wolfenstein branch and numerical
reconstruction
288
10. Comparison of the apex branches
289
11. Falsifiability and experimental targets 289
12. Scope and order of the claims
289
13. Final theorem ledger
290
14. Conclusion
290
AP. Corpus-Closure Theorems: the α57 Hierarchy,
the MOND Scale, and Rigor Discipline
291
1. Purpose and standard of proof
291
2. Notation and DFD primitives
292
3. Corpus-level rigor axioms
292
4. The α57 hierarchy from topology and
determinant scaling
293
a. Index count: kmax = 60
293
b. Primed determinant scaling
293
c. The cosmological invariant
294
5. The action-derived MOND prefactor
294

17
a. Gauge-emergence proof of
ka = 3/(8α)
294
b. The S 3 stationarity theorem
295
c. Action-level derivation of the MOND
prefactor
295
6. The nonlinear kinetic action and galaxy
law
295
7. Buckingham–π closure and the α58
acceleration invariant
296
a. Numerical prediction and
falsifiability
297
8. Black-hole sector: one base
temperature
297
9. Lensing and PPN branch discipline
297
10. Flavor/CKM sector: exact theorem
boundary
298
11. Inflation and early-universe sector
298
12. PDE/well-posedness sector
298
13. Theorem summary
298
14. Dimension checks
299
15. Failure modes and falsifiers
299
AQ. Executed Lattice-QCD Validation of the
DFD QCD Ledger (Program II)
1. Summary: what this appendix verifies
2. Frozen DFD QCD ledger
3. Methods
4. Validation: average plaquette
5. Static potential and string tension
6. Scale convention and the DFD
string-tension ledger
7. Scalar 0++ glueball
8. Executed Nf = 2 Wilson-HMC
validation
9. Claim discipline
10. Status boundary
11. Reproducibility

299
299
300
300
300
300
300
301
302
302
302
302

AR. A DFD-Native Yang–Mills Infrared Gap from
Loaded Spatial Frame Geometry
303
1. Purpose
303
2. DFD-native arena
303
3. Fluctuation operator
303
4. Main theorem
304
5. Deep-field DFD specialization
304
6. Why this supports DFD
304
7. Scope and non-claims
304
8. Falsifiers and completion checks
304
9. Conclusion
304
AS. Lattice Boundary Program: Staged
Execution and Provenance (Programs I–II) 305
1. Program I: frozen inputs, smoke test,
unquenched scaffold
305
2. Program I-A: first executed quenched 164
string tension
305
3. Status across the program
305

AT. Cross-Sector Exact Identities, Strong-Field
Theorems, and New Predictions
1. Exact cross-sector identities
2. Strong-field and gravitational results
3. Compact-star structure
4. Deep-sector conformal theorems
5. Strong-CP protection and the neutron
electric dipole
6. New predictions
7. A zero-parameter amplitude–shape
observable: the (ν, u) relation
8. Status and the remaining open lemma
AU. The Dark-Energy Sector from the α57
Clock
1. The dark-energy density as a pure
α-power
2. The Hubble–Planck seesaw and
holographic form
3. The “why now” coincidence
4. One dark energy: the equation of state
5. Look-elsewhere and status

306
306
307
308
309
309
309
312
313
313
313
314
315
315
316

AV. χ-Matter: the Weak-Sector Matter Field and
the True Definition of Dark Matter
316
1. Overview, grade, and what this appendix
claims
316
2. Step 1 — χ is the harmonic three-form
outside the rigidity classification
317
3. Step 2 — the gauge-adjoint dimension of
χ is dχ = 3
317
4. Step 3 — decay constant from the
per-mode determinant
318
5. Step 4 — see-saw mass: χ is light, cold,
and non-thermal
319
6. Step 5 — relic abundance by vacuum
misalignment
319
7. Step 5b — the relic amplitude is the
finite CS/WZW vacuum expectation (no
free angle)
320
8. Step 5c — the topological flux-density
form
322
9. Parity degeneracy of the CS measure, and
the vector count
323
10. Step 6 — the third peak, with no
postulated CDM
327
11. Step 6b — stability: χ has no photon
coupling
328
12. Step 7 — no photon coupling: E = 0
328
13. Why χ is the true definition of dark
matter, not adopted CDM
329
AW. The Electroweak Precision Program from
sin2 θW = 3/13
1. Exact electroweak identities (E)
2. The Z-pole suite from sin2 θW = 3/13
(P)

333
333
333

18
3. The W mass and the scale-ambiguity
theorem (T)
4. The ρ parameter (T)
5. The Higgs mass from the conjectured
quartic
6. Look-elsewhere and status
AX. The Rotating-Source Exterior
1. The stationary axisymmetric exterior
2. Frame dragging
3. Ergoregion structure
4. Spin-dependent ISCO
5. Spin-corrected QPO law
6. Photon ring and shadow displacement
7. Observational confrontation and
falsifiers
8. Status

334
335
335
335
336
336
337
337
337
337
337
337
338

AY. Cross-Sector Relations: Flavor Closures,
Hierarchy Rigidity, and Large-Number
Structure
338
1. AY.1 The Gatto–Sartori–Tonin closure
from locked integers
338
2. AY.2 The TM1 solar-mixing angle,
parameter-free
339
3. AY.3 Mass-scale hierarchy rank
339
4. AY.4 Large-number structure
339
SL. The Supernova–Lensing Sign
Discriminator
1. The line-of-sight optical screen and its
sign
2. The µ-law boost and the geometric
kernel
3. The standard (ΛCDM/GR) baseline
4. Pre-registered power forecast
5. Status and pre-registration

340
340
341
341
342
342

TP. The CMB Third Peak: a No-Go for
Zero-Dark-Matter, a Dark Sector with No
Fitted Density
343
1. What this appendix settles, and what it
does not
343
2. Part I — the height no-go
(theorem-grade)
343
3. Part II — the derived cold abundance 344
4. Part III — distinguishability: χ versus a
free-parameter CDM
345
5. Status and the one open tension
346
QM. Status of the Quantum-Mechanics Derivation
Program: What DFD Supplies and What
Remains Imported
346
1. Closures (what DFD supplies)
347
2. What remains imported, and why (the
precise obstruction)
347
3. The sharpened open problem (single
attackable object)
348

4. The J⋆ -uniqueness target via the physical
Hamiltonian (the one-H theorem)
349
5. Matter–ψ infrared class and dressed
(Faddeev–Kulish) asymptotic states
350
6. Status box
350
7. Falsifier
351
PK. The DFD-Native Production Galaxy P (k)
Pipeline
352
1. PK.1 Purpose: closing the pipeline half
of item F
352
2. PK.2 Stage 1 — linear matter power at
the frozen DFD cosmology
353
3. PK.3 Stage 2 — DFD growth carrier
(χ-matter, Q = 1)
353
4. PK.4 Stage 3 — nonlinear correction 353
5. PK.5 Stage 4 — galaxy redshift-space
multipoles
353
6. PK.6 Stage 5 — survey-window
convolution
353
7. PK.7 χ2 , covariance, and the validation
data
353
8. PK.8 Legacy-input run: DFD vs
ΛCDM
354
9. PK.9 Status: what is production-grade
and what remains
355
10. PK.10 Reproducibility
355
11. PK.11 Falsifier
356
YM. Status of the DFD Strong Sector:
Confinement Scale and the Yang–Mills Mass
Gap
357
1. The confinement scale from the
α-tower
357
2. The open item: string-tension
normalisation
357
3. The validated part: executed SU(3)
lattice
358
4. Two native gaps: an unconditional
compact-manifold gap and a conditional
loaded-frame gap
358
5. The ledger
358
6. Scope of the claim
359
7. Falsifier
359
FM. Charged-Fermion Mass Sector: Forced
Content and the Within-Multiplet
Degeneracy Obstruction
1. Scope and Summary
2. What is forced
3. The within-multiplet degeneracy:
theorem-grade obstruction
4. Scorecard
5. Reading the related appendices
Status

360
360
360
361
361
362
362

LS. Line-of-Sight Optical-Screen Amplitude: Full
2-Halo and Realization Computation
362
1. LS.1 Statement and Scope
362

19
2. LS.2 The two factors and what is firm 363
3. LS.3 Derived amplitude: central value
and band
363
4. LS.4 What the 2-halo calculation
settles
364
CL. Publication-Grade CMB Likelihood and
Bayesian Model Comparison
1. CL.1 Scope and the publication-grade
primary CMB
2. CL.2 Parameter count
3. CL.3 Bayesian model comparison
4. CL.4 Summary assessment
5. CL.5 Reconciliation of the item-4 S8
prediction
6. CL.6 Status and falsifier

365
365
366
366
367
368
368

FK. The Fitted Cosmological Knobs: Status and
Closure
369
1. The scalar amplitude As : one calibrated
input
369
2. The χ abundance normalization: from
open overshoot to
derived-with-conventions
371
3. Knob count: DFD vs ΛCDM
372
4. Inputs and axioms of DFD
374
5. Status and falsifier
374
SM. The “Scalaron Mass” is Not a DFD
Observable
1. Statement of the (mis-framed) problem
2. No native scalaron exists
3. Why the two routes disagree by ∼ 1011
4. The actual DFD primordial sector (no
scalaron)
5. Status
CE. The Improbability of the Pile: A
Combined-Evidence (No-Coincidence)
Bound
1. The surviving forced parameter-free
rows
2. Trials-Factor Audit and Conditional
No-Coincidence Bound
3. The conditional no-coincidence bound
4. Scope of the bound

378
378
379
379
380
381

382
382
383
385
386

RM. Falsifiable-Predictions Roadmap: How DFD
Is Confirmed or Killed
386
1. The complete bet list — master table
387
2. The four cleanest near-term decisive
tests
388
3. Tier 1 — near-term decisive (≲5–10 yr,
pass/fail)
388
4. Tier 2 — future decisive (next-generation
instruments)
389
5. Tier 0 — the already-passed bedrock
(would have killed DFD; did not)
389
6. Separation and open cracks
390

WS. Wave-Sector Closure: Elliptic Scalar, Derived
Gravitomagnetism, δrad = 0, and
Ringdown
391
1. Axiom E (Elliptic Constraint Reading) 391
2. The verified 1PN source and the
Standard-Gauge Coincidence Lemma
392
3. Vector-Sector Completion: g0i derived,
not matched
392
4. δrad = 0 with a mechanism; scalar
memory; falsifiers
396
5. Ringdown and echoes: the optical and
tensor channels
396
6. Consistency notes on other sections
396
7. What remains open in this sector
397
ZS. The Early-Clock CMB Fit: Conformal
Invariance Theorems, the Measured Clock
Coefficient, and the Shape Inputs
397
1. Three theorems: what a ψ screen cannot
do
397
2. The clock coefficient C: an eight-route
no-go, and its measurement
398
3. The CMB fit on the measured clock
399
4. The local arm and the acoustic arm, and
the decider
399
5. One universe, two clocks
400
ZT. The Domain-Local Screen Clock: A
One-Axiom Spatial Completion of the a0
Theorem
403
1. The axiom
403
2. The theorems
403
3. Price list and the declared δ̇ closure
403
4. Kinematic numbers
404
5. Predictions and pulls
404
6. The early-clock Hubble posture: SH0ES
is its named external falsifier
406
7. Falsifiers
406
8. Named open items
406
ZU. The Gravity Completion: the Exact Newton
Identity of the Quantized ψ Sector, and the
Kohn Null
407
1. The forced four-fermion vertex
407
2. Newton survives as an exact
zero-parameter identity
408
3. The gap in closed form; the carrier is
Hartree, not a condensate
408
4. Second-order control
408
5. The Kohn null
408
6. The Schrödinger–Newton term is the
completion’s falsifier
409
7. Scope and verification status
409
NL. The Scale-Additive Field Equation and Its
Test Record
1. Why a scale-resolved form is needed
2. Statement of the scale-additive
equation

409
409
409

20
3. The exact limits
410
4. Conservation
410
5. Action completion of the scale-additive
equation
411
6. The external-field sector: environmental
confinement
413
7. The test record
414
8. The distinctive signature: radial
414
9. The external-field effect: even anisotropy
dies
415
10. Two further constructions
416
11. No-go results that close the
alternatives
416
12. What is not derived
418
13. Effect on established results, and the
items still to be settled
419
BS. Single-Mode Bose Statistics and the Gravity
Law
420
1. The Partition Function
420
2. The Interpolation Function
421
3. The Magnitude of τ
422
4. Thermodynamics of the Mode
422
5. The Total Entropy of a Galaxy
423
6. The Observables
423
7. Confrontation, Including the Misses
424
8. Grades
425
Acknowledgments

427

References

427

I.

INTRODUCTION

A.

Theories of Gravity

Einstein’s general relativity (GR) has withstood a century of experimental scrutiny [5, 6]. Solar system tests,
binary pulsar timing, and gravitational wave observations
all confirm GR’s predictions to high precision. Yet the
theory’s success comes at a cost: explaining astrophysical and cosmological observations requires postulating
that 95% of the universe’s energy content consists of dark
matter and dark energy—components that have never
been directly detected despite decades of experimental
effort [7, 8].
Astrophysical anomalies relative to GR with visible matter alone form a coherent pattern. Spiral galaxy rotation
curves are flat rather than Keplerian [9]; low surfacebrightness galaxies follow tight scaling relations [10];
galaxy clusters require additional mass beyond their baryonic content [11]; and large-scale structure and supernova
data point to late-time accelerated expansion [12, 13].
The dominant response has been the ΛCDM paradigm,
which retains GR but postulates cold dark matter and a
cosmological constant.

An alternative approach modifies gravity itself. Modified Newtonian Dynamics (MOND) introduced a characteristic acceleration scale a0 ∼ 10−10 m/s2 governing
the transition between Newtonian and deep-field behavior
in galaxies [14, 15]. This single parameter successfully
predicts rotation curves, the baryonic Tully-Fisher relation, and the radial acceleration relation across galaxies
spanning five decades in mass [16].
A poorly understood coincidence is that a0 is numerically close to the cosmic acceleration scale aΛ ∼ cH0
inferred from the expansion rate [14]. This suggests a possible connection between galactic dynamics and cosmology
that ΛCDM treats as accidental.
TABLE I. Comparison of approaches to the gravitational
puzzle.
Theory
Key Feature
Status
DM/DE?
GR + ΛCDM Curved spacetime
Standard
Both
MOND
µ-crossover
Empirical Replaces DM
f (R)
Modified action
Various
Modified
TeVeS
Tensor-vector-scalar
Falsifieda
—
Brans-Dicke
Scalar-tensor
Constrained
Modified
DFD
Optical index
This work Derived DM;
DE
exponentb
a GW170817 speed constraint [17].
b DM as the derived χ component (App. AV); DE as the α57

vacuum with its measured coefficient R (App. AU).

Scalar-tensor theories have proliferated as alternatives
to GR [18, 19]. Brans-Dicke theory [20] introduced a dynamical scalar coupled to curvature. Bekenstein’s TensorVector-Scalar theory (TeVeS) [21] attempted to provide a
relativistic completion of MOND but was falsified by the
near-simultaneous arrival of gravitational waves and light
from GW170817 [17]. The f (R) family [22] modifies the
Einstein-Hilbert action directly. Each approach faces its
own challenges: additional parameters, instabilities, or
conflict with precision tests.
The theory presented in this review—Density Field
Dynamics (DFD)—takes a different path. Rather than
modifying GR’s geometric structure, DFD posits that
spacetime is fundamentally flat but contains a scalar field
establishing an optical refractive index. This approach
has historical precedent: in 1911-12, before completing
general relativity, Einstein himself explored gravity as a
variable speed of light [23, 24]. Gordon in 1923 showed
that electromagnetic wave propagation in a medium can
be described by an effective “optical metric” [25]. DFD
makes this optical perspective foundational rather than
emergent.
Table I summarizes how DFD relates to other approaches. The key distinction is that DFD reproduces
GR’s predictions where tested (solar system, gravitational
waves, binary pulsars) while making specific, falsifiable
predictions where not yet tested (laboratory LPI tests
and extended-channel clock anomalies).

21
B.

Core Idea: Gravity as an Optical Medium

The central insight of DFD is that gravity can be understood as a refractive medium. Just as light bends
when passing through glass because of a spatially varying
refractive index, light and matter in a gravitational field
respond to a cosmically varying index n = eψ , and this
index is the scalar dynamical content of the theory.
The formulation rests on two postulates that constitute
the Minimal Optical Equivalence principle:
a. Postulate P1 (Light). In a broadband nondispersive window, electromagnetic waves propagate according
to the eikonal of an effective optical metric
ds̃2 = −

c2 dt2
+ dx2 ,
n2 (x, t)

n(x, t) = eψ(x,t) .

(1)

This is the Gordon-Perlick optical geometry statement [25,
26], grounding ray optics in wave theory with a single
scalar field ψ determining the local refractive index.
b. Postulate P2 (Matter). Test bodies move under
the conservative potential
c2
c2
ψ,
a = ∇ψ = −∇Φ,
(2)
2
2
which fixes the weak-field normalization to match GR’s
classic optical tests (light deflection factor of two, Shapiro
delay coefficient, gravitational redshift). In weak hosts
the action completion of App. NL adds a reciprocal force
on bodies, which light does not feel (Cor. NL.17).
The exponential form n = eψ is not arbitrary but
follows from three requirements:
Φ≡−

(i) Positivity: n > 0 everywhere, ensuring light propagation is always defined.
(ii) Weak-field limit: For |ψ| ≪ 1, we have n ≈ 1 + ψ,
recovering the linear regime.
(iii) Multiplicative composition: Sequential media combine as ntotal = n1 n2 = eψ1 +ψ2 , matching the additive nature of gravitational potentials.
The factor-of-two deflection that matches GR emerges
automatically. In GR, light deflection receives equal contributions from spatial curvature and time dilation. In
DFD, the optical metric (1) encodes both effects: the
phase velocity c/n slows in the potential well, and wavefronts tilt toward the slower region. The result is precisely
4GM/(c2 b) at impact parameter b—the same as GR.
Figure 1 illustrates the conceptual difference. In GR,
gravity is geometry: mass curves spacetime, and particles
follow geodesics on a curved manifold. In DFD, spacetime
remains flat (Minkowski background), but a scalar field
creates a refractive medium. The observational predictions are identical in the weak-field regime and for black
holes, neutron stars and binary inspirals (App. WS); the
theories differ in their ontology, at galactic accelerations
and in specific laboratory contexts.
The connection between the two postulates is not coincidental. Both light and matter respond to the same field ψ,

(a) General Relativity

(b) Density Field Dynamics

Geodesic on curved manifold

Ray bent by n(x)

FIG. 1. Conceptual comparison of (a) General Relativity,
where gravity curves spacetime and particles follow geodesics
on a curved manifold, and (b) Density Field Dynamics, where
spacetime is flat but contains a refractive medium with index
n(x) = eψ(x) that bends light rays. Both yield identical weakfield predictions.

ensuring the Weak Equivalence Principle is satisfied: all
test masses fall with the same acceleration a = (c2 /2)∇ψ
regardless of composition. The universality of free fall is
built into the structure.

C.

What DFD Claims and What It Doesn’t

Before proceeding to the technical development, we
state explicitly what DFD claims and what it does not
claim.
a. Claim taxonomy. This review uses three claim
types. Core-derived statements follow from the DFD
field equations and actions presented in the main text.
Auxiliary-closure-derived statements follow from explicitly displayed supplemental structural postulates (for
example, the finite-symmetry closure used in the microsector). Empirical consistency statements summarize
benchmark calculations, fits, or data confrontations. This
taxonomy is used to keep the one-paper presentation logically unified without blurring the difference between core
theorems, closure-framework consequences, and benchmark evidence.
b. What DFD Claims:
1. Weak-field equivalence with GR: The optical
metric with n = eψ reproduces all Solar System
tests. The Parametrized Post-Newtonian (PPN)
parameters are γ = β = 1, and all ten PPN parameters match GR at first post-Newtonian order
(§IV).
2. Gravitational waves at speed c: A minimal
transverse-traceless sector propagates at the speed
of light with two tensor polarizations, consistent
with GW170817 and LIGO/Virgo/KAGRA observations (§V).
3. MOND-like phenomenology: At galactic scales
where |∇ψ|/a⋆ ≪ 1, a nonlinear crossover function

22
µ(x) produces flat rotation curves, the baryonic
Tully-Fisher relation, and the radial acceleration
relation without cold dark matter (§VII).
4. Channel-resolved clock and cavity residuals:
DFD’s minimal action predicts universal co-located
clock responses (Thm. XI.2); any departure belongs
to its extended non-metric sector (Sec. XV C) and is
channel-dependent rather than universal. Same-ion
optical ratios tightly constrain the pure α sector,
cross-species and nuclear clocks probe composition
and strong-sector channels, and cavity–atom comparisons reduce at tree level to a screened residual
rather than an order-unity slope (§XI, §XII).
5. Matter-wave T 3 signature: a null prediction:
DFD adds no contribution to GR’s own phase
through 1PN (§XIII).
6. Parameter-free α-relations: Three relations link
the fine-structure constant α to gravitational scales
without free parameters:
√
a0 = 2 α cH0 ,
(3)
ka = 3/(8α) ≈ 51.4,
2

(4)
−6

kα = α /(2π) ≈ 8.5 × 10

−10

.

(5)
2

The first gives a0 = 1.1966 × 10
m/s on boundmatter clocks and 1.1208 × 10−10 m/s2 on the early
clock (Sec. ZS 5); the third enters clock phenomenology (§VIII).
7. CMB from ψ-physics + derived χ: The analytic ψ calculation fixes the baryon-loading ratio
while the derived χ carrier supplies primordial cold
clustering without a postulated dark matter abundance. Peak ratio R ≈ 2.34 is the analytic baryonloading estimate; the peak location follows from the
measured-clock forward solution with no global ψ
screen: 100θ∗ = 1.04110 and ℓ1 ≈ 220.4. The exact
conformal invariance result of App. ZS rules out a
global ∆ψ ≈ 0.30 peak-location map. No postulated dark sector: galactic dynamics follow the
µ-law on the baryons with no fitted halo, and there
is no free dark energy; the cold dark matter is the
derived χ-matter particle (App. AV), which composes with the µ-law inside galaxies by the floor law
and assembles ΛCDM-like halos under the adopted
momentum-conserving energy (estimate), and dark
energy is the α57 geometric vacuum, whose coefficient is measured.
c.

Theoretical Completeness:

1. UV completion from topology: The CP 2 × S 3
gauge emergence framework is proposed as the UV
completion. Unlike GR, DFD has a flat arena
with absolute time, and that time is the uniformexpansion slicing, maximal around isolated bodies,
whose slices never reach the singularity of a collapsing star (App. WS, where the grade is stated);

ψ is classical (action ≫ ℏ). The topology fixes α,
the exponent of v and the leading fermion-mass
and mixing patterns; the other inputs are those of
App. FK (§XVIII C).
2. CMB forward closure: Peak ratio R = 2.34 is
the analytic baryon-loading result; peak location
ℓ1 ≃ 220 follows from the measured-clock no-screen
forward solution. CLASS/CAMB is used as the
explicit community transfer calculation on DFD’s
frozen current inputs; it is not a refit of free ΛCDM
densities.
3. Cluster mass: Under the µ-law alone clusters
fall short (Obs/DFD = 1.57 ± 0.08 relaxed, 1.99 ±
0.16 merging, at a0 = 1.20 × 10−10 m/s2 and at
the bound-matter a0 = 1.1966 × 10−10 m/s2 ); the
shortfall is the derived χ, 0.84 ± 0.12 of the cosmic
mix when χ adds its Newtonian pull and the relaxed
X-ray masses carry a hydrostatic bias b = 0.21. The
amplitude follows from the derived abundance under
ΛCDM-like assembly (1.21 ± 0.05 of the cosmic mix
predicted inside r500 , 1.28 ± 0.04 measured with
no hydrostatic bias; estimate). A no-χ closure by
literature correction factors does not hold. Galaxy
groups fall short in the same way on published X-ray
masses and need χ too (§XVI G, Appendix I).
4. Standard Model from topology: The gauge
emergence framework (§XVII) gives the electroweak
group as the isotropy group of CP 2 under one stated
identification, with colour, the family content and
Ngen = 3 as inputs (App. F), α = 1/137 from ChernSimons, the top and charm masses (the other seven
charged fermions rest on data-selected prefactors),
the CKM magnitudes and the PMNS angles with
the stated
√ inputs of App. FK, the exponent of v =
MP α8 2π, which accounts for the 17 orders of
magnitude of the hierarchy (its prefactor is fitted),
and, with θYM = 0 fixed by the spectral action given
that nothing is added to it for the gauge fields, θ̄ = 0
at tree level and loop-bounded (Theorem L.3; no
axion required). Its physical validity is conditional
on DFD gravity being confirmed experimentally.
5. Scope boundary: Loop corrections in the ψ-gauge
coupled system are bounded by power counting: ψ
couples with gravitational strength, so they are suppressed by (E/MP c2 )2 ≲ 10−30 at collider energies,
and the low-acceleration self-interaction becomes
strong only below 0.2 mm, far under every length
on which that sector is tested. The classical and
effective-field level is sufficient for all predictions.
DFD reproduces GR in all regimes where GR has been
confirmed, and makes specific, quantitative predictions in
regimes where decisive tests are experimentally accessible.

23
D.

Reader’s Guide

This review is organized to be readable both linearly
and as a reference. The structure follows a logical progression from foundations to frontiers, with each part
addressing a distinct aspect of the theory.
a. Part I: Foundations (Sections I–III). It establishes the mathematical framework: the optical metric, action principle, field equations, and proof of well-posedness
(existence, uniqueness, stability). This part is prerequisite
for all subsequent sections.
b. Part II: Contact with Known Physics (Sections IV–
V). It shows that DFD reproduces GR where tested.
Section IV presents the complete PPN analysis showing
γ = β = 1. Section V develops the gravitational wave
sector and verifies consistency with LIGO/Virgo/KAGRA
constraints.
c. Part III: Strong Fields (Section VI). It extends
the theory to strong-field regimes: spherically symmetric
solutions, photon spheres, optical horizons, and black hole
shadows. Comparison with EHT observations of M87*
and Sgr A* is presented.
d. Part IV: Galactic Dynamics (Section VII). It develops the deep-field regime where µ ̸= 1: rotation curves,
Tully-Fisher relation, and the radial acceleration relation.
The RAR/SPARC comparison is treated as validation of
the derived acceleration scale and kinetic crossover, not
as a calibration of a0 .
e. Part V: The α-Relations (Section VIII). It
presents the three parameter-free numerical relations linking α to gravitational phenomenology, with derivation
and verification.
f. Part VI: Laboratory Tests (Sections XI–XIII). It
details the laboratory tests: extended-channel atomic
clock anomalies (§XI), cavity-atom LPI tests (§XII), and
matter-wave interferometry (§XIII). These sections are
self-contained and can be read independently after Part I.
g. Part VII: Frontiers and Open Problems (Sections XVI–XX). It addresses cosmological implications
(§XVI), the conditional quantum/gauge sector (§XVII),
open problems and limitations (§XVIII), and conclusions
(§XX).
h. Dependencies.
• Sections I–III (Part I) are prerequisite for all subsequent sections.
• Section IV (PPN) is independent of galactic phenomenology (Section VII).
• Laboratory tests (Sections XI–XIII) require only
Part I.
• Strong fields (Section VI) requires Sections II–III.
i. Notation. Standard notation is defined in Appendix A and summarized here. The scalar field is
ψ; the refractive index is n = eψ ; the acceleration
is a = (c2 /2)∇ψ; the crossover function is µ(x) with
x = |∇ψ|/a⋆ ; the acceleration scale is a0 ∼ 10−10 m/s2 .
Key equations are numbered sequentially throughout.

j. A note on falsifiability. Every scientific theory
must specify conditions under which it would be falsified.
For DFD, the decisive tests are:
• Channel-resolved clocks and cavity residuals: If same-ion, cross-species, and nuclear-clock
data cannot be organized by the channel-resolved
structure of Eq. (291), the present clock mechanism is wrong. In particular, a high-precision null
in cross-species atomic ratios and in the surviving
229
Th/Sr nuclear window would remove the leading
live laboratory channels of the extended non-metric
sector.
• Cavity–atom residuals: After geometric cancellation, the cavity–atom observable is a screened
residual, not an order-unity discriminator. A future
dedicated null at the residual sensitivity target of
Sec. XII would constrain or remove that channel; a
null only at the δξLPI < 0.1 level would not.
• Gravitational waves: If ppE parameters deviate
from zero in the strong-field regime, the radiative
sector requires modification.

E.

Assumptions and Degrees of Freedom Ledger

We provide an explicit accounting of the inputs, outputs,
and falsifiers.
The µ(x) crossover function is not a continuous fit parameter.√Its single scale a0 is derived from the α-relation
a0 = 2 α cH0 ; the functional form is an exponential
response fixed by the S 3 Chern-Simons microsector composition law, with µ(x) = x/(1 + x) as its weak-response
(galactic) approximant (Appendix N, Appendix BS). With
H0loc fixed by the measured G (Eq. (479)), the galactic
sector carries no adjustable parameter.
a. Parameter structure. DFD has: (i) no continuous
parameter fitted to the shape of any observable it predicts
except the baryon density, and nothing analogous to Ωm ,
w, or CDM concentrations; (ii) a discrete input set, in
which Ngen = 3, the derived q1 = 3 and the five matter
types fix kmax = 60, listed with the adopted axioms in
Tables CXXVII and CXXVIII and priced in the trials
audit of App. CE; (iii) four continuous cosmological calibration constants: the vacuum coefficient R, which with
the empirical scale H0 (equivalently G) fixes all dimensionful quantities; the amplitude As of the primordial
perturbations, whose source the printed theory does not
contain; the coefficient of ωb h2 , the one shape among
them; and the reionization optical depth τ , an astrophysical quantity shared with ΛCDM (per-input grades:
App. FK, Rem. FK.8); the particle sector adds the fitted
Higgs prefactor, the imported factor ηq and the Higgs
quartic, conjectured to be 1/8 at a matching scale that
is not derived. Bound systems read H0loc = 72.09 (Postulate ZS.11), so a0 = 1.1966 × 10−10 m/s2 is a prediction,
not a fit.

24
TABLE II. Summary of DFD’s inputs, outputs and falsifiers. The inputs of the graded predictions, with where each is stated,
are listed in Tables CXXVII and CXXVIII (App. FK).
Category Item

Status

Core postulates (2)
n = eψ
Φ = −c2 ψ/2

Postulate
Postulate

Adopted axioms, premises and identifications
Gravity and dark sector: elliptic constraint, Rest-Mass Channel, Adopted
the floor law in energy form, environmental
confinement, the premises of the
Einstein–Hilbert term
Cosmology: clock and scalar dictionaries, two primordial inputs, Adopted
the flatness clause, flux transport,
the two-clock postulate, domain-local normalizer (optional)
Inner space and flavour: seven vacuum axioms, three structural Adopted
postulates, the baryon-number premise
Ground-state principle (ν = Z1 ), flux period of χ, twelve more Adopted
Background frameworks (shared)
Quantum mechanics: unit ℏ, interacting map, measurement
Standard thermal history; baryonic astrophysics

Shared
Shared

Discrete inputs
Ngen = 3, colour, family content
q1 = 3
(a, n) = (9, 5), kmax = 60
Weak-block count 158
Tilt ns = 1 − 2/60
Fermion prefactors, CKM counts, neutrino branch

As in the SM
Derived (electroweak identification)
From these; conditional count
Stated input
Stated number; mechanism open
Chosen by data

Scale and calibrations
G (equivalently H0loc )
Vacuum coefficient R
As , ωb coefficient, τ
Higgs prefactor; λH ; ηq

Measured unit
Measured
Calibrated
Fitted; conjectured 1/8, scale not derived; imported

Functional form
µ(x)

Ground-state principle; branch (SOURCE) chosen by data

Derived (no continuous shape parameters)
α−1 = 137.036
√
a0 = 2 αcH0loc
57
GℏH02 /c5 =
√Cα
v = MP α8 2π √
mt = (1 − α)v/ 2

CS quant. (conv.-locked); bracket coefficient and 1/e2 identification open
Derived (value under Postulate ZS.11)
Exponent derived; R measured
α8 derived; prefactor fitted
Derived

Forced mod. discrete selection / one postulate
Masses (below top and charm)
CKM
PMNS

Seven prefactors, chosen by data
Euler-proj. postulate
Forced (∼5%) up to one branch choice; θ13 a stated conjecture

Falsifiers
Cavity–atom residual null
Clock channel structure fails
cT ̸= c
RAR > 3σ off

F.

Completion Ledger

A sector is called closed when its postulates, theorem
chain, numerical implementation target, and falsifier are
all explicit. The following ledger summarizes the closure
status of each DFD sector.
a. The integrated-architecture statement. Its inputs
are listed in Tables CXXVII and CXXVIII. The same
S 3 saturation-union law that gives the weak-response
galactic interpolation function also fixes the deviationinvariant structure of the temporal sector, from which
the dust branch follows under the adopted identification
K ′ = µ (App. Q). The same finite Toeplitz algebra that
supplies the microsector cutoff also forces the regular

Cavity
Clocks
GW
Galactic

Hilbert module through real-bimodule consistency. The
theory is therefore overconstrained: changing one sector
generally breaks another.

II.

MATHEMATICAL FORMALISM

This section develops the complete mathematical structure of Density Field Dynamics: the optical metric governing light propagation, the action principle, field equations,
and the family of crossover functions. The presentation
aims for both rigor and physical transparency.

25
TABLE III. DFD completion ledger. A sector is closed when postulates, theorem chain, numerical implementation target, and
falsifier are all explicit.
Sector

Closure mechanism

Weak-field gravity Optical metric + physical metric
TT radiation
CP 2 × S 3 parent tensor
Galactic dynamics exponential response from S 3
composition; µ = x/(1 + x) its
weak-response form
√
cH0
MOND scale
a0 = 2 α√
Epoch scale
a0 (z) = 2 α cH(z)

Falsifier

PPN closed
Closed
Closed; SPARC tested

Any PPN parameter mismatch
cT ̸= c or non-GR ppE phase
RAR shape rejects n = 1

Closed
Theorem under Axiom E
(Thm. XIX.3); contested
(App. A0Z)
K ′ (∆) = µ(∆)
Closed given an adopted
identification
Qχ = 1 under the Rest-Mass
Closed by an adopted
Channel (App. GR)
postulate
Regular-module spectral trace
Closed to 21 ppm; bracket
coefficient open
Min faithful real-bimodule (App. AD) Closed
α57 finite-mode invariant
Exponent closed; coefficient
measured
Local, anisotropic refraction only; the Global use retired
global SN/CMB screen is retired
(App. LS)
Extended-channel residuals
Partially empirical

Dust branch
Linear growth
α value
α module
G–H0
ψ-screen

Clock sector

A.

Status

The Optical Metric and Geodesics
1.

Gordon’s Optical Metric

The optical metric approach has a distinguished history in relativity and optics. Gordon [25] showed in 1923
that electromagnetic waves propagating through a moving dielectric medium experience an effective spacetime
geometry. Perlick [26] systematically developed ray optics
in curved spacetimes, establishing the mathematical foundations for relating wave propagation to null geodesics.
DFD adopts this framework but makes a conceptual
inversion: rather than deriving an effective optical metric
from an underlying curved spacetime, the optical refractive index becomes the fundamental gravitational degree
of freedom on flat Minkowski spacetime.
The optical metric is defined by the single scalar field
ψ(x, t):
ds̃2 = −

c2 dt2
+ dx2 ,
n2 (x, t)

n(x, t) = eψ(x,t) .

(6)

The line element ds̃2 = 0 defines null rays—the trajectories of light. The refractive index n = eψ satisfies
n > 0 everywhere, ensuring light propagation is always
well-defined.

2.

Fermat’s Principle

Light rays extremize optical path length. For a path
x(s) parameterized by arc length:
Z
δ n(x) ds = 0.
(7)

Local a0 inconsistent with H0loc
High-z frozen a0 ; ∂z a0 ≤ 0
w ̸→ 0 or c2s ̸→ 0
Linear growth off Qχ = 1 beyond the
0.1–0.5% σ8 shift
Real-bimodule Branch B
counterexample
Faithful Cd bimodule construction
Dimensionless invariant fails
No local foreground correlation

Null in surviving channels

The Euler-Lagrange equations yield the ray equation:


d
dx
n
= ∇n,
(8)
ds
ds
which governs the bending of light in the refractive
medium. For small deflections, this reproduces Snell’s law
in differential form.
The connection to null geodesics is established by noting
that the optical metric (6) is a diagonal metric with
position-dependent lapse c/n; its null geodesics coincide
with extremals of Fermat’s principle.

3.

Phase and Group Velocities

The one-way phase velocity is
c
cphase = = c e−ψ .
(9)
n
In a gravitational potential well (ψ > 0), light slows:
cphase < c. The coordinate speed of light depends on
position, but the two-way speed—measured by local clocks
and rods—remains c.
For the group velocity in the nondispersive band (where
dn/dω = 0), group and phase velocities coincide: cgroup =
cphase .
a. Note on asymptotic propagation. This effectivemedium (optical metric) description does not imply an
asymptotic EM–GW speed split. The GW170817 constraint |cT /c − 1| < 10−15 is satisfied because (i) the TT
sector has no derivative mixing with ψ in its principal
part (§V A), and (ii) the leading propagation delay is
common-mode when EM and GW arrivals are compared
using receiver clocks.

26
B.

Action Principle

1.

Scalar Sector Action

The scalar field ψ is governed by a k-essence-type action
with a nonlinear kinetic term:

 2


Z
c2
a0
|∇ψ|2
−
Sψ = dt d3 x
W
ψ(ρ
−
ρ̄)
,
8πG
a2⋆
2
(10)
where:
• W (y) is a dimensionless potential with W (0) = 0,
W ′ (y) → 1 as y → ∞, and convexity W ′′ (y) ≥ 0.
• a⋆ is the characteristic gradient scale with [a⋆ ] =
1/m. It
√ relates to the MOND acceleration scale
a0 = 2 α cH0loc = 1.1966 × 10−10 m/s2 on boundmatter clocks (Postulate ZS.11; 1.1208 × 10−10 on
the early clock) via a⋆ = 2a0 /c2 . The argument
y = |∇ψ|2 /a2⋆ is then dimensionless.
• ρ is the local mass density; ρ̄ is the mean cosmic
density, ensuring proper cosmological boundary conditions.
The kinetic function W (|∇ψ|2 /a2⋆ ) interpolates between:
• High gradients (|∇ψ|/a⋆ ≫ 1): W ≈ y, yielding
linear (Newtonian) behavior.
• Low gradients (|∇ψ|/a⋆ ≪ 1): W ∼ 23 y 3/2 , producing MOND-like deep-field dynamics.
a. Dimensional verification. Note: In the Lagrangian, a⋆ has units of 1/m (a gradient scale), related
to the physical acceleration scale a0 by a⋆ = 2a0 /c2 .
This ensures |∇ψ|/a⋆ is dimensionless. The kinetic
prefactor a20 /(8πG) carries energy-density dimensions:
[a20 /G] = kg/(m · s2 ). The matter coupling c2 ψρ has
units:
3

• [c2 ψρ] = (m/s)2 · 1 · (kg/m ) = kg/(m · s2 ) (energy
density)
Both terms integrate to energy × time: [Sψ ] = J·s ✓
b. Comparison with AQUAL. The action (10) is the
scalar-field analogue of Bekenstein-Milgrom’s AQUAL
formulation [27]. The key differences are: (i) the fundamental field is ψ (determining refractive index n = eψ )
rather than the potential Φ directly; (ii) the coupling
to matter goes through the optical metric, not just the
potential; (iii) the µ-crossover is constrained by optical
consistency (positive n, well-posed wave propagation).
c. Status of Eq. (10). Equation (10) is the quasistatic spatial sector used for lensing, weak-field dynamics, and galactic phenomenology, at Newtonian order;
its µ-law term is written on the physical metric in
App. WS, as in the preferred-foliation theory of Blanchet
and Marsat [28]. The temporal completion is derived separately in Appendix Q, where the unique local temporal

invariant ∆ ≡ (c/a0 )|ψ̇ − ψ̇0 | is introduced and the dust
branch w → 0, c2s → 0 is proved. The full scalar-sector
action combining spatial and temporal sectors is
(


 

Z
a20
c
|∇ψ|2
3
Sψ = dt d x
+K
W
|ψ̇−ψ̇0 |
8πG
a2∗
a0
)
c2
ψ(ρ − ρ̄) .
−
2
(11)
where K is the temporal kinetic function with K ′ (∆) =
µ(∆), which acts on homogeneous deviations from the
background flow only (Axiom WS.1).
d. Convexity and stability. The function W must be
convex (W ′′ ≥ 0) to ensure:
1. Well-posed elliptic field equations
2. No ghost instabilities
This follows from standard variational theory: a convex energy functional has a unique minimizer, and small
perturbations about the minimum have positive energy.

2.

Matter Coupling

Matter couples to the physical metric g̃µν :


g̃µν = diag −c2 e−ψ , e+ψ , e+ψ , e+ψ .

(12)

This shares the same null cone as the Gordon eikonal
metric (6): setting ds̃2 = 0 gives |dx/dt| = c e−ψ =
c/n, so light propagation is identical. The exponential
structure n = eψ together with Postulate P2 fixes, to
first order in ψ, the relation between time and spatial
components (cf. the PPN derivation in §IV C). The lapse
e−ψ/2 and the volume det g̃ij = e3ψ are exact; the isotropy
of the spatial part holds through first post-Newtonian
order, and beyond it g̃ij = eψ (eh )ij , with hij the traceless
part of the parent strain (§V A 3; App. WS), and the
spatial part dx2 of Eq. (6) is then (eh )ij dxi dxj . For a
point particle of mass m, the action is:
r
Z
dxµ dxν
Spp = −mc dτ −g̃µν
.
(13)
dτ dτ
In the non-relativistic limit (v ≪ c, |ψ| ≪ 1):


Z
v2
Φ
Spp ≈ −mc2 dt 1 − 2 + 2 ,
(14)
2c
c
where Φ = −c2 ψ/2 is the effective Newtonian potential.
The equation of motion is:
d2 x
c2
= −∇Φ = ∇ψ = a,
(15)
2
dt
2
confirming that all test masses fall with acceleration a =
(c2 /2)∇ψ—the Weak Equivalence Principle is satisfied.
The unification of this acceleration law with the lightpropagation law c1 = c/n, as the timelike and null sectors
of a single optical-metric wave equation, is developed in
Appendix AN.

27
• Energy positivity: W convex ensures no negativeenergy modes.

TABLE IV. Action sectors and their physical content.
Sector Content
Degrees of Freedom
Sψ
Scalar refractive field 0 (constraint; Axiom WS.1)
Sh
TT gravitational waves 2 (tensor hTT
ij )
Sint
GW-matter coupling
—
Smatter Matter fields
Various

3.

• No ghosts below the cutoff: no propagating
scalar DOF (ψ is a constraint, Axiom WS.1); two
tensor DOFs in hTT
ij ; none in the constrained vector
and scalar-longitudinal pieces of the parent strain
(§V A 3); the Weyl-squared term of the spectral
action adds a spin-2 pole only above the cutoff
(App. WS).

Gravitational Wave Sector

• GW speed cT = c: Built into the TT action.
The transverse-traceless (TT) gravitational wave sector
is embedded with the standard linearized action:


Z
c4
1
2
TT 2
Sh =
dt d3 x 2 (∂t hTT
)
−
(∇h
)
. (16)
ij
ij
64πG
c
This is the canonical form for a massless spin-2 field on
flat spacetime. It is the quadratic part of the full a2
Einstein–Hilbert TT action, whose cubic vertex carries
the nonlinear gravitational-wave memory (App. AT). It
ensures:
• Propagation speed cT
GW170817)

=

c (consistent with

• Two tensor polarizations (+ and ×)
• No scalar or vector GW modes
The wave equation follows from variation:
16πG eff TT
□hTT
(Tij ) ,
(17)
ij =
c4
where □ = c−2 ∂t2 − ∇2 and (Tijeff )TT is the transversetraceless projection of the effective stress-energy tensor.

4.

Interaction and Complete Action

The gravitational wave sector couples to matter
through:
Z
1
ij
Sint =
d4 x hTT
(18)
ij Teff ,
2
with the effective stress-energy tensor:


ij
Teff
= ρv i v j + pδ ij + O v 4 /c4 .

(19)

The complete DFD action is:
SDFD = −Sψ ρ=0 + Sh + Sint + Smatter

• Newtonian limit: µ → 1 for large |∇ψ|/a⋆ .
• MOND limit: µ ∼ x for small |∇ψ|/a⋆ .
C.
1.

Field Equations

General Nonlinear Form

Variation of Sψ with respect to ψ yields the fundamental field equation:
 


|∇ψ|
8πG
∇· µ
(21)
∇ψ = − 2 (ρ − ρ̄),
a⋆
c
where the response (flux) function µ(x) is related to the
kinetic potential by:
|∇ψ|
µ(x) = W ′ (x2 ),
x=
.
(22)
a⋆
(The distinct combination W ′ (X) + 2XW ′′ (X) =
d[xµ(x)]/dx is the longitudinal eigenvalue of the flux Jacobian; it governs linear perturbations [App. AC] and
hyperbolicity, not the source law, and equals 2µ in the
deep-MOND limit.) Inside bound systems that contain χ
the adopted energy of App. AV, in which the µ-law acts
on the baryons’ field, governs instead of Eq. (21).
a. Derivation sketch. From action (10), compute:
 


2
a20
δSψ
2∇ψ
c2
′ |∇ψ|
=−
∇· W
(ρ − ρ̄)
−
δψ
8πG
a2⋆
a2⋆
2
=−

c4
c2
∇ · [W ′ (X)∇ψ] − (ρ − ρ̄),
16πG
2

(23)
where X = |∇ψ|2 /a2⋆ . Setting δS/δψ = 0 and identifying
µ(x) = W ′ (x2 ) gives Eq. (21).

(20)

where Smatter includes all matter field Lagrangians minimally coupled to the metric g̃µν of Eq. (12). In the Newtonian limit their coupling to ψ is the ρ term of Eq. (10), so
that term is counted once, in Smatter ; Eq. (10) is written
as an energy and enters with the opposite sign.
a. Key properties of the complete action:
• Explicit variational principle: All field equations derivable from δS = 0.

2.

The Optical Source Theorem: DFD’s replacement for Tµν

The source of ψ is fixed, not posited: it is the variational
response of the matter action to ψ, the exact analogue of
how Tµν arises in general relativity.
Theorem II.1 (DFD Optical Source Theorem). The
GR ten-component source Tµν = − √2−g δSmatter /δg µν is

28

— the Tolman active gravitational mass ρ + 3P/c2 , one
term per sector i (rest mass, kinetic, radiation, pressure, gauge, χ, vacuum). Because ψ enters every sector only through the single non-conformal optical metric g̃µν = diag(−c2 e−ψ , e+ψ , e+ψ , e+ψ ), the field equation
δSDFD /δψ = 0 [Eq. (20)] is ∇·[µ ∇ψ] = −(8πG/c4 )(ε +
3P )eψ , of which Eq. (21) is the non-relativistic (P ≪ ε,
ψ ≪ 1) limit −(8πG/c2 )(ρ − ρ̄). The source law holds
with the spatial part g̃ij = eψ (eh )ij as well; for static matter at µ = 1 the field equation is then ∂i [(e−h )ij ∂j ψ] =
−(8πG/c4 )(ε + 3P )eψ exactly, the time-time component
of the trace-reversed Einstein equation (App. WS).
Proof. With the standard definition Tµν
=
− √2−g δSmatter /δg µν ,
minimal
coupling
gives
√
1 √
δSmatter /δψ = + 2c
−g̃ T ab ∂ψ g̃ab with −g̃ = c eψ : the
lapse g̃tt = −c2 e−ψ contributes +ε, the three spatial
g̃ii = e+ψ contribute +3P , so ∂ψ Smatter = + 12 (ε + 3P )eψ
(verified symbolically, residual 0; the static-fluid special
case is App. AT). The scalar action of Eq. (10) is written
as an energy, so in Eq. (20) it enters with the sign
opposite to the matter Lagrangian, and at ρ = 0, since
its ρ term is the Newtonian limit of the ψ coupling in
Smatter ; with that sign and that count δSDFD /δψ = 0
is the field equation stated above. The rule is linear in
T ab , so it holds identically for the renormalized quantum
expectation ⟨T̂ ab ⟩ren , giving the semiclassical source
⟨ε̂ + 3P̂ ⟩ren .
Corollary II.2 (Nordström-avoidance). Had matter coupled conformally (g̃µν = e2ψ ηµν ) the source would be the
trace (ε−3P ), which vanishes for radiation (P = ε/3): radiation would not gravitate and light would not bend — the
reason Nordström’s 1913 scalar gravity is excluded. DFD’s
opposite-sign lapse/space exponents give (ε + 3P ) = 2ε
for radiation instead, so radiation gravitates at twice the
dust rate and light bends with γ = 1. The distinction is
a single sign in ∂ψ g̃µν ; the trace anomaly shifts the trace
(ε − 3P ) and cannot restore the traceless combination; the
stress it induces adds only a small ε + 3P , of order H 4
in the radiation era (Extended Derivations, App. AH).
Weak-field limit. For µ → 1, P ≪ ε, ψ ≪ 1 the theorem reduces to ∇2 ψ = −(8πG/c2 )ρ; with a = (c2 /2)∇ψ
and Φ = −(c2 /2)ψ this is Newtonian gravity, massindependent (WEP). Conservation. The optical stress
˜ µ T̃ µν = 0 (below), the on-shell-matter /
tensor obeys ∇
spatial-diffeomorphism Noether partner of Theorem II.1
— the DFD analogue of the contracted Bianchi identity,
confirmed at PPN order by ζ1 = ζ2 = ζ3 = ζ4 = 0.
a. Scope of the local form, and its scale-resolved completion. Equation (21) evaluates µ at a point. That is
the correct local reduction of DFD’s quasi-static scalar

1

µ(x)

replaced in DFD by a single scalar law for the opticaldensity field ψ,
X
δSmatter
= 12
Sψ :=
(εi + 3Pi ) eψ
(24)
δψ
i

0.5
Simple: x/(1 + x)
√
Standard: x/ 1 + x2
0
10−2

Deep-field

10−1

Solar

100
|∇ψ|/a⋆ = x

101

102

FIG. 2. The µ(x) crossover function interpolates between
deep-field (µ ∼ x) and solar (µ → 1) regimes. The transition
occurs at x ∼ 1, corresponding to |∇ψ| ∼ a⋆ . The “Standard”
form is shown for historical comparison; the S 3 microsector
composition law selects an exponential response (Appendix N,
Lemma N.6; Appendix BS), of which the “Simple” form is the
weak-response approximant (asymptotics forced).

sector, and it reproduces flat rotation curves, the baryonic Tully–Fisher relation and the deep-field limit exactly.
DFD’s derived response is, however, a mode statement —
ν = 1/µ is a single-mode partition function and τ an optical depth (App. N) — and the per-point and per-mode
readings agree only when all the sourcing mass lies at one
distance from the field point. Where they disagree there
is now a measurement: any functional whose argument is
the vector sum |∇ψint + ∇ψext | generates a static anomalous quadrupole Q2 in the planetary potential, bounded
observationally at (3 ± 3) × 10−27 s−2 by Cassini [29]
and at (1.6 ± 1.8) × 10−27 s−2 on the DE440 dataset [30].
Appendix NL constructs the scale-additive (nonlocal-inradius) completion of Eq. (21), of which Eq. (21) is the
exact co-distant limit (Thm NL.1, item 2), records its
forward evaluations against every existing measurement,
and proves five no-go results that close the obvious alternatives. In the Solar System, where the Sun and the
Galaxy are separated by 18.955 e-folds of vacuum, the
completion and the local form differ; in a galaxy, which
is matter all the way through, they agree to ≤ 0.018 dex.

D.

The µ(x) Crossover Function

The response function µ(x) must satisfy four physical
constraints:
1. Solar limit: µ(x) → 1 as x → ∞ (recover Poisson
equation).
2. Deep-field limit: µ(x) ∼ x as x → 0 (MOND-like
scaling for flat rotation curves).
3. Monotonicity: µ′ (x) > 0 for x > 0 (strict ellipticity of field equation).
4. Convexity: The associated W must be convex
(energy positivity, stability).

29

µα,λ (x)

1

rem II.1: its scalar projection along the ψ-flow reproduces
the source Sψ = δSmatter /δψ.

α = 1, λ = 1
α = 2, λ = 1
α = 3, λ = 1

2.

0.5

0

0

1

2
3
x = |∇ψ|/a⋆

4

5

Energy Conservation

In static configurations, the total energy functional:

 2


Z
c2
a0
|∇ψ|2
3
− ρψ
W
(26)
E[ψ] = d x
8πG
a2⋆
2
is minimized by solutions of the field equation.

FIG. 3. Constrained crossover functions µα,λ (x): linear at
small x (deep-field), saturating at large x (solar limit), monotone throughout.

3.

Local Conservation in PPN Framework

Within the PPN formalism (§IV), DFD satisfies local
energy-momentum conservation:
1.

Admissible Families

Table V catalogs the µ-functions used in the DFD literature. The “Simple” form µ(x) = x/(1 + x) is the
weak-response form (Assumption N.3) of the exponential
response derived from the S 3 microsector via a composition law (Appendix N, Lemma N.6, Theorem N.8; Appendix BS): its two asymptotic limits are forced, and it
tracks the derived response in its RAR form to ≲ 0.02
dex (below SPARC precision).

ζ1 = ζ2 = ζ3 = ζ4 = 0,

F.

4D-from-3D: Emergent Spacetime Structure

A distinctive feature of DFD is that the 4D optical metric is derived, not fundamental. The theory is intrinsically
3-dimensional.

TABLE V. Catalog of admissible µ(x) functions. The law of
record is the SOURCE branch of App. BS; the Simple form is
its weak-response approximant (App. N).
Name

Formula
µ(1)
Status
x
1/2
Approximant
1+x
√
x
√
Standard
1/ 2 Phenomenological
1 + x2
x
General
varies Phenomenological
(1 + λxα )1/α
−x
Exponential (TOTAL) 1 − e
1 − e−1
Admissible;
not selected
Simple

The general family µα,λ (x) satisfies all four constraints
only for λ = 1.

1.

The Fundamental Arena

DFD posits:
1. Space: Euclidean R3 with coordinates x
2. Time: Absolute parameter t (preferred foliation)
3. Field: Scalar ψ(x, t) on this arena
The “4D spacetime geometry” emerges as an effective
description of how light propagates and clocks tick in the
refractive medium.
2.

E.

(27)

where the ζi are PPN parameters measuring violation of
local conservation. This follows from the diffeomorphism
invariance of the optical metric coupling.

The 3D-to-4D Morphism

Conserved Quantities and Symmetries
1.

Diffeomorphism Invariance

The matter action is invariant under diffeomorphisms
that act on the matter fields and the optical metric together. On the matter equations of motion this gives a
conserved stress-energy tensor in the optical metric:
˜ µ T̃ µν = 0,
∇
(25)
˜ is the covariant derivative with respect to g̃µν .
where ∇
This is the Noether partner of the Optical Source Theo-

Theorem II.3 (Emergent Spacetime). There is a bijective correspondence:
{3D solutions ψ(x, t)} ←→ {4D optical intervals ds̃2 }
(28)
given by the Gordon-type optical interval:
c2 dt2
+ dx2 ,
n = eψ .
(29)
n2
Through first post-Newtonian order; beyond it the interval’s spatial part is (eh )ij , with hij the traceless part of
the parent strain (§V A 3), and the correspondence is with
the pair (ψ, hij ).
ds̃2 = −

30
a. Remark (auxiliary rescaled metric). For certain
calculations (gauge-sector derivations, Einstein-tensor
cross-checks), it is convenient to use an auxiliary metric
ĝµν = diag(−c2 e−2ψ , e2ψ , e2ψ , e2ψ ) that doubles the exponents relative to the physical metric (12). This is a computational device; the physical coupling is through (12),
the fundamental DFD arena remains flat R3 with absolute
time, and the Gordon interval (29) is its optical encoding.
The morphism to 4D curvature language is used only as
a “translation layer” for comparison with GR—it does
not promote 4D geometry to fundamental status.
b. Static check. The static field equation
8πGρ
(30)
∇2 ψ = − 2
c
can be repackaged as the (00)-component of the Einstein
tensor for the physical metric (12) at linear order. This is a
mathematical identity used for cross-checking. DFD’s Einstein dynamics comes instead from the Einstein–Hilbert
term of the spectral action acting on the physical metric
in absolute time (App. WS).
c. Physical consequences.
• Preferred foliation: DFD has absolute simultaneity (constant-t surfaces)
• No closed timelike curves: The 3D picture forbids them automatically

∇ · σ = −ρc2 : the divergence of the vacuum stress balances the energy loading from matter.
c. Reduced gravitational permittivity. The crossover
function µ(s) acts as a field-dependent gravitational permittivity. At high strain (s ≫ 1), µ → 1 and the vacuum
conducts gravitational flux at full Newtonian strength.
At low strain (s ≪ 1), µ ≈ s → 0: the vacuum becomes
a poor conductor of gravitational flux. By Gauss’s law,
the gradient |∇ψ| must then exceed the Newtonian value
to carry the same flux—yielding v 2 = ra = const (flat
rotation curves) without dark matter. The analogy is to a
nonlinear dielectric whose permittivity drops at low field
strengths.
d. Vacuum energy hierarchy. The loading picture
distinguishes three scales: the Planck density ρP c2 ∼
10113 J/m3 (naive QFT mode sum), the vacuum stiffness
K0 ∼ 1042 N (resistance to deformation), and the cosmological residual ρΛ c2 ∼ 10−9 J/m3 (residual strain of
order H02 /c2 ). The critical distinction is that K0 is a force
scale, not an energy density; the observed dark energy is
residual loading, not the stiffness itself. The α57 suppression from the finite microsector (Appendix O) provides
the exponent: 57 frozen KK modes, each suppressing by
1/137, give the 122 orders of magnitude between ρP and
ρc .
H.

• Fixed topology: Space is R3 forever
• Refractive interpretation: “Curved spacetime”
is refractive medium
This contrasts with GR, where 4D spacetime is fundamental. In DFD the arena is flat R3 with absolute
time, and the physical metric obeys the Einstein–Hilbert
dynamics in that time (App. WS).

G.

Physical Interpretation: Vacuum Loading

The mathematical formalism admits a direct physical
interpretation in which gravity arises from energy loading
of the quantum vacuum [31]. Mass—which is predominantly field energy (about 99% of the proton’s mass is
QCD field energy, not quark rest mass)—deposits a fractional loading ψ in the vacuum, modifying its refractive
index to n = eψ .
a. Vacuum stiffness. The coefficient K0 = c4 /(8πG)
is a force scale (units: newtons), not an energy density.
It is the vacuum’s resistance to deformation—the same
coefficient that appears in the Einstein field equations.
Via the master invariant (§XIX), its exponent is fixed and,
on bound-matter clocks where C = 1 (Postulate ZS.11),
K0 = ℏ(H0loc )2 /(8πα57 c).
b. Stress–strain interpretation. The field equation (21) has the structure of a nonlinear constitutive equilibrium. Defining the gravitational strain s ≡ |∇ψ|/a∗
and stress σ ≡ K0 µ(s) ∇ψ, the field equation reads

Summary of Section II

The mathematical structure of DFD is fully specified
by:
1. The optical metric ds̃2 = −c2 dt2 /n2 + dx2 with
n = eψ [Eq. (6)].
2. The scalar action with nonlinear kinetic term
[Eq. (10)].
3. The field equation ∇ · [µ(|∇ψ|/a⋆ )∇ψ] =
−(8πG/c2 )ρ [Eq. (21)]; inside bound systems that
contain χ, the adopted energy of App. AV.
4. The TT gravitational wave sector at speed c
[Eq. (16)].
5. The constrained µ(x) family satisfying solar, deepfield, monotonicity, and convexity conditions.
All dynamics derive from the action principle. The
theory has three degrees of freedom in its fields, of which
the two tensor ones (hTT
ij ) propagate; the scalar ψ is a
constraint (Axiom WS.1). Below the spectral cutoff there
are no ghosts and no negative-energy modes, and the field
equations are well posed (§III); the Weyl-squared term
implies a spin-2 pole above the cutoff (App. WS).
III.

MATHEMATICAL WELL-POSEDNESS

A physical theory must be mathematically well-posed:
given initial/boundary data, solutions must exist, be

31
unique, and depend continuously on the data. This section
establishes these properties for the DFD field equations
in both static and dynamic settings.

boundary data ψ = ψD on ∂Ω is:
Z
Z
a(∇ψ) · ∇v d3 x =
f v d3 x,
Ω

∀ v ∈ W01,p (Ω), (34)

Ω

where f = (8πG/c2 )(ρ − ρ̄) is the source term.
A.

Static Solutions: Elliptic Theory
1.

Theorem III.1 (Existence). Under assumptions (A1)–
(A4), for any f ∈ V ′ (the dual of the Sobolev space
W 1,p (Ω)), there exists a weak solution ψ ∈ W 1,p (Ω) satisfying (34) with the prescribed boundary data.

Assumptions on µ

The field equation (21) is a quasilinear elliptic PDE.
Well-posedness requires the following conditions on the
response function µ : [0, ∞) → [0, ∞):
(A1) Continuity: µ is continuous on [0, ∞).
(A2) Coercivity: There exist constants α > 0 and
β ≥ 0 such that
µ(|ξ|)|ξ|2 ≥ α|ξ|2 − β

∀ ξ ∈ R3

(31)

(we take p = 2). With a bounded domain this
makes the operator coercive on W01,2 (Ω), and the
energy functional is bounded below.
(A3) Growth bound: There exists β > 0 such that

Theorem III.2 (Uniqueness). If the flux operator a(ξ)
is strictly monotone [strict inequality in (A4)], then the
weak solution of Theorem III.1 is unique.
a. Proof sketch. The existence proof uses direct
methods in the calculus of variations. Define the energy
functional:
Z
Z
H(∇ψ) d3 x −

E[ψ] =
Ω

f ψ d3 x,

(35)

Ω

R1
where H(ξ) = 0 a(tξ)·ξ dt is the energy density satisfying
a(ξ) = ∇ξ H(ξ).

(32)

1. Coercivity (A2) ensures E[ψ] → +∞ as ∥∇ψ∥p →
∞, so minimizing sequences are bounded.

This controls the operator’s growth at large gradients.

2. Convexity of H (following from monotonicity) ensures E is weakly lower semicontinuous.

p−1

|µ(|ξ|)ξ| ≤ β(1 + |ξ|)

.

(A4) Monotonicity: For all ξ, η ∈ R3 ,


µ(|ξ|)ξ − µ(|η|)η · (ξ − η) ≥ 0.

(33)

3. By the direct method, a minimizer exists in
W 1,p (Ω).

Strict inequality (strict monotonicity) implies
uniqueness.

4. The Euler-Lagrange equation for the minimizer is
precisely (34).

a. Physical interpretation. Condition (A1) ensures
continuous transition between regimes. Condition (A2)
prevents the field from “running away” to arbitrarily large
values without cost in energy. Condition (A3) ensures
solutions have finite energy in bounded domains. Condition (A4)—monotonicity—is the ellipticity condition: it
ensures the linearized operator has the correct sign for
stable perturbations.
b. Verification for standard µ. The simple and standard forms from Table V satisfy (A1)–(A4):

Uniqueness follows from strict convexity: if two solutions ψ1 , ψ2 existed, convexity implies E[(ψ1 + ψ2 )/2] <
(E[ψ1 ] + E[ψ2 ])/2, contradicting minimality.

• Simple: µ(x) = x/(1 + x) is continuous, bounded
between 0 and 1, and strictly increasing.
√
• Standard: µ(x) = x/ 1 + x2 has the same properties with different asymptotic rates.
Both yield well-posed elliptic problems. For the exact law
the same holds with F (x) ≥ x − 0.648 (SOURCE) and
F (x) ≥ x − e−1 (TOTAL), and µ < 1.

3.

Theorem III.3 (Regularity). If f ∈ Lq (Ω) with q > 3/p′
(where 1/p + 1/p′ = 1), then any weak solution ψ is locally
0,α
Hölder continuous: ψ ∈ Cloc
(Ω) for some α > 0.
1
If additionally µ ∈ C and f ∈ C 0,γ (Ω), then ψ ∈
1,α
Cloc
(Ω).
Higher regularity follows by standard bootstrap arguments from quasilinear elliptic theory [32, 33]. For smooth
µ and smooth sources, solutions are C ∞ in the interior
away from points where ∇ψ = 0.

B.
2.

Existence and Uniqueness

Define the flux operator a(ξ) := µ(|ξ|)ξ. The weak
formulation of the field equation on a domain Ω with

Regularity

Exterior Domains and Boundary Conditions

For isolated gravitating systems, we work on exterior
domains Ω = R3 \ BR (the complement of a ball). Two
types of boundary conditions arise:

32
|∇ψ| → 0

1.

First-Order Symmetric Hyperbolic Form

Equation (37) can be rewritten as a first-order symmetric hyperbolic system. Introduce:
U = (ψ, ∂t ψ, ∂1 ψ, ∂2 ψ, ∂3 ψ)T .

(38)

The evolution takes the form:

Horizon

Ω
Asymptotic

A0 (U ) ∂t U + Ai (U )∂i U = S(U, x),
(39)


0
−2
where A (U ) = diag 1, c , M (∇ψ) is block-diagonal,
with Mij the flux Jacobian of §III D, the Ai (U ) are symmetric matrices depending on the state U , and S contains
source terms.
Symmetric hyperbolicity requires A0 to be uniformly
positive definite, i.e. for some λ ≥ 1
λ−1 |n|2 ≤ n · M (∇ψ) n ≤ λ|n|2

∀ n ∈ R3

(40)

[assumption (A1 ) of §III E]. The eigenvalues of M are
µ(x) (twice) and d[xµ(x)]/dx, x = |∇ψ|/a⋆ ; both vanish
at ∇ψ = 0 for every response with µ(0) = 0, so the
condition fails there for µ itself; it holds everywhere
p for the


regularised galactic approximant µs0 (x) = µ x2 + s20 ,
µ(x) = x/(1 + x), s0 > 0, with λ = 1 + 1/s0 .
′

FIG. 4. Domain structure for exterior problems. The solution
domain Ω (blue) excludes the optical horizon region (orange).
The field gradient vanishes at infinity.

a. Asymptotic condition. At spatial infinity, we require |∇ψ(x)| → 0 as |x| → ∞.
b. Optical horizon. At the optical horizon (where
n → ∞), an ingoing-flux Neumann condition is imposed:
a(∇ψ) · n = ghor ,

(outgoing flux = 0).

(36)

This asymmetric condition reflects the fact that light
cannot escape an optical horizon, a one-way membrane
in the optical metric. With the parent strain’s σ kept a
black hole’s event horizon is such a one-way membrane
(Sec. VI B): for an eternal static hole it is the optical
horizon, n → ∞, at the arena’s origin, and for a hole
formed by collapse it is the flow horizon of DFD’s maximal
slicing.
Theorem III.4 (Exterior well-posedness). Under assumptions (A1)–(A4) and the boundary conditions above,
1,p
there exists a weak solution ψ ∈ Wloc
(Ω) with the gradient vanishing at infinity. If the boundary operators are
strictly monotone, the solution is unique.
The proof extends standard techniques by using
weighted Sobolev spaces to handle the unbounded domain.

C.

Dynamic Solutions: Hyperbolic Theory

Under Axiom WS.1 the local scalar sector is an instantaneous constraint; its numerical hyperbolic relaxation,
which this subsection analyses, is:
1 2
8πG
∂ ψ − ∇ · [µ(|∇ψ|/a⋆ )∇ψ] = 2 (ρ − ρ̄).
(37)
c2 t
c
This is a quasilinear wave equation with nonlinear principal part.

2.

Local Well-Posedness

Theorem III.5 (Local existence). Let s > 5/2, and let
the response be the approximant µ(x) = x/(1 + x) or
either branch of the exact law. The flux a(ξ) = µ(|ξ|/a⋆ ) ξ
has Jacobian eigenvalues µ(x) (twice) and F ′ (x), F (x) =
xµ(x), which are positive for every x > 0 and bounded
by Λ = 1 (approximant), 1.034 (SOURCE) and 1.135
(TOTAL). Let g be a constant vector, and let ∇ψ0 − g ∈
H s−1 (R3 ), ψ1 ∈ H s−1 (R3 ) with inf x |∇ψ0 | > 0, and a
source in C([0, T0 ]; H s−1 ). Then there are T > 0 and a
unique solution with ∇ψ − g, ∂t ψ ∈ C([0, T ]; H s−1 (R3 ))
and inf |∇ψ| > 0 on [0, T ], depending continuously on
the data. It continues for as long as ∥∂ψ∥W 1,∞ stays
finite and |∇ψ| stays bounded away from zero. For the
regularised response µs0 the lower bound on |∇ψ0 | is not
needed and g = 0 is allowed.
The proof is the standard one for quasilinear secondorder hyperbolic equations (Hughes, Kato and Marsden
1977), applied where |∇ψ| > δ, since there the principal
part is smooth and uniformly hyperbolic. The state is
(∇ψ, ∂t ψ) rather than ψ, because the field of any body
falls off as 1/r and is not square integrable. By finite speed
of propagation the theorem also holds locally: data that
are smooth with |∇ψ0 | ≥ δ > 0 on a ball BR (x0 ) have
√ a
unique smooth solution in the cone |x − x0 | < R − c Λ t
for short times. Points where ∇ψ = 0 are excluded:
both characteristic speeds vanish there. The linearised
evolution at such points is covered by Theorem III.9.
a. Limitation: Global existence. Global existence (arbitrary long times) is not guaranteed. On a non-zero
uniform background gradient the smooth evolution has a
finite lifespan: gradient singularities form in finite time,

33
analogous to shock formation in nonlinear wave equations
(Sec. III F).
For small perturbations of amplitude ε in three dimensions the lifespan is at least eC/ε , beyond any astrophysical
time for small ε; plane-symmetric waves break on an algebraic timescale. The linearised and weak-field predictions
do not depend on it.
3.

Finite Speed of Propagation

This follows from the principal part of (37), which has
a00 = −c−2 and a0i = 0 (§III E): the surfaces t = const
are non-characteristic for every response, and each characteristic moves forward in t at a finite speed set by the
eigenvalues of M . A superluminal characteristic on a
background therefore violates light-cone containment, not
causality. This is the reading of superluminal propagation
on nontrivial backgrounds given by Babichev, Mukhanov
and Vikman [34]; it applies directly to DFD, whose preferred foliation supplies the global time function.

Theorem III.6 (Causality). Solutions of (37) with the
galactic approximant µ(x) = x/(1 + x), or with its regularisation µs0 , satisfy:
1. All characteristic speeds are ≤ c.
2. The domain of dependence of a point (t, x) is contained in the backward light cone {(t′ , x′ ) : |x − x′ | ≤
c(t − t′ )}.
3. No signal propagates faster than c.
This follows from the eigenvalues of the flux Jacobian M p
: the scalar sector has characteristic
speeds
p
v⊥ = c µ(x) (transverse) and v∥ = c d[xµ(x)]/dx
(longitudinal), x = |∇ψ|/a⋆ , so all of them are ≤ c
exactly when supx d[xµ(x)]/dx ≤ 1. For the galactic
2
approximant µ(x) = x/(1 + x), v⊥
/c2 = x/(1 + x) and
2 2
2
v∥ /c = x(2 + x)/(1 + x) ; for its regularisation µs0 both
speeds
are bounded by the same expressions evaluated at
p
x2 + s20 .
Causality is a crucial physical requirement. The TT
cone coincides with the optical cone, cT = cγ exactly: light
and gravitational waves share one metric (App. WS). With
the galactic approximant µ(x) = x/(1+x) the relaxation’s
characteristic speeds stay below c at every finite gradient,
1−v∥2 /c2 = (1+x)−2 , with v∥ → c only as x → ∞. For the
exponential responses of Appendix BS, Eqs. (BS8)–(BS9),
which are the SOURCE and TOTAL branches of the
exact µ-law (Theorem BS.9) and have x/(1 + x) as their
weak-response approximant, the same characteristic-speed
formula, applied to the µ-form of each branch (SOURCE
through x = y ν(y)), gives sup v∥ = 1.017 c (SOURCE,
near x ≈ 7) and 1.066 c (TOTAL, at x = 2) in the
transition regime.
Proposition III.7 (Causality of the exact law). For
the exact law, Eqs. (BS8)–(BS9), the characteristic speeds
of (37) are finite, with suprema 1.017 c (SOURCE) and
1.066 c (TOTAL) in the transition regime. The equation
is hyperbolic with finite propagation speed, on the same
terms as for the approximant (both degenerate only at
∇ψ = 0). Every characteristic advances the absolute time
t of the preferred foliation (§II F), so t is a global time
function for the scalar, tensor and light cones alike: no
signal reaches its own past, and no closed causal curve
can form. What fails for the exact law is containment in
the light cone, which items 1–3 of Theorem III.6 assert
for the approximant only.

D.
1.

Stability

Energy Minimization

Theorem III.8 (Energy minimization). If W is strictly
convex, then:
1. Static solutions are local energy minima.
2. There are no negative-energy (ghost) modes in the
linearized theory below the spectral cutoff.
a. Proof sketch. Convexity of W implies convexity
of the energy density H(ξ). The integral E[ψ] inherits
this convexity.
2.

Perturbative Stability

Consider small perturbations δψ about a static solution
ψ0 :
ψ = ψ0 + δψ,

|δψ| ≪ |ψ0 |.

The linearized equation for δψ is:
1 2
∂ (δψ) − ∇ · [Mij (∇ψ0 )∇j (δψ)] = 0,
c2 t
where the effective mass matrix is:
(∂i ψ0 )(∂j ψ0 )
Mij = µ(x0 )δij + µ′ (x0 )
,
|∇ψ0 | a⋆

(41)

(42)

(43)

with x0 = |∇ψ0 |/a⋆ . The denominator |∇ψ0 | a⋆ ensures
dimensional consistency: since [(∂i ψ0 )(∂j ψ0 )] = m−2 and
[|∇ψ0 | a⋆ ] = m−2 , the ratio is dimensionless.
Where ∇ψ0 ̸= 0, Mij is positive definite, and the linearized operator has only real, positive eigenfrequencies:
no growing modes and no instabilities. Theorem III.9
covers the general case.
Theorem III.9 (Linear well-posedness on any static
background). Let ψ0 be a static solution generated by a
bounded source, with ∇ψ0 bounded and continuous, and
let the set where ∇ψ0 = 0 have a boundary of zero volume. Put A(x)
R = Da(∇ψ0 (x)), so that 0 ≤ A ≤ Λ. The
form q(u) = ∇u · A ∇u d3 x on smooth compactly supported functions is closable in L2 (R3 ), and the self-adjoint
operator L ≥ 0 of its closure generates the linearised evolution c−2 ∂t2 u + Lu = 0. For data in the form domain

34
times
L2 the solution 
exists and is unique, its energy
R −2
1
c u2t + ∇u · A ∇u d3 x is conserved, no mode grows
2
√
exponentially, and disturbances travel no faster than c Λ.
This includes points where ∇ψ0 = 0, where A vanishes.
Proof. Where A > 0 the coefficient is locally uniformly elliptic, so a sequence that tends to zero in L2 and is Cauchy
for q has gradients tending to zero locally there; the limit
of A1/2 ∇un therefore vanishes, and q is closable. Kato’s
representation theorem
 √ gives L,√and the spectral
 √ theorem
−1
gives u(t) = cos ct L u0 + (c L) sin ct L u1 with
conserved energy and spectrum in [0,
√∞). The energy flux
through a sphere is bounded by c Λ times the energy
density, which uses only A ≤ Λ, so the energy in a ball
shrinking at that speed cannot grow. An isolated zero
of ∇ψ0 has zero capacity for q, so the result does not
depend on how the form is first defined there.
3.

No Ghosts

A ghost is a degree of freedom with wrong-sign kinetic
term, leading to negative-energy states. In DFD:
• The scalar ψ has kinetic term ∝ W ′ (X) > 0 by
(A4).
• The TT modes hTT
ij have standard positive kinetic
term from (16).
Propagating degrees of freedom below the spectral cutoff:
the two TT polarizations (under Axiom WS.1 ψ is a
constraint), all with positive kinetic energy, so there are
no ghosts there. The Weyl-squared term of the spectral
action (App. WS) implies an extra spin-2 pole at about
3.6 reduced Planck masses for f0 = 1, above the cutoff; it
moves no observable.
E.

where for the DFD equation (37) a00 = −c−2 , a0i =
0 and aij = Mij (∇ψ) is the flux Jacobian of §III D.
(A2′ ) Lower-order regularity: For |α| ≤ s (with s >
5/2), the derivatives ∂ α bµ , ∂ α c are continuous and
polynomially bounded in |ψ|, |∂ψ|.
(A3′ ) Source regularity: S(x) ∈ H s−1 on the spatial
domain.
These are satisfied by the DFD strong-field equation
with the regularised approximant µs0 whenever ψ and ∂ψ
remain bounded; for the unregularised µ, (A1′ ) fails at
∇ψ = 0 (§III C).
2.

IBVP Formulation

Let Ω ⊂ R3 be bounded with smooth boundary ∂Ω.
The Dirichlet IBVP is:


aµν (ψ, ∂ψ)∂µ ∂ν ψ + l.o.t. = S(x), (t, x) ∈ [0, T ] × Ω


ψ(0, x) = ψ (x),
x∈Ω
0

∂
ψ(0,
x)
=
ψ
(x),
x∈Ω
t
1



ψ(t, x) = g(t, x),
(t, x) ∈ [0, T ] × ∂Ω
(46)
3.

Compatibility Conditions

Regularity requires the initial and boundary data to be
compatible at {t = 0} ∩ ∂Ω:
• Zeroth order: ψ0 |∂Ω = g(·, 0)
• First order: ψ1 |∂Ω = ∂t g(·, 0)
• k-th order: ∂tk ψ|t=0,∂Ω = ∂tk g(·, 0)
For solutions in H s (Ω) with s > 5/2, compatibility is
required up to order ⌊s − 1⌋.

Initial-Boundary Value Problems

For laboratory experiments and numerical simulations
in finite volumes, we require well-posedness of the initialboundary value problem (IBVP). This is the natural
setting for terrestrial tests of DFD.

4.

Define the Sobolev energy:
X Z


Es (t) =
|∂ α ∂t ψ|2 + |∇∂ α ψ|2 d3 x.
|α|≤s

1.

Dynamic Structural Assumptions

The dynamic field equation can be written in the general
quasilinear form:
aµν (ψ, ∂ψ)∂µ ∂ν ψ + bµ (ψ, ∂ψ, x)∂µ ψ + c(ψ, ∂ψ, x) = S(x),
(44)
where aµν forms the principal symbol and bµ , c are lowerorder terms. Well-posedness requires:

(A1′ ) Uniform hyperbolicity: There exists λ ≥ 1 such
that, for all spatial covectors n ∈ R3 ,
λ−1 |n|2 ≤ aij ni nj ≤ λ|n|2 ,

(45)

Energy Estimates

(47)

Ω

Under assumptions (A1′ )–(A3′ ) with compatibility conditions:


d
Es (t) ≤ C(M ) Es (t) + ∥S∥2H s−1 (Ω) + ∥g∥2H s−1/2 (∂Ω) ,
dt
(48)
where C(M ) depends on bounds for ψ, ∂ψ in L∞ .
By Gronwall’s lemma:
Z t


 
Es (t) ≤ eC(M )t Es (0) +
∥S∥2H s−1 + ∥g∥2H s−1/2 dτ
0

(49)
This establishes continuous dependence on initial and
boundary data.

35
5.

8.

Main IBVP Theorem

Theorem III.10 (IBVP Well-Posedness). Let Ω ⊂ R3
be bounded with smooth boundary and s > 5/2. Under
assumptions (A1′ )–(A3′ ), given:
• Initial data (ψ0 , ψ1 ) ∈ H s (Ω) × H s−1 (Ω)
• Source S ∈ H s−1 (Ω)

Theorem III.13 (Continuous Dependence). Let ψ1 , ψ2
be solutions with data (f1 , BC1 ), (f2 , BC2 ) respectively.
If a is strongly monotone and locally Lipschitz:


∥∇(ψ1 − ψ2 )∥Lp (Ω) ≤ C ∥f1 − f2 ∥V ′ + ∥BC1 − BC2 ∥∂Ω .
(54)
This ensures physical stability: small changes in sources
or boundary conditions produce small changes in solutions.

• Boundary data g ∈ H s ([0, T ] × ∂Ω)
• Compatibility conditions up to order ⌊s − 1⌋
there exists T > 0 and a unique solution
ψ ∈ C 0 ([0, T ]; H s (Ω)) ∩ C 1 ([0, T ]; H s−1 (Ω))

9.

(50)

depending continuously on (ψ0 , ψ1 , S, g) in the natural
Sobolev norms.
a. Proof sketch. The proof uses standard techniques
for quasilinear hyperbolic IBVP. Linearization around an
approximate solution, energy estimates with boundary
multipliers, and Picard iteration in a suitable Banach
space yield existence and uniqueness. The compatibility
conditions control boundary terms in the energy estimates.
Finite Speed of Propagation

a. Regularization. At |∇ψ| → 0, the Jacobian may
become ill-conditioned.
A practical remedy is to replace
p
|∇ψ| by |∇ψ|2 + s20 with small s0 > 0.

for |x − x0 | ≤ R − cchar t.

(51)

This ensures causality: disturbances propagate at finite
speed. For the regularised approximant, to which (A1′ )–
(A3′ ) apply, that speed is bounded by c (Theorem III.6);
for the exact law it exceeds c in the transition regime,
and every characteristic advances the absolute time t
(Proposition III.7).
7.

Parabolic Extension

For dissipative problems or numerical relaxation
schemes, the parabolic extension is relevant:
∂t ψ − ∇ · [µ(|∇ψ|)∇ψ] = f (t, x).

1. Global existence for dynamic equations:
Smooth solutions do not persist in general: on a
non-zero uniform background gradient they lose regularity in finite time (the gradient-singularity item
below). Whether weak solutions exist for all time
and are unique is open, as it is for the equations of
gas dynamics in more than one dimension, where admissible weak solutions are known not to be unique
(Chiodaroli, De Lellis and Kreml 2015). In planar
symmetry entropy solutions exist for data whose
invariant region avoids ∇ψ = 0 (DiPerna 1983). No
prediction of this monograph uses a global weak
solution.
2. Horizon boundary conditions: With the parent
strain’s σ kept a black hole has general relativity’s event horizon (Sec. VI B), where the one-way
horizon condition of §III B applies; its uniqueness,
regularity and numerical convergence are open.

(52)

Theorem III.12 (Parabolic Well-Posedness). Under assumptions (A1)–(A4), there exists a unique evolution
ψ ∈ Lp (0, T ; W 1,p (Ω)) ∩ C([0, T ]; L2 (Ω)).

Open Mathematical Problems

Several mathematical questions remain open:

Theorem III.11 (Finite Speed). Let ψ and ψ̃ be solutions of (37) with initial data coinciding in a ball BR (x0 ).
There exists a characteristic speed cchar > 0 (depending
only on the hyperbolicity constant λ) such that
ψ(t, x) = ψ̃(t, x)

Numerical Implementation

The weak form (34) is directly implementable in finite
element packages. The Newton iteration Jacobian is:
∂i ψ ∂j ψ
Aij (∇ψ) = µ(|∇ψ|)δij + µ′ (|∇ψ|)
.
(55)
|∇ψ|

F.
6.

Stability Estimates

(53)

If f is time-independent and boundary operators are dissipative, solutions converge to a steady state as t → ∞.
This follows from Crandall–Liggett theory: the monotone operator Aψ = −∇ · a(∇ψ) generates a contraction
semigroup on L2 (Ω).

3. Gradient blow-up and singularity formation:
Partly settled. On a non-zero uniform background
gradient g ̸= 0 the smooth evolution of the vacuum
equation has a finite lifespan. In planar symmetry
every non-trivial small compactly supported perturbation reaches a gradient catastrophe at a finite
time T⋆ , at which ∇ψ stays bounded and continuous
while ∂ 2 ψ → ∞ (Lax 1964; John 1974). In three
dimensions the Klainerman null condition fails for
every g ̸= 0, and an open set of nearly radial small

36
data forms the same singularity at Tε ∼ eC(g)/ε ,
a statement conditional on the sketched extension
of Speck (2016) to principal parts that depend on
∂ψ. For the exact law the planar statement holds
on every background as well, with one refinement:
the quadratic nonlinearity along g vanishes on the
single background |g|/a⋆ = 7.18 (SOURCE; 2 for
TOTAL), where the longitudinal speed peaks, and
there plane waves along g break on the longer time
T⋆ ∝ ε−2 (Klainerman and Majda 1980). The
null condition fails at every g ̸= 0 for all three
responses, because µ′ > 0 everywhere. What remains open is nonlinear evolution through points
where ∇ψ = 0, where the principal part degenerates,
the continuation past T⋆ and the singularity structure; the linearised evolution there is well-posed
(Theorem III.9).
4. Coupling to quantum fields: The semi-classical
regime (quantum matter on classical ψ background)
is well-defined. Full quantization of ψ is unnecessary:
the action scales as Sψ ∼ (MP /a⋆ )2 ≫ ℏ, ensuring
quantum fluctuations are negligible. The gauge
emergence framework provides the connection to
particle physics (§XVII).
These technical open problems do not affect the physical predictions in §IV–§XIII, which operate in wellunderstood weak-field or linearized regimes.
G.

Summary of Section III

The DFD field equations are mathematically wellposed:
TABLE VI. Well-posedness summary. The Dyn. and IBVP
entries hold for the exact law and its approximant on data
with ∇ψ bounded away from zero (Theorem III.5), and for the
regularised approximant µs0 without that restriction (§III C);
the light-cone bound cchar ≤ c of the Causality row holds
for the unregularised approximant µ(x) = x/(1 + x) as well
(Theorem III.6). For the exact law cchar exceeds c in the
transition regime, and causality rests on the absolute time t
(Proposition III.7).
Property
Static
Dyn.
IBVP
Existence
✓
✓(loc.)
✓(loc.)
Uniqueness
✓(str. mon.) ✓(loc.) ✓(compat.)
1,α
Regularity
Cloc
H s pres. H s pres.
Stability
✓(convex W )
✓
✓(Gron.)
Causality
—
cchar ≤ c cchar ≤ c
No ghosts (below the cutoff)
✓
✓
✓

The mathematical foundations are solid: existence and
uniqueness theorems, regularity results, stability guarantees, causal propagation, and explicit energy estimates.
The IBVP formulation enables rigorous treatment of
laboratory-scale experiments in bounded domains. This
places DFD on equal footing with GR for the static and
relaxation problems treated here.

IV.

PARAMETRIZED POST-NEWTONIAN
ANALYSIS

Having established DFD’s mathematical structure in
Part I, we now demonstrate that the theory reproduces
General Relativity in all precision tests of gravity conducted within the Solar System. This section presents a
complete Parametrized Post-Newtonian (PPN) analysis,
showing that DFD’s ten PPN parameters are identical
to those of GR. The critical result—γ = β = 1 with
all preferred-frame and conservation-violation parameters
vanishing—ensures compatibility with the most stringent
experimental constraints on gravitational physics.

A.

The PPN Framework

The PPN formalism provides a systematic method for
comparing metric theories of gravity in the weak-field,
slow-motion regime characteristic of the Solar System [5,
6]. Any theory predicting a metric gµν can be expanded in
powers of the Newtonian potential U/c2 ∼ ϵ2 and velocity
v/c ∼ ϵ, with coefficients parametrized by dimensionless
constants.
a. Newtonian potential and matter variables. For a
perfect fluid with density ρ, pressure p, specific internal
energy Π, and velocity v, define the Newtonian potential
Z
ρ(x′ ) 3 ′
U (x) = G
d x.
(56)
|x − x′ |
Additional potentials capture velocity-dependent effects:
Z
Z
ρ(v · R)Ri 3 ′
ρvi 3 ′
d x,
Wi = G
d x,
Vi = G
R
R3
(57)
Z
Z
2
′
ρv 3 ′
ρU (x ) 3 ′
Φ1 = G
d x , Φ2 = G
d x,
(58)
R
R
Z
Z
ρΠ 3 ′
p 3 ′
Φ3 = G
d x , Φ4 = G
d x,
(59)
R
R
where R = x − x′ and R = |R|.
b. The PPN metric template. The general PPN metric in isotropic coordinates takes the form [6]:
2U
U2
g00 = −1 + 2 − 2β 4
c
c
(60)


1h
+ 4 − 2ξΦW + 2γ + 2 + α3 + ζ1 − 2ξ Φ1
c




+ 2 3γ − 2β + 1 + ζ2 + ξ Φ2 + 2 1 + ζ3 Φ3
(61)



 i
+ 2 3γ + 3ζ4 − 2ξ Φ4 − ζ1 − 2ξ A + O(c−6 ),
R
where A = G ρ [v · R]2 /R3 d3 x′ is the standard
anisotropic kinetic potential, and the template is written
in the mean rest frame of the universe (the preferredframe wi -dependent terms are omitted). At the GR values γ = β = 1, ξ = α3 = ζ1,2,3,4 = 0 the bracket reduces

37
to 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4 .

1 
g0i = − 3 4γ + 3 + α1 − α2 + ζ1 − 2ξ Vi
2c
−


1 
1
+
α
−
ζ
+
2ξ
Wi ,
2
1
2c3

gij =

U
1 + 2γ 2
c

C.

(62)

(63)


δij .

(64)

The ten PPN parameters {γ, β, ξ, α1 , α2 , α3 , ζ1 , ζ2 , ζ3 , ζ4 }
have the following physical interpretations:
• Curvature/nonlinearity (γ, β, ξ): γ measures
the amount of spatial curvature produced by unit
rest mass; β measures nonlinearity in the superposition of gravitational potentials; ξ is the Whitehead
parameter for anisotropic stress contributions.
• Preferred-frame effects (α1 , α2 , α3 ): These
parametrize preferred-frame effects that would arise
if gravity selects a cosmologically preferred rest
frame.
• Conservation laws (ζ1 , ζ2 , ζ3 , ζ4 ): These
parametrize violations of total momentum and energy conservation.
General Relativity predicts γ = β = 1 and all other
parameters zero. Table VII summarizes current experimental constraints.

B.

DFD Physical Metric in PPN Form

In the nondispersive regime, DFD’s dynamics are governed by the physical metric (12) (Sec. II B):
g00 = −e−ψ ,

gij = e+ψ δij ,

(65)

where the scalar field ψ satisfies the field equation (21). In
the weak-field limit relevant to Solar System
√ tests, ψ ≪ 1
− aN /a0
and µ(|∇ψ|/a⋆ ) → 1, with 1 − µ = e
exactly
on the derived exponential response (SOURCE branch;
App. BS, Cor. BS.14; the weak-response approximant
x/(1 + x) of Sec. VII does not apply here), so the field
equation reduces to the Poisson equation:
8πG
2U
∇2 ψ = − 2 ρ ⇒ ψ = + 2 + O(c−4 ).
(66)
c
c
The crucial observation is that the exponential physical metric (65) determines the PPN parameters through
Taylor expansion.

Parameter Extraction: γ = β = 1

a. Spatial metric and γ. Expanding gij = e+ψ δij to
first order in ψ:


ψ2
+ψ
+ · · · δij
gij = e δij = 1 + ψ +
2
(67)


2U
−4
= 1 + 2 δij + O(c ).
c
Comparing with the PPN template (64), which has coefficient 2γU/c2 , immediately yields
γ=1.

(68)

b. Temporal metric and β. Expanding g00 = −e−ψ
to second order:


ψ2
g00 = −e−ψ = − 1 − ψ +
+ ···
2
ψ2
(69)
+ O(c−6 )
2
2U 2
2U
= −1 + 2 − 4 + O(c−6 ).
c
c
The coefficient of −U 2 /c4 in the PPN template (61) is
2β. Since DFD gives exactly −2U 2 /c4 , we have
= −1 + ψ −

β=1.

(70)

c. 1PN completion: the Φ-sector coefficients derived,
and ξ = 0. The c−4 sector of g00 is not matched to
GR term by term; it is derived from DFD’s own field
equation with the standard perfect-fluid closure. Varying
the matter action with respect to ψ (the same variation
that yields the Tolman source law S = 12 (ε + 3p) eψ ) and
expanding through 1PN order gives


Π
2v 2
2U
3p
ρc2
S=
1 + 2 + 2 + 2 + 2 + O(c−4 ), (71)
2
c
c
c
ρc
verified
algebra; the 2U/c2 term arises solely
√ by computer
ψ
from −g̃ = e . Writing ψ = 2U/c2 + ψ2 /c4 and solving
with the flat Green function term by term against ∇2 Φk =
−4πG (densities),
ψ2 = 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4 − ∂t2 χ,
(72)
R
3 ′
where χ = −G ρ R d x is the PPN superpotential of
Eq. (79) below.
Gauge Lock. The single time redefinition ξ 0 = ∂t χ/2c3
produces δg00 = +∂t2 χ/c4 and δg0i = + 12 ∂i ∂t χ/c3 : one
coordinate change simultaneously removes the −∂t2 χ
term here and the longitudinal gradient discarded in
the vector-sector split of Eq. (80). One parameter, two
cancellations—the scalar and vector sectors are gaugeconsistent, and the metric lands in the standard PPN
gauge. Expanding g00 = −e−ψ then yields the coefficient
set
s1 = 4, s2 = 4, s3 = 2, s4 = 6, sW = 0, sA = 0

(73)

for (Φ1 , Φ2 , Φ3 , Φ4 , ΦW , A) in the template (61). The
match delivers β = 1 a second, independent time: ψ2
contains 4Φ2 but no U 2 piece, so −2U 2 comes solely

38
TABLE VII. Current experimental bounds on PPN parameters. GR predicts γ = β = 1 and all others zero.
Parameter GR Value Experimental Bound Primary Constraint
γ−1
β−1
ξ
α1
α2
α3
ζ1
ζ2
ζ3
ζ4

0
0
0
0
0
0
0
0
0
0

(2.1 ± 2.3) × 10−5
|β − 1| < 3 × 10−4
|ξ| < 10−3
|α1 | < 4 × 10−5
|α2 | < 2 × 10−9
|α3 | < 4 × 10−20
|ζ1 | < 2 × 10−2
|ζ2 | < 4 × 10−5
|ζ3 | < 10−8
—

from e−ψ = 1 − ψ + ψ 2 /2 − · · · . With γ = 1 and the
conservation results of §IV F, the Φ1 coefficient also fixes
α3 = 0 directly.
Whitehead Locality (ξ = 0 structurally). DFD’s kinetic
operator is exactly linear (flat); the field equation’s only
nonlinearity is the local, multiplicative eψ in the source.
A local source convolved with the flat Green function can
never generate the non-local Whitehead double integral
ΦW . Hence
sW = 0 ⇒ ξ = 0

(74)

to all 1PN orders by construction—a structural statement,
not an order-by-order accident.
Kinetic Lock (internal falsifier). Had the spatial kinetic
operator carried a factor ebψ instead of being flat, the
(∇ψ)2 cross term would shift β at O(b); the LLR bound
|β −1| < 3×10−4 enforces the flat master-equation branch.
The 1PN sector thus ties the action’s kinetic structure
directly to lunar ranging.
Remark (elliptic reading). On the elliptic constraint
reading of the scalar sector (Axiom E, Appendix WS), the
instantaneous solve produces ψ2 with no ∂t2 χ term at all,
and the metric lands on the standard PPN gauge directly,
with no gauge transformation (Standard-Gauge Coincidence Lemma). Both routes give the identical 1PN metric;
the derivation above is unconditional, while the elliptic
route is conditional on Axiom E as named there. The elliptic solve’s slicing is the zero-mean-curvature (maximal)
slicing at leading order, and with the parent strain’s σ
kept it is maximal for isolated systems (Appendix WS,
Proposition WS.6).
d. Physical interpretation. The result γ = β = 1 is
not a coincidence but a direct consequence of the exponential structure n = eψ . The optical refractive index n
determines both the light propagation speed
(c/n) and
√
the gravitational time dilation (dτ = dt/ n, the lapse of
the physical metric (12)). The exponential ensures that
these effects are related by exact exponentiation rather
than independent parametrizations, automatically reproducing the GR relation between spatial curvature and
time dilation.

Cassini [35]
LLR [36]
Geophysical
Binary pulsars [37]
Millisecond pulsars [5]
Pulsar spin-down [6]
Combined tests
Lunar/planetary
Lunar acceleration
Not directly tested

D.

Vector Sector: α1 = α2 = α3 = 0

To complete the PPN analysis, we must determine the
gravitomagnetic sector g0i . Introduce a shift vector Ni
such that, through first post-Newtonian order,
ds2 = −e−ψ c2 dt2 + e+ψ δij (dxi + N i dt)(dxj + N j dt).
(75)
The term e+ψ N 2 that this form adds to g00 is of second
post-Newtonian order. Beyond first order this line element
and the form of Appendix WS, condition (b) extended
beyond first order, with g00 = −e−ψ and the shift in
g0i alone, differ only in how the shift enters g00 ; with
the Einstein–Hilbert couplings evaluated order by order
they give the same second-order binding energy, so any
difference enters at third order at the earliest (computed).
Working to 1PN order, the weak-field equation for the
shift is the Poisson problem
16πG
∇2 Ni =
ρvi ,
(76)
c3
sourced by the full matter momentum current ρvi (since
∇2 Vi = −4πGρvi , this sign convention yields Ni < 0
for prograde currents, matching the template (63)). The
residual longitudinal coordinate freedom Ni → Ni + ∂i λ
is fixed below by the superpotential split, consistently
with the Gauge Lock of §IV C.
a. Solution. The solution of Eq. (76) vanishing at
infinity is the pure-V gravitomagnetic potential
4
Ni = − 3 Vi ,
(77)
c
the standard GR result (coefficient −4; recall that Vi as
defined in §IV A already contains G).
Transverse projection. The solution (77) does not solve
the transverse-projected equation ∇2 Ni = −16πG ji⊥ ,
ji⊥ = (δij −∂i ∂j ∇−2 )(ρvj ). Pairing the two is inconsistent
(projection, overall sign and normalization): the transverse part of Vi is Vi⊥ = 12 (Vi + Wi )—from the continuity
identity ∂i Vi = −∂t U together with ∇2 ∂t χ = −2 ∂t U and
the identity (79)—so the transverse-gauge solution is


2
Ni⊥ = − 3 Vi + Wi ,
(78)
c
not the pure-V form. The two solutions differ only by

39
the longitudinal gradient −(2/c3 ) ∂i ∂t χ—precisely the
coordinate freedom exploited in the split below—so the
far-zone coefficient (−4) and every physical quantity are
unchanged; the pairing of Eq. (76) with its solution (77)
is consistent.
Provenance (derived, at a conditional grade). At this
order Eq. (76) coincides with the linearized-GR momentum constraint. Within DFD its derivation is supplied by
the Vector-Sector Completion theorem

 (Appendix WS),
which obtains g0i = − 72 Vi + 12 Wi /c3 directly in the
physically fixed DFD gauge, conditional on two named
inputs: (I) the elliptic constraint reading of the scalar
sector (Axiom E) and (II) the spectral Einstein–Hilbert
zero-mode reduction—the same input the TT sector already carries. At that conditional grade—and only at that
grade—the coefficient −4, the Lense–Thirring precession
(§IV H 5), and the leading spin terms of Appendix AX
become consequences rather than template matches (beyond linear order the Einstein–Hilbert coupling adds the
general-relativistic O(JM ) terms, App. AX); input (II)
is Proposition WS.5, the spectral Einstein–Hilbert reduction, and the vector sector’s derivation carries its
premises, G1 and G2, explicitly; G1 follows from the
Einstein–Hilbert static-energy identity with one G (Proposition WS.6) and the Rest-Mass Channel. To compare
with the PPN template, which is written in the canonical Vi /Wi gauge, we use the superpotentialR identity.
Define the PPN superpotential χ(x) = −G ρ R d3 x′ ,
which obeys ∇2 χ = −2U . Differentiating with the masscontinuity relation ∂t ρ = −∂j (ρvj ) and integrating by
parts gives the standard Will (1993) identity
∂i (∂t χ) = Vi − Wi

⇐⇒ Vi = Wi + ∂i (∂t χ) . (79)
7
Writing −4Vi = − 2 Vi − 21 Vi and replacing the single residual − 12 Vi via the identity (79), Vi = Wi + ∂i (∂t χ), splits
the pure-V form into the standard V /W representation,


4
1
1
7
1
Ni = − 3 Vi = 3 − Vi − Wi − 3 ∂i (∂t χ) , (80)
c
c
2
2
2c
where the final gradient term is pure gauge: it is cancelled
exactly by the same time redefinition ξ 0 = ∂t χ/2c3 that
removes the −∂t2 χ term from g00 (the Gauge Lock of
§IV C—one parameter, two cancellations), and it drops
out of all physical quantities. Since e+ψ = 1 + O(c−2 )
multiplies an O(c−3 ) quantity, g0i = e+ψ Ni retains these
coefficients to the order computed:


1
7
1
DFD
g0i
= 3 − Vi − Wi .
(81)
c
2
2

The far-zone sum is preserved: −7/2 − 1/2 = −4, matching the coefficient of Eq. (77) (where Wi → Vi in the far
zone), so the gauge rewrite changes the V /W split but
not the physical gravitomagnetic field.
b. Extraction of preferred-frame parameters. Matching Eq. (81) to the PPN template (63) (with γ = 1, ξ = 0)
supplies only the two coefficient equations α1 −α2 +ζ1 = 0
(from the Vi term) and α2 − ζ1 = 0 (from Wi ), i.e. α1 = 0
and α2 = ζ1 ; the parameter α3 does not enter g0i at
this order and so cannot be read off from this match

alone. The remaining vanishings follow from covariant
conservation (§IV F, Eq. (89): ζ1 = ζ2 = ζ3 = ζ4 = 0)
together with the absence of a dynamical preferred-frame
field (below), which also removes the semi-conservative
parameter α3 . With ζ1 = 0 the g0i match then forces
α2 = 0, so that
α1 = α2 = α3 = ζ1 = 0 .

(82)

Physically, DFD’s preferred foliation induces no preferredframe effects above the bounded residual of Thm. IV.1: no
dynamical preferred-frame vector field couples to the field
equation—the sole gravitational degree of freedom is the
scalar ψ (sourced by a flat Poisson equation), and at this
order g0i is generated by the matter momentum current
ρvi , as in GR, up to the suppressed temporal-completion
current of Eq. (86). The parameters α1,2,3 therefore have
no dynamical preferred-frame field to couple to.
c. Far-zone consistency check. For a rigid rotator
with angular momentum J, the far-zone behavior has
Wi ≃ Vi , so g0i ≃ (dV + dW )Vi /c3 . With α1,2 = ξ = ζ1 =
0 and γ = 1, the PPN template demands g0i = −4Vi /c3 ,
requiring dV + dW = −4. Equation (81) satisfies this
identically: −7/2 − 1/2 = −4. This confirms the LenseThirring gravitomagnetic field has the correct GR form.

E.

Preferred-Frame Invisibility: Why the Foliation
Decouples at 1PN

DFD has a genuine preferred foliation: absolute time
t defines a cosmological rest frame, identified physically
with the CMB frame (§II, Appendix Q). A preferred
frame is exactly the structure that, in Lorentz-violating
gravity theories, produces nonzero α1 , α2 , α3 . Binarypulsar and millisecond-pulsar observations bound these
to |α1 | < 4 × 10−5 , |α2 | < 2 × 10−9 [5, 37], consistent
with zero. We must therefore prove that DFD’s preferred
frame is observationally invisible at 1PN. The vectorsector match of §IV D already gives α1 = 0 and α2 = ζ1
from g0i alone; what remains is to show that the one
place the preferred frame could enter the local dynamics—
the temporal-completion invariant ∆ = (c/a0 ) |uµ ∇µ ψ −
uµ ∇µ ψ0 | of Eq. (Q4), built from the preferred-frame flow
uµ —cannot inject an unsuppressed preferred-frame term
into the 1PN metric. Axiom WS.1 confines the temporal
completion to homogeneous deviations, which removes
this term from local systems altogether; the argument
below bounds it without using that scope.
Theorem IV.1 (Preferred-frame invisibility at 1PN).
In the nondispersive (solar-system) regime, DFD’s full
scalar sector—the spatial AQUAL term plus the temporal
completion Eq. (11), whose temporal invariant ∆ is built
from the preferred-frame flow uµ —produces a 1PN metric
with
α1 = α2 = α3 = 0

(83)

up to the boost-induced residual of Eq. (86), bounded by

40
Eq. (85). The preferred foliation is observationally invisible at first post-Newtonian order.
Proof. The static (γ = β = 1, ξ = 0) sector is established
in §IV C; the gravitomagnetic match Eq. (81) fixes α1 = 0
and α2 = ζ1 (§IV D); covariant conservation gives ζ1 = 0
(§IV F), hence α2 = 0. It remains to show that the
preferred-frame flow uµ , which enters only through ∆,
contributes no unsuppressed preferred-frame PPN term
to g00 or g0i , and that α3 = 0. We give two independent
suppression arguments for the temporal completion; either
one bounds the residual.
In the temporal completion, uµ enters through the
contracted scalar ∆ = (c/a0 )|uµ ∇µ ψ − uµ ∇µ ψ0 |, and
in a boosted frame this term does carry a shift current,
the O(w) current of Eq. (86) below, together with a
boost-induced g00 contribution at O(w2 ); both carry the
suppressed coefficient αeff of Eq. (85), bounded by arguments (ii)–(iii) below to lie far under every experimental
threshold.
(i) Scalar, not vector, coupling (heuristic placement in the LLN classification). The Lee–Lightman–
Ni theorem [5, 6] states that a Lagrangian-based metric theory whose static 1PN metric coincides with GR
(γ = β = 1, ξ = 0) has α1 = α2 = α3 = 0 only when
it contains no prior-geometric absolute element and no
dynamically propagating preferred-frame vector/tensor.
This is a heuristic guide, not an automatic proof for DFD:
DFD carries a prior-geometric absolute vector uµ (the
non-varied CMB-frame foliation), which places it in LLN’s
preferred-frame-capable class (Rosen/Ni/aether-type theories), and the AQUAL nonlinearity µ(∆) lies outside
LLN’s smoothness hypotheses. Hence the αi must be
controlled directly, as done in (ii)–(iii). What the classification does supply is the reason DFD lands harmlessly:
because uµ is non-dynamical and enters only through the
fully-contracted scalar ∆, it carries no independent radiative or constraint degree of freedom—unlike a propagating
aether vector, which would source g0i directly. The residual preferred-frame coupling is therefore confined to the
temporal-completion channel and bounded by (ii)–(iii).
(ii) Wrong post-Newtonian order. The preferredPF
frame metric terms have the canonical form g0i
∼
PF
α1 (wi /c)(U/c2 ) and g00
∼ (α1 − α2 )(w2 /c2 )(U/c2 ),
where wi is the system velocity relative to the preferred
frame. The temporal term enters the ψ field equation as
−c−2 ∂t [ µ(∆) K ′ · · · ]. When the sources are at rest in the
preferred frame, ψ̇ = 0 identically and the temporal term
vanishes; it switches on only at O(w), through ψ̇ ∼ wi ∂i ψ.
Its contribution to g0i , relative to the GR gravitomagnetic
term (4G/c3 )ρv, is suppressed by the dimensionless ratio
rtemp =

a2∗ ∆
4 a0 w
= 3
,
|∇ψ|2
c |∇ψ|

(84)

using a∗ = 2a0 /c2 . Dividing out the explicit boost w/c
gives the induced effective coefficient
4a0
2a0
αeff ≲ 2
=
,
(85)
c |∇ψ|
|aN |

since |aN | = 12 c2 |∇ψ| is the Newtonian acceleration. This
is the same MOND ratio that governs every DFD departure from Newtonian gravity: in any high-acceleration
system (|aN | ≫ a0 ) it is driven to zero.
a. Explicit variation. The result above is confirmed
by varying the temporal Lagrangian directly with respect
to the shift Ni that carries g0i = e+ψ Ni . Writing Ltemp =
K(∆) with ∆ = (c/a0 )|ψ̇ − ψ̇0 | and ψ̇ = uµ ∇µ ψ →
N i ∂i ψ + O(w) in the boosted frame, a single chain-rule
variation gives


δLtemp
c
= µ′ (∆)
wj ∂j ψ ∂ i ψ + O(w2 ),
(86)
δNi
a0
with an explicit O(1) coefficient (µ(∆) = K ′ (∆)). The
induced shift current is therefore µ′ (∆)(c/a0 ) w |∇ψ|suppressed, and dividing out the boost w/c reproduces
exactly the prefactor of Eq. (85), αeff ≤ 2a0 /|aN |, now carrying the additional saturation factor µ′ (∆) of argument
(iii).
(iii) Saturation freezes the response. A genuine
preferred-frame force requires the constitutive function
to respond to the boost, i.e. a factor µ′ (∆). The regime
selector here is the acceleration ratio aN /a0 , not ∆ itself:
in every PPN-testing system aN ≫ a0 , which drives ∆ =
(aN /a0 )(2w/c) large and freezes the response. At 1 AU,
aN /a0 ∼ 5 × 107 , so with 2w/c ∼ 2.5 × 10−3 one has
∆ ∼ 1.2×105 , hence µ(∆) → 1 and µ′ (∆) = 1/(1+∆)2 ∼
6.6 × 10−11 ≪ 1. The temporal current is then a spatially
uniform cosmological background (µ = 1), whose gradient
and curl—the only quantities that can source a metric
perturbation—vanish. Hence the temporal-completion
coefficient is suppressed below Eq. (85) by the additional
factor µ′ (∆) = 1/(1 + ∆)2 ≪ 1 in the solar system.
Arguments (ii)–(iii) independently bound the temporalcompletion residual of Eq. (86) to lie far below experimental thresholds. The semiconservative parameter α3
requires both a preferred frame and momentum nonconservation; the latter is excluded by covariant conservation
(§IV F, ζi = 0), and the former is decoupled by (ii)–(iii),
so α3 = 0.
Corollary IV.2 (Conservative bound, assumption-light).
Even if the saturation argument (iii) is set aside and only
the post-Newtonian power-counting (ii) is retained, the
worst-case effective preferred-frame parameter obeys
2a0
αeff (system) ≲
.
(87)
|aN |
Evaluated where the published bounds are actually set, this
gives αeff ≲ 4.0 × 10−8 in the solar system (|aN | ∼ 6 ×
10−3 m s−2 ) and αeff ∼ 3×10−12 in binary pulsars (|aN | ∼
80 m s−2 , the orbital relative acceleration of systems such
as PSR J1738+0333 and the Hulse–Taylor binary), to be
set against |α1 | < 4×10−5 and |α2 | < 2×10−9 [5, 37]. In
this maximally conservative reading the solar-system value
sits ≈ 1000 below the α1 bound; the α2 bound is set by
millisecond-pulsar spin precession, at pulsar accelerations,
where the binary-pulsar value sits ≈ 700 below it.

41
β −·10
1 −4
4

Cassini + LLR

2
GR, DFD

BD (ω → ∞)

γ−1
−0.5

0.5
−2

·10−4

−4

FIG. 5. PPN parameter space in the (γ − 1, β − 1) plane. The
shaded ellipse sketches the Cassini and LLR bounds, centred
on GR. DFD (red point) sits exactly at the GR location (0, 0).

F.

Conservation Laws: ζ1 = ζ2 = ζ3 = ζ4 = 0

In any metric theory with minimal matter coupling to
a single metric, covariant conservation of the total stressenergy tensor follows from the matter action’s invariance
under coordinate changes:1
˜
∇µ T µν = 0.
(88)
DFD in its nondispersive band is precisely such a theory: the dynamics is entirely encoded in the physical
metric (65) with standard minimal coupling to matter
(Sec. II B). Consequently, the PPN parameters that would
signal violations of momentum or energy conservation
must vanish:
ζ1 = ζ2 = ζ3 = ζ4 = 0 .

Table VIII presents the complete PPN benchmark comparing DFD, GR, and experimental constraints.
Key Result: PPN Equivalence
DFD reproduces GR at 1PN order. All ten PPN
parameters match GR predictions (the preferred-frame
set to within the bounded residual of Thm. IV.1):
ξ = α1 = α2 = α3 = ζ1 = ζ2 = ζ3 = ζ4 = 0.

Classic Solar System Tests

With γ = β = 1, DFD makes identical predictions
to GR for all classic tests of gravity. We verify each
explicitly.

1.

Summary: DFD Equals GR at 1PN

γ = β = 1,

H.

(89)

Combined with Eqs. (68), (70), and (82), this completes
the ten-parameter PPN map for DFD.

G.

The PPN parameter space can be visualized by considering the (γ − 1, β − 1) plane (Fig. 5). DFD sits exactly at
the GR point (0, 0), well within the experimental ellipse
defined by Cassini and Lunar Laser Ranging constraints.
a. Second order. Outside a static source, with the
scalar-longitudinal piece σ of the parent strain kept
(§V A 3), the exterior is the Schwarzschild metric in DFD’s
coordinates, lapse e−ψ/2 and det gij = e3ψ (App. WS):
with U = GM/(c2 r), ψ = 2U − 43 U 3 + . . . and gij =
eψ [δij + U 2 (δij − 3ni nj ) + . . . ], so −g00 = 1 − 2U +
2U 2 + O(U 4 ) and the spatial metric is anisotropic at
second order. The second-order observables are general
relativity’s: the deflection coefficient 15π/4, no Shapiro2
delay excess, the periastron ratio Ωϕ /Ωr = 1 + 3x + 27
2 x
3 2/3
with x = (GM Ω
 ϕ /c ) , and
 the circular-orbit energy
2
x
E/µc2 −1 = − x2 1− 34 x− 27
(computed). With σ set to
8
zero the exterior would be the isotropic exponential metric
studied by Boonserm, Ngampitipan, Simpson and Visser
(2018), with ϵ = 4/3 in gij = (1 + 2γU + 32 ϵU 2 )δij and the
deflection coefficient 4π; under the Einstein–Hilbert term
that metric needs a negative-energy scalar (App. AA). No
experiment has yet measured a second-order coefficient;
sub-microarcsecond astrometry at the solar limb would
test it.

Light rays follow null geodesics of the optical metric.
For a spherically symmetric source with n(r) = eψ(r) and
ψ(r) = 2GM/(c2 r), the conserved impact parameter is
b = n(r) · r sin θ. The total deflection angle for a ray with
closest approach r0 ≫ rg = 2GM/c2 is [5]:
(1 + γ) 4GM
4GM
· 2 = 2 ,
(91)
2
c b
c b
where the second equality uses γ = 1.
a. Numerical verification. At the Sun’s limb (b =
R⊙ = 6.96 × 108 m, M = M⊙ = 1.99 × 1030 kg):
δθ =

δθ =
(90)

This ensures compatibility with all Solar System tests at
their current precision.

Light Deflection

4 × 6.67 × 10−11 × 1.99 × 1030
(3 × 108 )2 × 6.96 × 108

= 8.5 × 10

−6

(92)

′′

rad = 1.75 .

This matches the GR prediction precisely, consistent with
VLBI observations at the 10−4 level [38].

2.

Shapiro Time Delay

1 DFD’s preferred foliation does not spoil this: the conservation law

˜ µ T µν = 0 depends only on the matter sector’s coupling to g̃µν ,
∇
not on whether the gravitational sector is generally covariant.

The coordinate time for a photon traveling from point
r1 to r2 near a mass M is increased by the gravitational

42
TABLE VIII. Complete 1PN PPN benchmark for DFD: exact equality with GR across all ten parameters.
Parameter GR DFD Experimental Bound Consistent?
γ
β
ξ
α1
α2
α3
ζ1
ζ2
ζ3
ζ4

1
1
0
0
0
0
0
0
0
0

1
1
0
0
0
0
0
0
0
0

1 ± 2.3 × 10−5
1 ± 3 × 10−4
< 10−3
< 4 × 10−5
< 2 × 10−9
< 4 × 10−20
< 2 × 10−2
< 4 × 10−5
< 10−8
—

time delay [39]:

4.



(1 + γ)GM
(r1 + r1 · n̂)(r2 − r2 · n̂)
ln
, (93)
c3
d2
where d is the impact parameter and n̂ is the unit vector
along the unperturbed ray. With γ = 1, this becomes:


4r1 r2
2GM
ln
.
(94)
∆t =
c3
d2
a. Cassini constraint. The Cassini spacecraft measured the Shapiro delay during solar conjunction with
unprecedented precision, yielding [35]:
γ − 1 = (2.1 ± 2.3) × 10−5 .

(95)

DFD’s prediction γ = 1 lies comfortably within this
bound, representing a consistency test at the 10−5 level.

3.

Gravitational Redshift



∆t =

Perihelion Precession

The PPN prediction for orbital perihelion advance per
revolution is [5]:
2 + 2γ − β
6πGM
·
.
(96)
∆ω = 2
c a(1 − e2 )
3
With γ = β = 1, the prefactor becomes (2 + 2 − 1)/3 = 1:
6πGM
∆ω = 2
.
(97)
c a(1 − e2 )
a. Mercury. For Mercury (a = 5.79 × 1010 m, e =
0.2056):
∆ω =

✓
✓
✓
✓
✓
✓
✓
✓
✓
✓

6π × 6.67 × 10−11 × 1.99 × 1030
(3 × 108 )2 × 5.79 × 1010 × (1 − 0.20562 )

(98)

= 5.02 × 10−7 rad/orbit.
Over 100 years (415 orbits), this accumulates to
42.98′′ /century, matching the observed anomalous precession after accounting for planetary perturbations [5].

The gravitational redshift of a photon climbing from
potential Φ1 to Φ2 is:


∆ν
Φ1 − Φ 2
GM 1
1
=
= 2
−
.
(99)
ν
c2
c
r2
r1
In DFD, this follows directly from ν ∝ e−ψ/2 ≈ 1 + Φ/c2
(Sec. II B).
a. Experimental verification.
• Pound-Rebka (1960): Measured redshift over
22.5 m in Earth’s gravitational field, confirming
Eq. (99) at ∼ 10% precision.
• Gravity Probe A (1976): Hydrogen maser comparison over 10,000 km altitude yielded agreement
at 7 × 10−5 [40].
• ACES (on the ISS since April 2025): The
Atomic Clock Ensemble in Space aims for 2 × 10−6
precision.
DFD predicts the standard gravitational redshift, consistent with all observations.
5.

Frame Dragging and Lense-Thirring Effect

The gravitomagnetic field generated by a rotating mass
with angular momentum J causes precession of test gyroscope spin and orbital plane precession of satellites. The
Lense-Thirring precession rate is [41]:
2GJ
Ω̇LT = 2 3
.
(100)
c a (1 − e2 )3/2
DFD reproduces this effect exactly because the gravitomagnetic sector g0i (81) has the correct GR form. Experimental confirmations include:
• LAGEOS satellites: Measured Ω̇LT due to
Earth’s rotation at ∼ 10% precision [42].
• Gravity Probe B (2011): Directly measured
frame-dragging of orbiting gyroscopes, confirming
GR at 19% precision [43].

43
I.

Where DFD Differs from GR

The exact PPN match means that Solar System tests
cannot distinguish DFD from GR. This is a structural
consequence: DFD’s µ-function reduces to µ → 1 in the
high-acceleration Solar System regime, and the weak-field
expansion of n = eψ automatically produces the correct
PPN parameters from DFD’s own field equations—GR is
not assumed at any step.
The discriminating tests for DFD lie in two regimes:

A.

Two Gravitational Sectors on Flat R3

Before presenting the technical details, we establish the
conceptual framework for gravitational radiation in DFD.
This framework preserves DFD’s core identity—flat Euclidean space R3 with absolute time t—while accounting
for the observed tensor polarization structure of gravitational waves.
1.

1. Galactic scales (Sec. VII): Where |a|/a0 ∼ 1, the
µ-crossover produces MOND-like phenomenology
absent in GR.
2. Laboratory clock tests (Sec. XI): The DFD
clock sector is channel-resolved: the simplified
α
KA ≈ kα SA
scaling captures only the pure-α leading term, while the full structure includes strongsector and composition-dependent contributions
of the extended non-metric channel (null under
the minimal action, Thm. XI.2), testable with colocated atomic and nuclear clocks. Matter-wave
interferometry is a null check rather than a discriminator: DFD and GR coincide through 1PN
(§XIII).
Summary: Solar System Compliance
DFD passes all Solar System tests of gravity:
• Light deflection: δθ = 4GM/(c2 b) (matches GR)
• Shapiro delay: Cassini bound satisfied
(|γ − 1| < 2.3 × 10−5 )
• Perihelion precession: ∆ω = 6πGM/(c2 a(1 − e2 ))
(Mercury: 42.98”/cy)
• Gravitational redshift: Standard formula confirmed
to 10−4
• Frame dragging: Lense-Thirring precession
matches LAGEOS/GP-B
The theory’s distinguishing predictions emerge in galactic
dynamics and laboratory clock tests.

V.

GRAVITATIONAL WAVES

Gravitational wave astronomy provides stringent tests
of gravity in the strong-field, dynamical regime. The
direct detection of binary black hole and neutron star
mergers by LIGO, Virgo, and KAGRA has opened a
new window for testing alternative theories. This section
demonstrates that DFD reproduces GR’s gravitational
wave predictions at leading order, satisfying all current
observational constraints while providing a framework for
quantifying potential deviations through the parameterized post-Einsteinian (ppE) formalism.

The Optical Sector (DFD Core)

DFD posits a scalar field ψ(x, t) on flat R3 with absolute
time t. The optical sector defines a refractive index n = eψ
and an effective optical interval:
c2 2
dt + dx2 .
(101)
n2
We introduce ds̃2 as a compact encoding of how ψ rescales
light propagation; it is not a dynamical spacetime geometry and no curvature field equations are assumed for
it. The fundamental arena remains (R3 , t); the physical
metric (12), to which matter couples, obeys the Einstein–
Hilbert dynamics of App. WS.
Local observers in regions with √
different ψ compare
clock rates by dτ = e−ψ/2 dt = dt/ n, the lapse of the
physical metric (12) to which matter couples (Thm. XI.2),
while light follows the null cone of (101), coordinate speed
c/n. In DFD, n = eψ sets both; it does not introduce
an asymptotic subluminal EM signal speed relative to
the shared far-zone cone. Observable light-bending and
gravitational time delay are encoded via an effective traveltime functional (Fermat principle) built from the null cone
of (101); this is used as a bookkeeping device for clockrate comparisons and Fermat/eikonal propagation, not as
a dynamical metric with curvature equations.
ds̃2 = −

2.

The Radiative Sector (Tidal Disturbances)

Compact-binary mergers exhibit gravitational radiation
with two tensor polarizations. A scalar field ψ alone
cannot reproduce this polarization structure. DFD’s spectral completion on CP 2 × S 3 derives both sectors from a
single parent object:
• Optical gravity (ψ): scalar field governing clock
rates, refractive bending, and quasi-static matter
dynamics
• Radiative gravity (hTT
ij ): transverse-traceless tensor field describing propagating tidal disturbances
The TT field is defined on R3 by the standard conditions:
∂i hTT
ij = 0,

δ ij hTT
ij = 0,

(102)

and obeys a wave equation on the flat background:


1 2
16πG TT
2
∂ − ∇ hTT
Πij ,
(103)
ij =
c2 t
c4

44
where ΠTT
ij is the TT projection of the source stress.
This is not an appeal to curved spacetime: both ψ and
3
hTT
ij are fields on the same flat (R , t) arena, derived as irreducible components of the same zero-mode parent tensor
on K = CP 2 × S 3 , whose vector and scalar-longitudinal
pieces are constrained, non-radiative fields of the same
arena (§V A 3).
The radiative sector does not alter the optical-sector
derivations of lensing, clocks, or MOND phenomenology.
Within the full CP 2 × S 3 spectral completion of DFD,
the TT sector is derived as the spin-2 irreducible component of the same zero-mode parent tensor whose trace
yields ψ. Both sectors emerge from a single parent metric
perturbation hµν on the internal manifold K = CP 2 × S 3 ,
expanded in harmonics and restricted to the zero mode
(m20 = 0). The 3 + 1 decomposition of hµν under O(3)
gives the trace ψ (1 DOF) and the TT tensor hTT
ij (2
DOF) as its irreducible components, with constrained vector and scalar-longitudinal pieces (no radiative DOF). A
Lichnerowicz analysis on K proves no unwanted massless
tensor or vector modes arise from internal deformations
(see §V A 4 below). The absence of derivative mixing between trace and TT sectors is a structural consequence
of O(3) rotational symmetry on flat R3 , not an ad hoc
postulate.

3.

Parent Strain Field and Irreducible Decomposition

Define a symmetric strain field on flat R3 :


1
1
TT
2
Ψij = ψ δij + hij + ∂(i Vj) + ∂i ∂j − δij ∇ σ,
3
3
(104)
where ψ = δ ij Ψij is the trace (scalar), hTT
is
the
ij
transverse-traceless piece (tensor), Vi is a transverse vector, and σ is a scalar-longitudinal auxiliary. The trace
ψ governs optical and quasi-static gravity and the TT
piece hTT
ij governs gravitational radiation; the vector and
scalar-longitudinal pieces are constrained non-radiative
auxiliaries, whose dynamics is the part of the Einstein–
Hilbert term (App. WS) that the trace and TT actions
do not contain, which fixes Saux in Eq. (106). The vector piece is removed by the spatial diffeomorphisms that
preserve the arena’s volume form; σ is not set to zero
but solved from the traceless part of the Einstein equation. The physical metric has lapse e−ψ/2 and spatial
part gij = eψ (eh )ij , with hij ≡ Ψij − 13 ψ δij the traceless
part. The scalar-longitudinal piece σ does not enter at
first post-Newtonian order; beyond it, keeping it makes
DFD’s rest-mass gravity general relativity’s in DFD’s
slicing (Proposition WS.6).
a. No-mixing theorem. For any isotropic quadratic
principal symbol built from Ψij , the trace and TT pieces
are orthogonal. Any isotropic cross-term between trace
and TT reduces to one of the forbidden contractions:
δij hTT
ij = 0,

∂i hTT
ij = 0.

(105)

TT
So terms like ∂k ψ ∂k hTT
ii and ∂i ψ ∂j hij vanish identically.
The principal symbol is therefore automatically blockdiagonal between trace and TT sectors:

Sgrav =

c4
64πG

Z

"
dt d3 x

#
2
(∂t hTT
ij )
TT 2
−
(∇h
)
+ Str [ψ] + Saux .
ij
c2

(106)
By irreducible decomposition of an isotropic parent
strain field, the principal symbol is automatically blockdiagonal between trace and TT sectors—the absence of
derivative mixing is a structural consequence, not a separate assumption. The two helicity-zero pieces, ψ and σ,
are not separated by this argument: the Einstein–Hilbert
term couples them, and Saux carries that coupling.
4.

Spectral-Geometry Origin of the Two-Sector Structure

The two-sector structure (ψ + hTT
ij ) is a consistent
completion, and within the CP 2 × S 3 spectral action
framework, it is derived [1]. The spectral action SB =
Tr f (D2 /Λ2 ) on R3,1 × K produces a 4D Einstein–Hilbert
action from the a2 Seeley–DeWitt coefficient. A metric
perturbation Hµν (x, Y ) on the total space, expanded in
scalar harmonics on K, has a massless zero mode hµν (x)
(constant on K). Its 3 + 1 decomposition yields ψ (trace)
and hTT
ij (spin-2) as siblings in the same multiplet.
A Lichnerowicz analysis on K verifies the mode count
is clean:
• CP 2 is Einstein-rigid:
no TT zero modes
(Koiso 1980; spectral gap λmin = 8/R12 ).
• S 3 is Einstein-rigid: no TT zero modes (spectral
gap λmin = 12/R22 ; Higuchi 1987).
• b1 (CP 2 ) = b1 (S 3 ) = 0: no harmonic 1-forms on
either factor, eliminating mixed zero modes.
• On a continuous K one scalar zero mode survives,
the squashing modulus controlling R1 /R2 ; on the
truncated microsector there is none.
a. The Einstein product condition. The internal geometry enters the spectral action through the volume and
the curvature of K. The α constraint does not fix the
2
shape ratio τ ≡ R2 /R1 : the CP
does
R factor of α’s 2trace
not depend on its radius, since CP 2 F ∧F = 8π q 2 is topological (§X C), and in a joint reading 1/G (from a2 ) and
1/α (from a4 ) both scale with Vol(K). TheR shape enters
through the internal-curvature part of a2 , K RK dvol =
π 8/7 Vol(K)5/7 Φ(τ ) with Φ(τ ) = 24τ 6/7 + 6τ −8/7 , which
belongs to the four-dimensional√vacuum term. At fixed
volume it is stationary at τ∗ = 1/ 3, the Einstein product
condition 6/R12 = 2/R22 , with Φ′′ /Φ = 2.94, a curvature
ratio rather than a mass; this is a minimum of the potential for the connection Laplacian that fixes α (§X B)
and a maximum for an internal Dirac operator, and the
overall volume is not fixed at this order. On the truncated microsector, which the closed form for α and the

45
α57 derivation use, the internal space is finite and carries
no size or shape modulus.
b. Constitutive interpretation. With the traceless
pieces included, the generalized optical metric becomes
ds̃2 = −c2 dt2 /n2 + (eh )ij dxi dxj , with hij the traceless
part of Ψij (δij + hTT
ij at linear order in the radiative
sector). The Tamm–Plebanski construction gives tensor
ij
ij,TT
constitutive relations εij
). The vaceff = ε0 n (δ − h
uum medium has compression stiffness K0 = c4 /(8πG)
and shear stiffness K0 /4: gravity as electromagnetic vacuum loading [31].

5.

Why cT = c (Structural Requirement)

Radiative Sector: O(3) Irrep Block-Diagonality
The TT principal part is the flat wave operator, with no
(∂ψ)(∂hTT ) mixing. This is not a free choice: it follows
from the irreducible decomposition of the parent strain
field Ψij under the isotropic O(3) symmetry of flat R3 .
(Any derivative mixing would require breaking the isotropy
of the principal symbol, which is excluded by the flat-space
construction.)

The action for the radiative sector takes the form:
"
#
Z
TT 2
(∂
h
)
c4
t
ij
2
dt d3 x
− (∇hTT
+ Sint ,
STT =
ij )
64πG
c2
(107)
where Sint [ψ, hTT , ρ] contains no terms that modify the
principal part of STT : gravitational waves and light
share one metric (App. WS). (This normalization yields
TT 4
TT
eff TT
□hTT
.)
ij = 16πG Πij /c , with Πij ≡ (Tij )
Under this condition, the characteristic cone of hTT
ij is
the optical cone, flat on the background:
cT = cγ

(one metric for light and gravitational waves).
(108)
Since both EM and GW share the same far-zone causal
cone (cT = cγ ) and we impose no derivative mixing that
would alter the tensor principal part, any additional ψdependent timing effects enter identically (or negligibly)
in the eikonal limit for both channels. The observed
≲seconds coincidence over ∼ 40 Mpc (GW170817) therefore constrains only differential coupling, which this completion sets to zero.
Any alternative completion that introduces (∂ψ)(∂hTT )
mixing or additional radiative degrees of freedom generically predicts cT ̸= cγ and is immediately constrained by
multimessenger observations.
Remark V.1 (Scope of the no-mixing inference). The irrep
argument forbids the bilinear mixing (∂ψ)(∂hTT ); it does
not forbid ψ-dependent coefficients of the TT bilinear
2 2
TT 2
itself, A(ψ)(∂t hTT
ij ) /c − B(ψ)(∇hij ) , which are O(3)
scalars. What fixes them is the requirement that the TT
sector obey the same local relativity as matter at constant
ψ0 : the map t = e+ψ0 /2 T , x = e−ψ0 /2 X of Thm. XI.2
pulls the physical metric (12) back to Minkowski, and the

TT operator is the flat wave operator in (T, X) if and
only if A/B = e2ψ . The coordinate tensor cone is then
c e−ψ , the optical cone of Eq. (101), and Eq. (106) is its
ψ0 = 0 reference-gauge form. The common normalisation
F (ψ) (A = F e2ψ , B = F ; F = 1 under the constantG assumption) is open. The same coefficient carries a
2
rotationally invariant cubic vertex ψ (∂t hTT
ij ) that the
bilinear no-mixing statement does not exclude (App. AT,
Rem. AT.10).
a. Corollary (standard-siren null). On the opticalcone reading of the TT sector (App. WS) photons and
TT waves share the eikonal equation g̃ µν kµ kν = 0 of the
physical metric
(12), hence the same coordinate travel
R
time (1/c) eψ ds along the same Fermat path: the same
Shapiro delay, the same lensing rays and ∆tGW-EM = 0;
on a homogeneous background both amplitudes obey the
same eikonal transport on the same cone, the same 1/(a r)
GW
EM
decay with a the background scale factor, so DL
= DL
— a standard-siren null, with any difference confined to
the open normalisation F (ψ) of Rem. V.1.

6.

Adiabatic Limit and GW Speed in the Unified Picture

The parent strain field Ψij of Eq. (104) naturally accommodates the trace and TT sectors as complementary irreducible pieces. If µ-type nonlinearity from the trace sector
couples to the tensor sector, the far-zone propagation of
hTT
ij remains effectively luminal in the WKB/adiabatic
regime, because µ varies only on a macroscopic scale Lµ
set by the background (e.g. galactic/cluster potentials),
while gravitational waves have wavenumber k satisfying
kLµ ≫ 1. A natural estimate for any correction to the
tensor characteristic cone is
|∇ ln µ|
1
λ
ϵ ∼
∼
=
.
(109)
k
kLµ
2πLµ
For LIGO/Virgo-band waves (f ∼ 102 –103 Hz, so λ =
c/f ∼ 3 × 105 –3 × 106 m) and a conservative astrophysical
variation scale Lµ ≳ 1–10 kpc (≈ 3 × 1019 –3 × 1020 m),
one finds
3 × 106
ϵ ≲
∼ 10−14 –10−15 , (110)
2π (3 × 1019 –3 × 1020 )
to be compared with the GW170817 bound |cT /c − 1| ≲
10−15 [17].
This adiabatic estimate applies to any completion in
which slowly varying µ-dependent coefficients enter outside the principal part. DFD’s completion is the blockdiagonal completion of Eq. (106), for which cT = c exactly:
gravitational waves and light share one metric (App. WS),
so cT = cγ on every background.
7.

Falsifiability

If observations ever require:

46
• a TT–EM cone split, cT ̸= cγ on any background,
or

C.

• Scalar or vector polarization modes in far-zone GWs,
or

The previous subsection established the DFD-native
3
framework: hTT
ij is a field on flat (R , t) with no derivative mixing with ψ in its principal part. Here we verify
this structure and connect to standard scalar-tensor formalisms for readers familiar with that literature.

• a confirmed gravitational-wave echo train (absent:
the remnant has a Kerr horizon, argued beyond first
order in the spin; App. WS),

Verification: cT = c from No Derivative Mixing

then DFD’s tensor sector is falsified.

B.

1.

The Minimal Transverse-Traceless Sector

Having established the conceptual framework, we now
present the technical details. DFD’s gravitational wave
sector is constructed to respect GW170817’s tight constraint on the GW propagation speed: |cT /c − 1| < 10−15
[17].
a. TT action. The radiative sector is a massless
transverse-traceless tensor field propagating at speed c
on the background. Its quadratic action, the leading part
of the full a2 Einstein–Hilbert term (whose cubic vertex
carries the nonlinear memory, App. AT), is:


Z
1
c4
2
TT 2
dt d3 x 2 (∂t hTT
)
−
(∇h
)
Sh =
. (111)
ij
ij
64πG
c
This is identical to the linearized GR action for tensor perturbations on flat spacetime. The TT constraint
eliminates the trace (hi i = 0) and longitudinal modes
(∂i hij = 0), leaving exactly two polarization degrees of
freedom:
+
×
hTT
ij = h+ eij + h× eij ,

(112)

where e+,×
are the plus and cross polarization tensors for
ij
propagation along the z-axis:




1 0 0
0 1 0
0 −1 0 ,
1 0 0 .
e+
e×
(113)
ij =
ij =
0 0 0
0 0 0
b. Key properties.
tion guarantees:

The minimal TT sector construc-

1. cT = cγ on every background: light and gravitational waves share one metric (App. WS), satisfying
GW170817.
2. Only tensor (+, ×) polarizations—no scalar or vector modes in the far zone.
3. Standard GR amplitude scaling with distance: h ∝
1/r.
The only departure from GR enters through the µcrossover of the conservative source dynamics, where accelerations approach a0 ; the propagation and emission of
gravitational waves are GR’s.

The Flat-Background Wave Equation

In DFD, the TT field satisfies the flat-space wave equation (Eq. 103):


1 2
16πG TT
2
∂t − ∇ hTT
Πij .
(114)
ij =
2
c
c4
i(ωt−k·x)
For a plane wave hTT
, the dispersion relation
ij ∝ e
is:

ω 2 = c2 k 2

⇒

cT = c.

(115)

This result is structural : it follows from the O(3) irrep
block-diagonality of the parent strain field (Sec. V A 3),
which forbids terms like (∂ψ)(∂hTT ) in the kinetic sector. Scope: Rem. V.1. Any such mixing would require
breaking the isotropy of the principal symbol.

2.

Why No Derivative Mixing is Natural in DFD

In DFD’s flat-arena formulation:
1. Tensor-scalar decoupling: The TT perturbation
hTT
ij is traceless and transverse, coupling only to
the traceless part of the source. The scalar ψ governs time dilation and scalar gravitational effects,
ensuring the two sectors do not mix at leading order.
2. No scalar higher-derivative terms beyond
the Weyl term: Unlike general Horndeski theories, DFD contains no terms involving (□ψ)2 or
curvature-scalar couplings; its only four-derivative
term beyond the Gauss–Bonnet total derivative,
the a4 Weyl-squared term of the spectral action
(App. WS), has its spin-2 pole at 3.6 M̄P for f0 = 1,
above the cutoff, and moves no observable; on the
scalar sector, evaluated on the matter metric, it
adds only a spatial four-derivative term in δψ, the
helicity-0 part of that pole, smaller than the Planckstrength stiffness by 43 (k/3.6 M̄P )2 at wavenumber
k. The absence of any other scalar higher-derivative
term is equivalent to:
d ln c2T
=0
d ln a
(identically: one metric for light
αT ≡

and gravitational waves; App. WS).

(116)

47
D.

F.

Wave Equation and Source Coupling

The TT field couples to matter through the effective
stress tensor derived from the optical metric:
Z
1
ij
Sint =
dt d3 x hTT
(117)
ij Teff [ψ; ρ, v].
2
Variation of Sh + Sint with respect to hTT
ij yields the wave
equation:
16πG eff TT
1 2 TT
2 TT
(Tij ) , (118)
□hTT
ij ≡ 2 ∂t hij − ∇ hij =
c
c4
where the superscript TT denotes projection onto the
transverse-traceless part.
a. Effective stress tensor. The source (Tijeff )TT depends on the matter distribution and its motion in the
ψ-mediated potential. At leading (Newtonian) order:
Tijeff = ρvi vj + (pressure and binding energy corrections).

(119)
The ψ-dependence enters through the conservative dynamics: orbital parameters are determined by the effective
potential Φ = −c2 ψ/2.

E.

a. Far-zone solution. The standard retarded solution
to Eq. (118) in the far zone (r ≫ λGW ) is:
2G ¨TT
hTT
(120)
ij (t, x) = 4 Iij (tret ),
c r
where tret = t − r/c is the retarded time and Iij is the
mass quadrupole moment tensor:


Z
1
Iij = ρ(x, t) xi xj − δij r2 d3 x.
(121)
3
b. Energy flux. The gravitational wave luminosity
follows from the standard Isaacson stress-energy tensor
averaged over several wavelengths:
G D ... ...ij E
dE
= − 5 I ij I
[1 + δrad ],
(122)
dt
5c
where the angle brackets denote time averaging and δrad
parametrizes any small DFD-specific departure from the
GR prediction. The factor [1 + δrad ] captures potential
radiative inefficiencies in the DFD framework.
c. DFD prediction. In the high-acceleration regime
relevant to compact binary inspirals, µ → 1 and the conservative dynamics reduce to Newtonian gravity. Since
the TT sector is derived from the same CP 2 × S 3 spectral geometry as the scalar sector (§V A 4), reproduces
linearized GR as output, and the scalar sector is an instantaneous constraint that does not radiate (Axiom WS.1),
we have:
(leading order).

The parameterized post-Einsteinian (ppE) framework
provides a systematic way to constrain deviations from
GR using gravitational wave observations [44]. DFD maps
naturally onto this framework through its conservative
and dissipative departure parameters.

1.

Conservative and Dissipative Parametrization

Following [44], parametrize departures from GR in the
binary orbital dynamics:


E(v) = EGR (v) 1 + ε0 + ε2 v 2 + · · · ,
(124)


F(v) = FGR (v) 1 + φ3 v 3 + · · · ,

(125)

where v = (πM f )1/3 is the characteristic orbital velocity,
M = m1 + m2 is the total mass, and f is the gravitational
wave frequency. Here E(v) is the binding energy and F(v)
is the gravitational wave flux.
a. Physical interpretation.
• ε0 : Leading (0PN) conservative correction to orbital
energy.

Quadrupole Formula and Energy Flux

δrad = 0

Post-Newtonian and ppE Framework

(123)

Corrections to δrad enter only through µ-function effects
in systems where |∇ψ|/a⋆ is not asymptotically large:
where µ → 1 the source dynamics is GR’s (App. WS).

• ε2 : 1PN conservative correction.
• φ3 : 1.5PN dissipative correction to energy flux.
2.

Phase Coefficients

The inspiral waveform phase accumulation, computed
via stationary phase approximation, takes the form:
Ψ(f ) = ΨGR (f ) + β−5 u−5 + β−3 u−3 + β−2 u−2 + · · · ,
(126)
where u = (πMf )1/3 with chirp mass M =
(m1 m2 )3/5 /(m1 + m2 )1/5 , and η = m1 m2 /M 2 is the symmetric mass ratio.
The explicit dictionary relating (ε0 , ε2 , φ3 ) to the ppE
phase coefficients is:
5
β−5 = −
ε0 ,
(127)
128η
β−3 =

3
C1 (η)ε2 ,
128η

(128)

β−2 =

3
D3 (η)φ3 ,
128η

(129)

where C1 (η) = 743/336 + 11η/4 and D3 (η) = −16π are
standard GR coefficients.
a. DFD mapping. Equations (127)–(129) enable direct translation between DFD theory parameters and
LVK catalog bounds without requiring bespoke waveform
models.

48
G.

Comparison with LIGO-Virgo-KAGRA
Observations

1.5

GM
2
∼ 1010 –1011 m/s ,
r2

(130)

while the µ-crossover scale is a0 ∼ 10−10 m/s2 . The ratio:
a0
∼ 10−20 –10−21 .
(131)
abinary
In this regime, a/a0 ≫ 1, so µ(x) → 1 and DFD reduces
to standard Newtonian/GR dynamics. Therefore:
ε0 = ε2 = 0

(at leading PN order).

(132)

b. Radiative sector. The quadrupole flux formula (122) with δrad = 0 matches GR exactly, implying:
φ3 = 0

(at leading order).

(133)

c. GW propagation speed. Light and gravitational
waves share one metric (App. WS), so cT = cγ exactly,
satisfying the GW170817 bound.

2.

GR/DFD prediction
GWTC-3 90% CI

0.5

DFD does not predict specific non-zero values for
(ε0 , ε2 , φ3 ) in the compact binary regime. Rather, in systems where the µ-crossover is negligible, the leading-order
dynamics reduce exactly to GR.
a. Conservative sector. For stellar-mass black hole
binaries at LIGO frequencies, the characteristic acceleration is:
abinary ∼

ppE Constraints on GW Phase Deviations (GWTC-3)
DFD: k = 0 for all k
All GWTC-3 bounds consistent with GR/DFD

1.0

DFD Predictions for Compact Binaries
k

1.

2.0

Comparison with LVK O3 Bounds

The GWTC-3 tests of GR [45] provide the most stringent constraints on ppE deformation parameters. Table IX compares DFD expectations with LVK bounds.
a. Notes on the table.
• The δ φ̂k are fractional deviations in PN phase coefficients; GR predicts 0 for all.
• LVK bounds are from combined GWTC-3 analysis
using hierarchical inference.
• The graviton mass bound assumes a dispersive propagation correction.
• The GW speed bound from GW170817/GRB
170817A is the most stringent constraint on cT .
Key Result: GW Consistency
Through second post-Newtonian order DFD’s
two-body dynamics is general relativity’s, and
gravitational waves travel at c (App. WS). The phase
terms, the graviton-mass bound and the speed bound are
consistent with current observations.

0.0
0.5
1.0
1.5
2.0

1

0

1

2

Post-Newtonian order

3

4

FIG. 6. Parameterized post-Einsteinian (ppE) constraints
from GWTC-3 [45]. Points with error bars: schematic intervals
(the GWTC-3 bounds are in Table IX) on fractional phase
deviations δ φ̂k at each post-Newtonian order. Red line: GR
prediction, which DFD shares (Table IX).

3.

Falsifiability and Future Tests

The ppE mapping serves a forward-looking purpose:
it enables future observations to be translated directly
into DFD parameter constraints if deviations from GR
are ever detected. Falsifiability requires either:
1. Detection of ppE deviations: Any non-zero
β−5,−3,−2 would constrain DFD parameters via
Eqs. (127)–(129).
2. µ-crossover regime observations: If GW
sources exist in the low-acceleration regime where
|∇ψ|/a⋆ ∼ 1, DFD would predict detectable deviations. Such sources (e.g., extremely wide binaries
or primordial backgrounds) are not currently accessible.
H.

Binary Pulsar Verification

Binary pulsars provide precision tests of gravitational
radiation in the weak-field but highly relativistic regime.
The Hulse-Taylor binary (PSR B1913+16) remains the
canonical verification of the quadrupole formula.
1.

The Hulse-Taylor System

The observed parameters [46, 47] are:
The observed orbital decay, after correcting for the
Shklovskii effect and Galactic acceleration, is:
Ṗbint = (−2.398 ± 0.005) × 10−12 s/s.
2.

(134)

DFD Prediction

a. Why δrad = 0 for compact binaries. The µcrossover is completely negligible for the Hulse-Taylor

49
TABLE IX. Comparison of DFD predictions with LVK O3 ppE bounds. DFD’s deviations are zero and consistent: through
second post-Newtonian order its two-body dynamics is GR’s (App. WS).
Parameter PN Order DFD Prediction LVK O3 Bound (90% CL) Consistent?
δ φ̂−2
δ φ̂0
δ φ̂1
δ φ̂2
δ φ̂3
δ φ̂4
mg
|cT /c − 1|

Parameter

−1PN
0PN
0.5PN
1PN
1.5PN
2PN
—
—

[−7.3, 7.3] × 10−4
[−0.15, 0.15]
[−0.5, 0.5]
[−0.3, 0.3]
[−0.2, 0.2]
[−0.5, 0.5]
≤ 1.27 × 10−23 eV/c2
< 10−15

0
0
0
0
0
0
0
0

one of the most precise tests of the quadrupole formula.

Symbol Value

Pulsar mass
m1
Companion mass m2
Total mass
M
Orbital period
Pb
Eccentricity
e
Semi-major axis
a
Periastron distance rp

✓
✓
✓
✓
✓
✓
✓
✓

1.4398 ± 0.0002 M⊙
1.3886 ± 0.0002 M⊙
2.8284 ± 0.0003 M⊙
27906.98 s
0.6171340
1.95 × 109 m
7.5 × 108 m

4.

Other Binary Pulsars

Multiple binary pulsar systems confirm the same result:
TABLE XI. Binary pulsar orbital decay tests.
Ṗbobs /ṖbGR

System

system:
GM
(6.67 × 10−11 )(5.6 × 1030 )
2
∼
∼ 670 m/s .
2
rp
(7.5 × 108 )2
(135)
The ratio a0 /abinary ∼ 10−13 , so crossover corrections are
suppressed by a0 /abinary ∼ 10−13 or more.
b. Explicit prediction. The orbital period decay from
quadrupole radiation is:

5/3
73 2
4
e + 37
1 + 24
192π 2πGM
96 e
Ṗb = −
[1 + δrad ],
3
2
7/2
5
c Pb
(1 − e )
(136)
where M = (m1 m2 )3/5 /M 1/5 is the chirp mass.
With δrad = 0:

Consistent with DFD?

PSR B1913+16
0.9983 ± 0.0021
PSR J0737-3039A 0.999963 ± 0.000063

✓
✓

abinary ∼

ṖbDFD = ṖbGR = (−2.402531 ± 0.000014) × 10−12 s/s.
(137)

All binary pulsar systems show orbital decays consistent
with the GR quadrupole formula, which is identical to
the DFD prediction in the high-acceleration regime.

5.

Bounds on DFD Parameters

The Hulse–Taylor data constrain the radiative inefficiency parameter:
δrad =

Ṗbobs − ṖbGR
= −0.0017 ± 0.0021.
ṖbGR

At 95% confidence:
|δrad | < 0.006.

3.

Quantitative Comparison

TABLE X. Hulse-Taylor binary orbital decay comparison.
Quantity

Value

ṖbGR (quadrupole formula) (−2.402531 ± 0.000014) × 10−12 s/s
Ṗbint (observed, corrected) (−2.398 ± 0.005) × 10−12 s/s
ṖbDFD (predicted)
(−2.402531 ± 0.000014) × 10−12 s/s
Ratio Ṗbobs /ṖbGR
0.9983 ± 0.0021
Ratio Ṗbobs /ṖbDFD
0.9983 ± 0.0021

Agreement: The observed orbital decay agrees with
the GR/DFD prediction at the 0.2% level, representing

(138)

(139)

DFD predicts δrad = 0 exactly in this regime, fully consistent with observations.

I.

Numerical Evolution for Compact Binaries

For systems whose accelerations reach the µ-crossover
we outline the DFD-consistent numerical evolution
scheme; where µ → 1 the waveforms are those of general
relativity in DFD’s slicing (App. WS).

50
1.

J.

Evolution System

With ψ the elliptic constraint of Axiom WS.1, the first
two equations below are its numerical relaxation, not a
propagating field; the system evolves as:
∂t ψ = Π,

(140)



 
|∇ψ|
∇ψ
∂t Π = c ∇ · µ
a⋆
2

− Γψ Π + Sψ (ρ, v),

(141)

16πG eff TT
(Tij ) ,
(142)
c2
with matter following the conservative potential Φ =
−c2 ψ/2:
2 2 TT
∂t2 hTT
ij − c ∇ hij =

Summary and Implications

Summary: Gravitational Wave Tests
DFD passes all gravitational wave tests:
• Two-sector origin: ψ and hTT
ij derived as trace
and TT components of the same zero-mode parent
tensor on CP 2 × S 3 (§V A 4)
• GW speed: cT = cγ on every background: light
and gravitational waves share one metric
(App. WS)—structural, not fine-tuned
• Polarizations: Two tensor modes only (+, ×);
Lichnerowicz rigidity excludes extra modes
• ppE bounds: All phase deviations consistent with
zero

(143)

• Binary pulsars: Orbital decay matches GR at
0.2%

where aRR enforces energy balance via the quadrupole
formula.

• Radiative efficiency: |δrad | < 0.006 (95% CL)

v̇A = −∇Φ(xA ) + aRR [hTT ],

2.

Boundary Conditions

For total mass M , stationary tails obey the Gauss-law
Robin condition:


|∂r ψ|
2GM
2
Rout
µ
(144)
∂r ψ = − 2 .
a⋆
c
Use sponge/characteristic outflow for hTT . Time-stepping
via RK4 with CFL from max(c, vphase,ψ ); Kreiss-Oliger
damping Γψ stabilizes high-k modes.
3.

AMR Strategy

Refine where the µ-crossover is active: |∇ψ| ∈ [0.3, 3] ×
a⋆ . For stellar-mass binaries, this shell lies far from the
strong-field region; for galactic-scale problems, it requires
targeted resolution. Two FAS V-cycles per macro timestep suffice for weak-to-moderate fields.

4.

Validation Tests

1. Single static mass: Stationary ψ with correct 1/r
tail from Robin BC.
2. Circular inspiral: Leading phase agrees with GR
0PN/1PN; deviations quantified by (ε0 , ε2 , φ3 ).
3. Grid convergence:
Order ≈ 4; energy balance
Rt
|Eorb (t) + 0 F dt′ | small and decreasing with refinement.

a. Physical interpretation. DFD passes the binary
pulsar test, as expected. The theory was constructed to
reproduce GR in strong-field situations. The physical
reason is that the µ-crossover scale a0 ∼ cH0 ∼ 10−10
m/s2 is 13 (binary pulsars) to 21 (LIGO binaries) orders
of magnitude below typical accelerations in neutron star
and black hole binaries.
b. Distinguishing tests. The linear GW waveform
does not distinguish DFD from GR — both make identical inspiral–merger–ringdown predictions in the currentdetector regime. The distinguishing tests for DFD are:
1. Clock tests (Sec. XI): Channel-resolved crossspecies and nuclear-clock comparisons probe the full
coupling structure of Eq. (291); cavity–atom comparisons (Sec. XII) test only the screened residual
after geometric cancellation. Matter-wave interferometry is a null check, since DFD and GR coincide
through 1PN (§XIII).
2. Galactic dynamics (Sec. VII): The µ-crossover
produces MOND-like behavior where a ∼ a0 .
3. Clock anomalies: species-dependent residuals of
the extended non-metric channel (Sec. XI); the
common-sector coupling at the 10−5 level cancels
in co-located ratios (Thm. XI.2).
Gravitational-wave memory is not among these tests.
The transverse-traceless action is the full nonlinear a2
Einstein–Hilbert term (§V A 4), so DFD’s tensor memory, linear and nonlinear (Christodoulou), equals GR’s
(App. AT, Prop. AT.9). LISA massive–black-hole mergers
therefore test memory for consistency: a measurement
consistent with zero where GR requires a permanent strain
offset (∼20–27% of the edge-on peak for an equal-mass
merger) would falsify DFD.
The GW verification demonstrates that DFD is not
falsified by strong-field dynamics; neither the linear wave-

51
form nor the nonlinear memory can confirm DFD over
GR.

VI.

STRONG FIELDS AND COMPACT
OBJECTS

Sections IV and V demonstrated that DFD reproduces
GR in the weak-field Solar System and gravitational-wave
regimes. We now examine compact objects where gravitational effects are strong. The key results are: (1) DFD’s
optical metric defines the correct variational condition for
photon spheres and optical horizons; (2) with the scalarlongitudinal piece σ of the parent strain kept (Sec. V A 3),
the exterior geometry is Schwarzschild’s in DFD’s coordinates (App. WS), so a black hole has Schwarzschild’s event
horizon, at the areal radius 2GM/c2 , and Schwarzschild’s
shadow; and (3) current Event Horizon Telescope observations of M87* and Sgr A* are consistent with DFD at
present precision.

A.

Static Spherical Solutions

Consider a static, spherically symmetric mass distribution with density ρ(r) = 0 for r > R⋆ (the stellar radius
or horizon scale). The DFD field equation (21) reduces
to:

 ′  
1 d
|ψ |
8πG
2
(145)
r
µ
ψ ′ = − 2 ρ(r).
r2 dr
a⋆
c
a. Exterior vacuum solution. For r > R⋆ with ρ = 0,
Eq. (145) integrates to:
 ′ 
|ψ |
2GM
r2 µ
(146)
ψ ′ = − 2 = const.
a⋆
c
In the strong-field regime around compact objects,
|ψ ′ |/a⋆ ≫ 1 so µ → 1, yielding the Newtonian/GR result:
ψ(r) =

2GM
+ ψ∞ ,
c2 r

with ψ∞ = 0 (asymptotic flatness).

(147)
This corresponds to the effective potential Φ = −c2 ψ/2 =
−GM/r. Equation (147) is the exterior at the orders
where the spatial metric is isotropic. With the scalarlongitudinal piece σ of the parent strain kept (Sec. V A 3)
the full exterior of a static star is the Schwarzschild metric
in DFD’s coordinates, lapse e−ψ/2 
and det gij = e3ψ
(App. WS): ψ = − ln 1 − 2GM/c2 R at areal radius R,
which in the radius r of Eq. (147) reads ψ = 2GM/(c2 r)−
4
2 3
3 (GM/c r) + · · · , the same function through second
order.
b. Existence and uniqueness. The operator in
Eq. (145) is elliptic when µ′ > 0 and W is convex (conditions (A1)–(A4) from Sec. III A). Standard PDE methods
establish:
1. Existence: Weak solutions exist for any bounded
source ρ with suitable decay.

2. Uniqueness: Strict monotonicity of µ guarantees
uniqueness.
3. Regularity: Solutions are C 1,α away from sources;
smooth where ψ ′ ̸= 0 if µ ∈ C ∞ .
4. Maximum principle: ψ achieves extrema only at
boundaries or source locations.

B.

Optical Causal Structure

DFD’s optical metric (Sec. II A) defines the causal
structure for light propagation:
ds̃2 = −

c2 dt2
+ dx2 ,
n2 (x)

n(x) = eψ(x) .

(148)

Light travels at the coordinate phase velocity cphase = c/n,
which varies with position. The spatial part dx2 is exact
through first post-Newtonian order; with the traceless
part hij of the parent strain kept it is (eh )ij dxi dxj , of
unit determinant (Sec. V A 3; App. WS).
a. Optical horizons. An optical horizon is a surface
where n → ∞ (equivalently ψ → +∞), causing cphase →
0. At such a surface, light cannot propagate outward—it
becomes “trapped” in the refractive medium.
With the parent strain’s σ kept a black hole has general
relativity’s event horizon, at the areal radius 2GM/c2
(App. WS); the µ-function is unity there to one part in
1013 or better (1022 for a stellar-mass hole; Sec. VI D 1).
How the horizon appears in n depends on the slicing, as
follows.
b. Comparison with GR. For the Schwarzschild geometry, the event horizon at rg = 2GM/c2 corresponds
to g00 → 0 and grr → ∞. In DFD’s optical metric (148)
the analogous surface is an optical horizon, where n
 → ∞.
On the static exterior, ψ = − ln 1 − 2GM/c2 R , that
happens at R = 2GM/c2 , which DFD’s coordinates place
at the arena’s origin; the Newtonian-regime solution (147)
is its large-radius expansion. A hole formed by collapse
is described in DFD’s absolute time by the maximal
slicing (App. WS), whose late-time slices cross the horizon and stop at a limiting surface inside it: the event
horizon, at R = 2GM/c2 , is a flow horizon, where the
inflow of the slices equals the local speed of light; there
n = eψ = 256/27, and n diverges only on the limiting
surface, R = 3GM/(2c2 ), which sits at the arena’s origin.

C.

Photon Spheres

The photon sphere is the surface of unstable circular
photon orbits—rays that neither escape to infinity nor
fall into the horizon. Its location determines the black
hole shadow boundary.
a. Derivation from Fermat’s principle. Null
geodesics of the optical metric (148) satisfy Fermat’s
principle. For a static spherically symmetric exterior the

52
conserved impact parameter is
b=e

ψ/2

2.

R sin θ,

(149)

with R the areal radius; for a flat spatial part R = eψ/2 r
and b = n(r) r sin θ. Circular orbits occur where b is
stationary with respect to R:
2
d  ψ/2 
dψ
e R
=−
= 0 ⇐⇒
.
dR
dR Rph
Rph
R=Rph
(150)
The condition (150) determines the photon sphere radius
Rph .
b. Critical impact parameter. Photons with impact
parameter b > bcrit escape to infinity; those with b < bcrit
fall inward. The critical value is:
bcrit = eψ(Rph )/2 Rph .

√ GM
GM
bcrit = 3 3 2 ≈ 5.20 2 ,
c
c

Black Hole Shadows: EHT Comparison

The Event Horizon Telescope has imaged the shadows
of two supermassive black holes: M87* and Sgr A*. These
observations provide direct tests of strong-field gravity.

1.

DFD in the Strong-Field Regime

For black hole environments, the characteristic acceleration vastly exceeds a0 :
aBH ∼

GM
c4
2
=
∼ 1012 m/s
rg2
4GM

(stellar mass BH),

(154)
giving a/a0 ∼ 10 . In this regime, µ(x) → 1 and DFD
reduces exactly to GR.
a. Key result. DFD’s shadow is Schwarzschild’s
(Eq. (153)), so the EHT comparison below is GR’s. The
correction from µ-function effects at the photon sphere
scale is of order a0 /aph ∼ 10−13 for M87* and 10−16 for
Sgr A* (10−22 for a stellar-mass hole) and completely
negligible.
22

System parameters [48].

Parameter

Symbol Value

Mass
M
Distance
D
Angular grav. radius θg
b.

(6.5 ± 0.7) × 109 M⊙
16.8 ± 0.8 Mpc
3.8 ± 0.4 µas

Predictions.

√
GR
θsh
= 3 3 θg = (19.7 ± 2.1) µas,
diameter 39.5 µas;

DFD
GR
θsh
= θsh

(Eq. (153)).

(155)
(156)

c. EHT observation. The observed ring diameter is
(42 ± 3) µas. After calibrating the relationship between
the photon ring and the shadow boundary it agrees with
the common prediction (Table XII). Verdict: DFD is
consistent with M87* observations, with the same margin
as GR.

3.

Sgr A* Shadow

DFD
GR
θsh
= θsh
,

(153)
Schwarzschild’s photon sphere and shadow. The critical
impact parameter is a property of the geometry, the same
in every slicing, so a hole formed by collapse casts the
same shadow. The shadow therefore tests DFD exactly
as it tests GR and does not distinguish them.

D.

a.

(151)

c. Shadow angular radius. For an observer at distance Do ≫ Rph , the angular radius of the black hole
shadow is:
bcrit
eψ(Rph )/2 Rph
=
.
(152)
θsh =
Do
Do
d. DFD strong-field prediction.
On the static exte

rior, ψ = − ln 1 − 2GM/c2 R (Sec. VI A), the condition (150) gives
3GM
Rph =
,
c2

M87* Shadow

a.

System parameters [49].

Parameter

Symbol Value

Mass
M
Distance
D
Angular grav. radius θg
b.

(4.0 ± 0.2) × 106 M⊙
8.1 ± 0.1 kpc
5.0 ± 0.3 µas

Predictions.

√
GR
θsh
= 3 3 θg = (26.0 ± 1.6) µas,

DFD
GR
θsh
= θsh

(Eq. (153)).

(157)
(158)

c. EHT observation. The observed ring diameter is
(51.8 ± 2.3) µas, consistent with the common prediction
(Table XII). Verdict: DFD is consistent with Sgr A*
observations, with the same margin as GR.

4.

Summary Comparison

Key Result: EHT Consistency
DFD’s shadow is Schwarzschild’s (Eq. (153)): with
the parent strain’s σ kept the static exterior is the
Schwarzschild metric in DFD’s coordinates (App. WS).
Current EHT observations of M87* and Sgr A* are
consistent with it, and the shadow does not distinguish
DFD from GR.

53
TABLE XII. Black hole shadow comparison: DFD predictions vs. EHT observations.
Object Property

GR

DFD

EHT Observation

Consistent?

M87* 2θsh
39 ± 4 µas 39 ± 4 µas
42 ± 3 µas
M87* dsh /dGR
1.00
1.00
1.00 ± 0.17
sh
Sgr A* θsh
26 ± 2 µas 26 ± 2 µas
25.9 ± 1.2 µas
+0.09
(Keck),
0.92 ± 0.09 (VLTI)
Sgr A* dsh /dGR
1.00
1.00
0.96
−0.10
sh

E.

Constrained µ-Function Family

While DFD predicts µ → 1 in the strong-field limit, a
parametric family of crossover functions enables systematic exploration of potential deviations.

1.

The Constrained Family µα,λ (x)

We impose physical constraints on any admissible µ:
1. Solar limit: µ(x) → 1 as x → ∞ (recover Newtonian dynamics)
2. Deep-field branch: µ(x) ∼ x as x → 0 (flat
rotation curves)

a. DFD-TOV equations. The structure of a spherically symmetric, static star in hydrostatic equilibrium
is governed by the Tolman-Oppenheimer-Volkoff (TOV)
equations. In DFD, with the parent strain’s σ kept, a
static star obeys the TOV system (App. WS):
 a i
dP
G(ρ + P/c2 )(m + 4πr3 P/c2 ) h
0
1
+
O
,
=−
dr
r2 (1 − 2Gm/(c2 r))
a
(162)
Rr
where r is the areal radius, m(r) = 4π 0 ρ(r′ )r′2 dr′ is
the enclosed mass and P (r), ρ(r) are the pressure and
density profiles.
b. Strong-field limit. Inside neutron stars, the characteristic acceleration is:
GMNS
aNS ∼
2
RNS

3. Monotonicity: µ′ (x) > 0 for ellipticity

∼

4. Convex W : Energy positivity and stability

µα,λ (x) ≈ x

x ≫ λ−1/α :

µα,λ (x) ≈ λ−1/α

(deep-field)
(saturation)

(160)
(161)

The minimal case α = 1, λ = 1 gives the simple µ(x) =
x/(1 + x).
b. Physical interpretation. The parameter α controls
the sharpness of the crossover transition, while λ sets its
location relative to a⋆ . At a black-hole photon sphere
a/a0 ≳ 1013 (1013 for M87*, 1016 for Sgr A*, 1022 for a
stellar-mass hole; Sec. VI D 1), where the solar limit puts
every member with λ = 1 at unity, so shadow radii do
not constrain the family.

F.

(163)

2

Asymptotic behavior.

x ≪ λ−1/α :

(1.4 × 2 × 1030 kg) · 6.67 × 10−11
(104 m)2

∼ 1012 m/s .

A two-parameter family satisfying the last three, and
the solar limit for λ = 1, is:
x
µα,λ (x) =
, α ≥ 1, λ > 0.
(159)
(1 + λxα )1/α
a.

✓
✓
✓
✓

Compact Star Structure

Neutron stars provide additional tests of strong-field
gravity through their mass-radius relation and maximum
mass.

With a0 ∼ 10−10 m/s2 , the correction factor in Eq. (162)
is O(a0 /aNS ) ∼ O(10−22 )—utterly negligible.
c. Implications. The a0 /a correction to the TOV
pressure gradient is negligible inside neutron stars, and
the compact-star sector is general relativity’s: static
stars obey the TOV equations (App. WS). For the reference Γ = 2, K = 100 polytrope the maximum mass is
Mmax = 1.637 M⊙ , at central density 1.97 × 1015 g cm−3
and areal radius 11.3 km (App. AT, Theorem AT.13); its
dependence on the equation of state and the causal ceiling
are those of GR. Massive pulsars (e.g., PSR J0740+6620
at 2.08 ± 0.07 M⊙ ) are consistent with DFD as they are
with GR.

G.

Potential DFD-Specific Signatures

DFD’s strong-field observables are GR’s. Departures
could come only from the µ-function, and the ringdown
tests the tensor sector:
a. Strong-field µ-closure. If the µ-function deviates
from unity at extremely high gradients, shadow sizes
would shift. EHT data constrain:
∆θsh
< 0.17 (from M87*).
(164)
θsh
This bounds any strong-field modifications at the O(10%)
level.

54
b. Quasi-normal modes. The ringdown phase of binary black hole mergers probes the near-light-ring effective
potential of the tensor sector. GW250114 constrains the
ˆ
fundamental-mode frequency shift to δf
220 = 0.02 ± 0.02
(90% C.L.; GWTC-4.0 hierarchical 0.00 ± 0.06). With
the parent strain’s σ kept the remnant is a Kerr hole
in DFD’s coordinates and its perturbations obey the
Einstein–Hilbert equations (App. WS; argued beyond
first order in the spin), so DFD’s ringdown is Kerr’s,
δf220 = 0, consistent with both.
Summary: Strong-Field Behavior
DFD strong-field scorecard:
• Photon sphere / shadow: Schwarzschild’s
(Eq. 153), with Schwarzschild’s event horizon at
the areal radius 2GM/c2 (App. WS)
• Black hole shadows: EHT observations consistent
(M87*, Sgr A*)
• Neutron stars: static stars obey the TOV
equations; the reference polytrope has
Mmax = 1.637 M⊙ (App. AT, Thm. AT.13)
• Constraints: photon-sphere/shadow and NS sectors
consistent; tensor ringdown Kerr’s, consistent with
GW250114 (+2 ± 2%, 90% C.L.; App. WS; argued
beyond first order in the spin)

Key Result: the µ-law from the S 3 microsector
The interpolation function
√ µ(x) = x/(1 + x) and the
acceleration scale a0 = 2 α cH0 are not
phenomenological inputs—they are determined by the
S 3 Chern-Simons microsector (Appendix N): the
acceleration scale, the two asymptotic limits of µ (µ → 1
Newtonian, µ ∼ x deep-MOND) and the exponential
form of the response (Lemma N.6; Appendix BS) are
fixed, and µ(x) = x/(1 + x) is its weak-response form
(Assumption N.3), tracking the derived response in its
RAR form (Eq. (BS8)) to ≲ 0.02 dex, below SPARC
precision. The same topology that gives α = 1/137 also
produces flat rotation curves.

This section demonstrates that DFD, with no free
theory parameter in the galactic sector—the acceleration scale a0 is derived from fundamental constants (Appendix AP), not fit to data—explains: (1) flat galaxy
rotation curves, (2) the baryonic Tully-Fisher relation,
and (3) the tight empirical correlation between observed
and baryonic accelerations. As in any baryonic rotationcurve analysis, observational nuisance inputs such as distance, inclination, and stellar mass-to-light assumptions
enter through the data reduction rather than through
new theory parameters.

The µ → 1 limit at high accelerations ensures GR
recovery. Distinguishing tests require laboratory LPI
measurements or galactic-scale dynamics.

VII.

GALACTIC DYNAMICS

The previous sections established that DFD reproduces
GR in high-acceleration environments: the Solar System (Sec. IV), gravitational waves (Sec. V), and compact
objects (Sec. VI). We now turn to the regime where
DFD predicts new physics—galactic scales where the µcrossover produces MOND-like phenomenology without
requiring dark matter particles. DFD’s derived cold dark
matter χ (App. AV) is also present in galaxies. Its pull
composes with the µ-law by the adopted floor law and a
stated filling rule (App. AV, App. GR); the fits in this
section are those of the µ-law on the baryons, which DFD
retains if χ inside galaxies stays thinner than the baryons’
phantom.

A.

The Deep-Field Limit

The µ-function interpolates between Newtonian gravity
(µ → 1 for |∇ψ|/a⋆ ≫ 1) and a modified regime at low
accelerations. In the deep-field limit where |∇ψ|/a⋆ ≪ 1:
|∇ψ|
≪ 1.
(165)
a⋆
a. Implications for the field equation. In the deepfield regime, the DFD field equation (21) becomes:


|∇ψ|
8πG
∇·
∇ψ = − 2 ρ.
(166)
a⋆
c
For spherical symmetry with enclosed mass M :
µ(x) → x

for x =

|ψ ′ |2
8πGM
· 4πr2 =
,
a⋆
c2

(167)

yielding:
r
′

|ψ | =
b.

2GM a⋆
.
c2 r 2

(168)

Logarithmic potential. Integrating Eq. (168):
√
 
2GM a⋆
r
ψ(r) = −
ln
+ const,
(169)
c
r0
where r0 is an integration constant. The effective Newtonian potential Φ = −c2 ψ/2 is:
 
r
cp
2GM a⋆ ln
.
(170)
Φ(r) =
2
r0
This logarithmic potential produces flat rotation curves—
the hallmark of MOND phenomenology.

55
160

Rotation velocity (km/s)

140

NGC 2403

SPARC Data with DFD Fit

C.

The Tully-Fisher relation is a tight empirical correlation between galaxy luminosity (or baryonic mass) and
rotation velocity. In the deep-field limit, DFD predicts
this relation exactly.
a. Derivation. From Eq. (174), the asymptotic flat
rotation velocity satisfies:

120
100
80
60
40

Baryonic ( = 0.81)
Gas
Disk
DFD prediction
Observed (SPARC)

20
0
0.0

The Baryonic Tully-Fisher Relation

a0 = 1.2e 10 m/s2
Data: Lelli+ 2016

2.5

5.0

7.5

10.0

12.5

Radius (kpc)

15.0

17.5

20.0

FIG. 7. NGC 2403 rotation curve from SPARC data [50].
Black points: observed rotation velocity with error bars. Blue
dashed: baryonic contribution (stellar disk + gas) with fitted
mass-to-light ratio Υ⋆ = 0.81. The red curve uses the empirical
round benchmark a0 = 1.2 × 10−10 m/s2 ; the bound-matter
DFD normalization (Postulate ZS.11) is 1.1966 × 10−10 m/s2 .

GM a⋆ c2
.
2
Solving for the baryonic mass:
vf4 =

vf4
2vf4
=
,
(176)
Ga⋆ c2
Ga0
where we define the MOND acceleration scale:
a⋆ c2
2
a0 ≡
= 1.1966 × 10−10 m/s (bound-matter clock).
2
(177)
b. The BTFR. Equation (176) is the baryonic TullyFisher relation (BTFR):
Mbar =

Mbar ∝ vf4
B.

Galaxy Rotation Curves

The circular velocity of a test mass orbiting at radius
r is determined by centripetal balance:
c2
vc2
= |∇Φ| = |ψ ′ |.
(171)
r
2
a. High-acceleration (Newtonian) regime. Where
|∇ψ|/a⋆ ≫ 1, we have µ → 1, ψ ′ = 2GM/(c2 r2 ), and:
r
GM
GM
2
vc =
⇒ vc =
∝ r−1/2 (Keplerian).
r
r
(172)
b. Low-acceleration (deep-field) regime. Using
Eq. (168):
r
r
c2 r 2GM a⋆
c2 r ′
GM a⋆ c2
2
vc =
|ψ | =
=
. (173)
2
2
2
2
c r
2
Thus:

1/4
GM a⋆ c2
vc =
= const (flat rotation curve).
2
(174)
c. Physical interpretation. In the deep-field regime,
the circular velocity becomes independent of radius—
rotation curves flatten. This occurs without dark-matter
halos; it is a direct consequence of the µ-crossover. The
asymptotic velocity depends only on the enclosed baryonic
mass M and the characteristic scale a⋆ .
d. Transition region. Real galaxies transition
smoothly from Newtonian inner regions to deep-field
outer regions. The full rotation curve is obtained by
solving the µ-modified field equation (21) with the actual
baryonic mass distribution (stellar disk + gas).

(175)

(178)

with normalization fixed by a0 . This is a prediction:
a⋆ = 2a0 /c2 is set by the bound-matter acceleration scale
(Postulate ZS.11), not fit to the data.
c. Observational verification. The SPARC database
[10, 16] confirms Eq. (178) with high precision. For
SPARC disk galaxies spanning five decades in mass:
• The observed BTFR has slope 3.98±0.10, consistent
with 4.
• The scatter about the relation is only 0.1 dex, much
smaller than expected from measurement errors plus
astrophysical variance.
• The empirical normalization is centered near a0 ≈
1.2 × 10−10 m/s2 ; the bound-matter DFD value is
1.1966 × 10−10 m/s2 .
The tightness of the BTFR is difficult to explain in
ΛCDM, which predicts significant scatter from variations
in halo concentration, spin, and assembly history. In
DFD, the relation follows directly from the field equation
with no free parameters beyond a0 .
D.

The Radial Acceleration Relation

The radial acceleration relation (RAR) is a point-bypoint correlation between the observed centripetal acceleration gobs = vc2 /r and the Newtonian (baryonic)
acceleration gbar = GMbar (< r)/r2 at each radius in each
galaxy.
a. DFD prediction. The RAR follows directly from
the µ-function. From the field equation:
gbar
gobs =
.
(179)
µ(gobs /a0 )

56
Baryonic Tully-Fisher Relation

SPARC Data

30

SPARC galaxies
DFD: slope = 4.00
Data fit: slope = 3.97

Radial Acceleration Relation
8.0
8.5

1010

log10 (gobs) [m/s2]

Mbar (M )

1011

109
108
N = 153 galaxies
Obs. slope = 3.97
Scatter = 0.11 dex
DFD slope = 4.00

107
Statistics from Lelli+ 2016

102

9.0

20

9.5

15

10.0
10.5
11.0

a0

12

FIG. 8. Baryonic Tully-Fisher relation from SPARC data [16].
Blue points: 153 galaxies with carefully calibrated baryonic
masses. The red line shows the slope 4. The normalization
is a0 in Eq. (176). Blue dashed: observed best fit with slope
3.97 ± 0.10.

Inverting this relation:

gobs = gbar · ν

gbar
a0


,

(y ≫ 1),

ν(y) → y −1/2

11

10

log10 (gbar) [m/s2]

10
N = 2693 points
Scatter = 0.13 dex
Data: McGaugh+ 2016

9

8

5

FIG. 9. Radial acceleration relation from SPARC data [10].
Hexagonal bins show density of 2693 data points from 153
galaxies. The red curve uses the empirical round benchmark
a0 = 1.2×10−10 m/s2 ; the bound-matter DFD value is 1.1966×
10−10 m/s2 . Dashed black: Newtonian expectation (gobs =
gbar ). Dotted green: deep-field asymptote.

(180)

where ν(y) is the inverse interpolation function satisfying:
ν(y) → 1

25

11.5

Vflat (km/s)

SPARC Data

DFD prediction
Newtonian (1:1)
Deep-field: g gbar

Points per bin

1012

(y ≪ 1). (181)

b. DFD prediction from the weak-response approximant µ(x) = x/(1 + x). Algebraic inversion of gbar =
gobs µ(gobs /a0 ) with µ(x) = x/(1 + x) gives the quadratic:
p
2 + 4g
gbar + gbar
bar a0
.
(182)
gobs =
2
This is the DFD radial acceleration relation on the weakresponse approximant (App. N, Assumption N.3); the
derived law is the exponential response of App. BS
(Thm. BS.9, Eq. (BS8)), which Eq. (182) tracks to ≲ 0.02
dex. The scale is not fit here: on bound-matter clocks
a0 = 1.1966 × 10−10 m/s2 (Postulate ZS.11).
c. Relation to the McGaugh empirical form. The
commonly
used empirical fitting function gobs = gbar /(1−
√
e− gbar /a0 ) [10] closely approximates Eq. (182) and is
the derived exponential response itself in its RAR form
(App. BS, Thm. BS.9, Eq. (BS8)). The two forms agree to
better than 4.5% everywhere and are observationally indistinguishable at current SPARC precision. Throughout
this paper, Eq. (182) is used as the weak-response (galactic) approximant of that law (App. N, Assumption N.3).
d. Observational verification. McGaugh et al. (2016)
[10] demonstrated that all 2693 data points from 153
galaxies follow a single RAR with:
• Observed scatter of only 0.13 dex, most of it observational error.
• No dependence on galaxy type, size, surface brightness, or gas fraction.

• Empirical normalization near 1.2 × 10−10 m/s2 , consistent with the bound-matter DFD value 1.1966 ×
10−10 m/s2 .
Key Result: RAR Match
The RAR (182) has a fixed a0 : 1.1966 × 10−10 m/s2 on
bound-matter clocks (Postulate ZS.11). The SPARC
extraction (1.20 ± 0.02stat ± 0.24sys ) × 10−10 is consistent
with it within its systematic error. This is a theory
prediction with ordinary observational nuisance inputs,
not a fitted DFD acceleration scale.

E.

Predicted acceleration scale and empirical
validation

The galactic sector of DFD carries no free theory parameter. The characteristic acceleration a0 is not fit √
to rotation curves and then frozen; it is derived as a0 = 2 α cH0
at the action level (Appendix N). On bound-matter
p
√
clocks this is a0 = 2 α cH0loc = 2α29 c7 /(Gℏ) =
1.1966 × 10−10 m/s2 (Postulate ZS.11, Appendix AP,
Thm. AP.32 at C = 1), with no measured coefficient
and never fit to rotation curves. SPARC therefore remains an independent empirical validation. The only
inputs entering a rotation-curve comparison are the observational nuisance parameters shared by all baryonic
analyses (distance, inclination, gas normalization, stellar
mass-to-light).
a. Empirical-validation procedure.
1. Fit the RAR (182) to the SPARC database.

57
= (1.20 ± 0.02stat ± 0.24sys ) × 10−10
2. Extract: aemp
0
2
m/s .
3. Compare
to the bound-matter value a0 =
√
2 α cH0loc = 1.1966×10−10 m/s2 (Postulate ZS.11):
the residual is within the published systematic uncertainty, which dominates.
4. No tuning step: no galactic parameter is adjusted
for the predictions below.

b. Results: DFD has the lower χ2 in every galaxy
tested. Across all galaxies tested:
Comparison

DFD preferred Percentage

DFD vs Newton
DFD vs Standard MOND
Newton best overall
c.

16/16
16/16
0/16

100%
100%
0%

Key examples.

b. No free parameter. With a0 set by the boundmatter value (Postulate ZS.11), no galactic prediction
carries a fitted theory parameter:

• DDO154 (dwarf irregular): Newton predicts V =
14 km/s; DFD predicts V = 47 km/s; observed
V = 47 km/s. DFD matches exactly.

• Individual rotation curves: predicted from baryonic
mass distribution.

• IC2574 (gas-rich dwarf): Newton predicts V =
21 km/s; DFD predicts V = 65 km/s; observed
V = 66 km/s. DFD within 2%.

• Baryonic Tully-Fisher: slope = 4 and normalization
fixed.
• Dwarf galaxies, low surface brightness galaxies:
same a0 .
• Vertical disk dynamics: same a0 ; where χ fills the
µ-law’s demand, the vertical force is the Newtonian
force of the baryons and χ (App. AV).
c. The α-relation derivation. DFD derives a0 from
fundamental constants (Sec. VIII; Appendix AP):
√
2
a0 = 2 α cH0 = 1.1966 × 10−10 m/s ,
(183)
where α ≈ 1/137 and H0 = H0loc = 72.09
p km/s/Mpc.
The equivalent closed form is a0 = 2α29 c7 /(Gℏ), the
C = 1 value of the clock identity, with no measured coefficient (Postulate ZS.11), consistent with SPARC within
its systematic error. On the early clock the same law
gives 1.1208 × 10−10 .

F.

Quantitative SPARC Validation

To rigorously test whether the DFD interpolation function µ(x) = x/(1 + x) outperforms alternatives, we performed a systematic head-to-head comparison using published SPARC galaxy parameters [10, 50].
a. Methodology. For each galaxy, we:
1. Computed baryonic circular velocities from stellar
mass (exponential disk + bulge) and gas distributions.
2. Predicted rotation curves for 16 galaxies using four
interpolation functions: DFD
(µ = x/(1 + x)),
√
Standard MOND (µ = x/ 1 + x2 ), the RAR empirical√form, which is DFD’s SOURCE law (µ =
1 − e− gbar /a0 ), and Newton (µ = 1).
3. Calculated χ2 against observed flat rotation velocities for each model.

• NGC3198 (spiral): Newton predicts V = 48 km/s;
DFD predicts V = 124 km/s; observed V =
150 km/s. DFD captures the enhancement.
√
The McGaugh function (1 − e− gbar /a0 ) often achieves
marginally lower χ2 ; it is DFD’s law of record (App. BS).
The DFD quadratic (182) is the weak-response approximant of the McGaugh form, which is DFD’s derived
exponential response (App. BS), and differs from it by at
most 4.5% in the transition region. The fair test is DFD
(a derived prediction) versus Newton (no modification).
Newton has lower χ2 in 0 of the SPARC galaxies under
the fixed-Υ⋆ comparison.
Validation Result: SPARC Database
DFD beats Newton in 100% of SPARC galaxies
tested.
The weak-response approximant µ(x) = x/(1 + x)
successfully explains galaxy rotation curves without
dark-matter halos, outperforming both Newton and
Standard MOND.

G.

Model-Independent Interpolation-Function
Shape Test

A stronger and more discriminating SPARC result
comes from a dedicated model-independent scan of the
interpolation-family
x
µn (x) =
,
(184)
(1 + xn )1/n
performed across all 175 SPARC galaxies. This test asks
a narrower question than the usual MOND-vs-Newton
confrontation: which transition shape the rotation-curve
data prefer.
The data-optimal index is
nopt = 1.15 ± 0.12

(95% CI : [1.00, 1.50]),

(185)

so DFD’s weak-response approximant n = 1 lies inside
the confidence interval, while the Standard MOND form
n = 2 lies well outside the preferred region. In the

58
TABLE XIII. DFD galactic acceleration scale: derived value vs. independent empirical extraction.
Parameter

Value

Source
Status
p
a0 (α-derived, bound-matter clock) 1.1966 × 10−10 m/s2
2α29 c7 /Gℏ (Postulate ZS.11) Derived (C = 1)
a0 (empirical)
(1.20 ± 0.26) × 10−10 m/s2 SPARC RAR extraction
Validation

free-Υ⋆ scan, DFD’s n = 1 incurs only a small penalty
above the optimum, whereas Standard’s n = 2 is strongly
disfavored. When the comparison is repeated at fixed
Υ⋆ = 0.5—so that the interpolation-function shape is
tested with zero compensation freedom—the preference
for the DFD/Simple shape strengthens rather than weakens.
This result matters because it isolates a common loophole in rotation-curve fitting: a model with the wrong
transition shape can hide part of its deficiency by pushing
Υ⋆ to astrophysically implausible values. The dedicated
SPARC shape study shows that Standard MOND benefits far more from this compensation freedom than DFD
does. In equal-budget and fixed-Υ⋆ tests, DFD remains
preferred, and its best-fit universal Υ⋆ stays close to the
stellar-population-synthesis expectation, whereas Standard’s optimum is pushed noticeably high.
The rotation-curve evidence is therefore stronger than
a comparison with Newton alone: the data require a
MOND-like departure from Newtonian dynamics and
prefer a specific shape very close to DFD’s weak-response
approximant µ(x) = x/(1 + x).
SPARC Shape Result
Across the full 175-galaxy SPARC sample, the preferred
interpolation-family index is nopt = 1.15 ± 0.12 with
bootstrap 95% confidence interval [1.00, 1.50]. DFD’s
approximant n = 1 lies inside this interval; Standard
MOND’s n = 2 does not. In the free-Υ⋆ scan, Standard
incurs a 9.2× larger penalty than DFD; in the stricter
fixed-Υ⋆ = 0.5 comparison (zero free parameters) the
penalty ratio strengthens to 29×. The preference holds in
equal-budget and systematics-marginalized tests.

H.

Wide Binary Stars

Wide binary stars separated by > 1000 AU probe the
MOND regime locally, providing a test independent of
galaxy-scale assumptions. This is currently one of the
most active areas of observational testing.
a. DFD prediction. For a binary with total mass M
and separation s, the Newtonian acceleration is aN =
GM/s2 . The acceleration ratio is:
aN
GM
x=
= 2 .
(186)
a0
s a0
For solar-mass binaries, the Newtonian ratio yN = aN /a0
reaches unity at s ≃ 7040 AU on bound-matter clocks,
the clock of every wide-binary and external-field value
below. The DFD acceleration ratio x = a/a0 is obtained

from x2 /(1 + x) = yN . The velocity enhancement factor
is:
s
r
VDFD
1
1
=
= 1+ .
(187)
VNewton
µ(x)
x
b. Quantitative values (external field included). In
the wide-binary regime we use the DFD-acceleration convention x = a/a0 (consistent with the asymptotic-velocity
treatment above). For an isolated pair of total mass 1 M⊙
the internal acceleration reaches x = 0.99 at 104 AU
4
and
p 0.42 at 2 × 10 AU, and the isolated-pair formula
(1 + x)/x would give velocity boosts of 42% and 84%.
That formula is an upper envelope only: the Galactic
field at the Sun, gext = 1.79 a0 (§VII I), exceeds a0 , so a
Solar-neighbourhood binary never enters the deep-field
regime and its boost stays below that envelope. With the
external field included, the completed scale-additive law
(App. NL, §NL 5) gives a force boost γ ≡ gobs /gN of 1.37–
1.52 at 104 AU and 1.59–1.74 at 2 × 104 AU (equal-mass
to test-body; velocity +17% to +32%), rising toward 1.87
beyond 3 × 104 AU; the forward response alone would
saturate at ν(τext ) = 1.36, and the local vector-sum reading (§VII I, simple-µ approximant: orientation-averaged
γ = 1.41 at gext = 1.79 a0 ) is the reading excluded by the
Solar-System quadrupole. These values are not realised
at the Sun. The Galaxy’s Newtonian baryonic field there,
1.19–1.33 a0 , exceeds (ln 2)2 a0 ≃ 0.48 a0 , so its mode is
typically empty and, by the environmental-confinement
axiom (App. NL, Axiom NL.14), a local pair is exactly
Newtonian, γ = 1 at every separation. The Sun’s comets
require it: every reading that boosts the Sun’s own field is
excluded by the Oort spike (Status NL.12). An isolatedpair table (41.6% velocity boost at 104 AU, 84.0% at
2 × 104 AU) omits the external field and is not a DFD
prediction.
c. Comparison with Gaia DR3 analyses. Published
boost parameters are acceleration ratios and must be compared with γ, not with a velocity ratio. Chae (2023) [51]
reports γ ≃ 1.43 ± 0.06 at gN ≃ 10−10.15 m s−2 ; Banik
et al. (2024) [52] report no boost. DFD predicts γ = 1
exactly for local pairs, with the null analyses (Banik et
al. 2024, αgrav = −0.021+0.065
−0.045 , −0.3σ; Saad & Ting 2026,
0.0σ) and against Chae (2023) at +7.2σ, Chae et al. (2026)
at +4.3σ and Yoon et al. (2025) at +2.9σ.
d. Controversy and status. Banik et al. (2024) [52]
dispute the Chae findings, citing systematics in binary
sample selection. This debate is ongoing, and Gaia DR4
will provide decisive data. Regardless of the outcome:
• If Chae confirmed: a boost at the Sun falsifies
the environmental-confinement axiom

59
• If Banik confirmed: DFD’s prediction holds, and
the test moves to pairs beyond R ≃ 16 kpc, which
keep the boost, with a step at R ≃ 13–16 kpc (the
outer-halo globular clusters show about a third of
that boost, 0.33 ± 0.08, App. NL)
Status: Wide Binaries
DFD predicts no wide-binary boost near the Sun: γ = 1
exactly, with no orientation term, because the Galaxy’s
mode at the Sun is typically empty and the
environmental-confinement axiom then makes the Sun’s
neighbourhood Newtonian (App. NL). This sides with
the null analyses and against the detections (Chae 2023,
+7.2σ; Chae et al. 2026, +4.3σ). A confirmed boost of
the Chae type in local pairs falsifies the axiom. Pairs
beyond the edge of the frozen zone, R ≃ 13–16 kpc, keep
the boost, and pairs on either side of the edge should
show a step where MOND predicts a smooth decline; the
outer-halo globular clusters show about a third of the
predicted boost (App. NL). A 41.6% velocity figure omits
the external field and is not a DFD prediction. Gaia DR4
decides the local test.

e. The orientation null: a systematics-robust widebinary test. Amplitude tests of wide binaries are in
a poor state: the published boost parameters span
γ ∈ {1.43, 1.37, 1.60, 1.12, 0.99, 1.00} with sample standard deviation 0.251—the field disagrees with itself —and
the dominant systematics are mass-to-light ratio, contamination and eccentricity. DFD offers a test immune to
all three. Let θ be the angle between a pair’s separation
vector and the Galactic-centre direction. Under the
completed scale-additive field equation (App. NL)
DFD predicts exactly zero even (cos2 θ) dependence of the force boost γ on θ, at every separation
and mass ratio (Thm NL.10, Falsifier NL.20; vobs /vN
inherits an even term ≲ 7 × 10−4 , zero for equal masses),
because its depth never forms the vector sum gint + gext
and its energy is linear in |∇ΦN |2 ; the one allowed term
is odd, ∝ (m1 − m2 ) cos θ, zero for equal-mass pairs. The
pairs of this catalogue lie inside the frozen zone, where
the environmental-confinement axiom removes the odd
term as well: DFD predicts no modulation and no boost.
AQUAL/MOND, by contrast, predicts a 6.88% peak-topeak modulation at s = 104 AU, rising to 46.65% at
s ≈ 5 × 103 AU where gint = gext and the vector sum can
cancel. The measurement is a pure geometry cut on the
1.1 × 106 pairs with > 99% bound probability already
on disk in the Gaia eDR3 catalogue of El-Badry, Rix &
Heintz [53]. An even-harmonic null for DFD, a signal for
MOND, and no new observation required.

I.

External Field Effect

In non-linear theories like MOND and DFD, the internal
dynamics of a system can depend on its external gravitational environment—the external field effect (EFE).

a. Physical origin. The DFD field equation (21) is
non-linear in ∇ψ. When a dwarf galaxy or satellite orbits
within the gravitational field of a larger host, the total
gradient |∇ψtot | = |∇ψint + ∇ψext | may exceed a⋆ even
if |∇ψint | < a⋆ internally. This can “turn off” the µcrossover enhancement.
b. Observational signatures.
• Satellite galaxies near the Milky Way may show less
enhanced dynamics than isolated dwarfs.
• The correlation depends on the satellite’s position
relative to the host’s gravitational gradient.
• Recent observations of Crater II, Antlia 2, and other
diffuse satellites probe this effect.
c. DFD prediction. On the local (vector-sum) reading, which the completed law of App. NL replaces, the
EFE follows the same structure as in AQUAL/MOND.
Defining the total dimensionless acceleration ratio:
|aint + gext |
c2
,
with a = ∇ψ,
(188)
a0
2
the µ-function argument becomes xtot rather than xint
alone. Quantitative predictions require numerical integration of the non-linear field equation in specific configurations.
d. Magnitude and anisotropy are separate questions.
The above is the local reading, in which µ evaluates
the vector sum and the external-field effect is therefore anisotropic. Its coefficient is closed-form (App. NL,
Prop. NL.19): Lext = (τ /2) nBE (τ ) with nBE = 1/(eτ −1),
giving G∥ /G⊥ = 1/(1 + Lext ). At the solar circle the
depth is set by the Newtonian Galactic field, gN,e ≃ 1.19–
1.33 a0 for an observed 1.79–1.94 a0 , so τext = 1.09–1.15;
this gives Lext = 0.27–0.28 and G∥ /G⊥ = 0.78–0.79, an
anisotropy of about 21% (computed). Wherever this corpus enters 1.793 a0 as the argument of τext or in ∇ΦN ,
the value is the observed solar-circle field, i.e. the boosted
total; the Newtonian field is gN,e ≃ 1.2 a0 (App. NL,
Status NL.11, which quotes the shift for the wide-binary
entries). Under the scale-additive completion the external field is instead a separate scalar mode (a depth,
never a vector-sum argument), so the even anisotropy
vanishes identically: G∥ /G⊥ = 1 exactly for the axisaveraged ratio, for every geometry, separation and mass
ratio (App. NL, Thm NL.10; an odd cos θ term proportional to the mass asymmetry remains). Two MOND
consequences are therefore forbidden outright in that
reading: isophotal elongation of EFE-dominated dwarfs
anti-aligned with the host, and a Solar-System quadrupole
Q2 ̸= 0. The last of these is measured, and is a theorem of
the completed force law (Thm NL.8). Inside a host whose
baryonic Newtonian field exceeds (ln 2)2 a0 a subsystem’s
own field is Newtonian (Axiom NL.14); every Milky Way
satellite lies below that threshold, Sagittarius (at 18 kpc)
by a factor of 1.3–2.1.
xtot ≡

60
J.

Dwarf Spheroidal Galaxies

Dwarf spheroidal galaxies (dSphs) provide important
tests of modified gravity theories due to their low internal
accelerations and proximity to the Milky Way. The classical dSphs (Fornax, Sculptor, Draco, Carina, Sextans,
Leo I, Leo II, Ursa Minor) span a range of stellar masses
105 –107 M⊙ and distances 76–254 kpc.
1.

Jeans Analysis with EFE

The spherical Jeans equation relates velocity dispersion
to the gravitational field:
1 d(ρ∗ σr2 ) 2β(r)σr2
+
= −g(r),
(189)
ρ∗ dr
r
where ρ∗ (r) is the stellar density, σr is the radial velocity
dispersion, and β = 1−σt2 /σr2 is the anisotropy parameter.
In DFD, the gravitational acceleration includes the
µ-enhancement:
q
gN (r)
gDFD (r) =
,
xtot = x2int + x2ext ,
(190)
µ(xtot )
2
where xint = GM (< r)/(r2 a0 ) and xext = VMW
/(D a0 )
with VMW ≈ 220 km/s.

2.

Two-Regime Model

Classical dSphs naturally divide into two limiting
regimes:
a. 1. Isolated regime (xint ≫ xext ): For systems like
Leo I at D = 254 kpc, the internal field dominates. The
velocity dispersion follows the deep-MOND scaling:
1
4
4
σlos
≈ 81
GM∗ a0 ,
Ψiso = √
.
(191)
xint
b. 2. EFE-dominated regime (xint ≪ xext ): For systems like Draco at D = 76 kpc, the Milky Way’s external
field dominates. The dynamics become quasi-Newtonian
with enhanced effective gravity:
G
1
1 + xext
Geff =
,
ΨEFE =
=
. (192)
µ(xext )
µ(xext )
xext
3.

Comparison with Data

A spherical Jeans model on this reading gives:
This Jeans model uses the vector-sum form of the external field written above. The printed law of satellites adds
the Newtonian fields of the satellite and its host (App. NL).
On its forward form, with population-synthesis mass-tolight ratios, the µ-law alone falls 0.2–0.3 dex (3–4σ) short
of the eight dispersions (computed). Nearly all of the
shortfall sits in Draco, Ursa Minor, Sextans and Carina,
where the Galaxy’s field dominates and which are not
tidally safe; the four that are tidally safe fit the µ-law

TABLE XIV. Classical dwarf spheroidals on the vector-sum
reading of the external field, which the printed satellite law
does not use; their status is Open (Table XVIII).
dSph

M∗ /M⊙ D (kpc) xint /xext
7

Fornax 2.8 × 10
Sculptor 2.8 × 106
Leo I
6.8 × 106
Leo II 1.2 × 106
Draco
4.4 × 105
UMi
4.0 × 105
Sextans 8.2 × 105
Carina 4.8 × 105

147
86
254
233
76
76
86
105

1.5
0.5
4.9
1.6
0.12
0.17
0.03
0.14

Regime
Isolated
Transition
Isolated
Isolated
EFE
EFE
EFE
EFE

alone to about 1σ (computed). The floor law’s χ subhalos or tidal disequilibrium can supply the difference
(App. AV), so these four dwarfs test χ and tides as much
as the µ-law.

4.

Ultra-Faint Dwarfs: Systematic Effects

Ultra-faint dwarfs (Segue 1,
Willman 1,
Coma Berenices, etc.)
show extremely high inferred mass-to-light ratios (M/L ∼ 100–1000). Before
attributing this to dark matter, systematic effects must
be assessed.
The observed velocity dispersion σobs can be systematically inflated by:
TABLE XV. Systematic effects inflating ultra-faint σ measurements.
Effect

Factor on σ Factor on M/L

Binary stars (fb ≈ 40%,
vorb ∼ 12 km/s)
Tidal heating (rh ∼ rtidal )
Velocity anisotropy (β ∼ 0.5)
Small-N bias (N ∼ 25 stars)

1.8–2.5

3–6

1.5–3.0
1.1–1.3
1.1–1.2

2–9
1.2–1.7
1.2–1.4

Combined

3–10×

10–100×

For an intrinsic σtrue ∼ 2.5 km/s (DFD prediction
for EFE-dominated ultra-faints), these systematics can
inflate the apparent M/L by factors of 10–100.
a. Evidence for systematic origin:
• Of the systems cited for extreme M/L, Tucana III
has confirmed tidal tails and Willman 1’s kinematics may not be in equilibrium; Segue 2 has only
an upper limit on its dispersion and may be the
remnant of a tidally stripped galaxy.
• Multi-epoch binary characterization systematically
lowers σ estimates.
• Better membership selection systematically lowers
M/L.

61
• The correlation “worse data → higher M/L” is
opposite to the dark matter expectation.

K.

Cluster-Scale Phenomenology

Galaxy clusters provide tests at scales intermediate
between galaxies and cosmology. This section analyses
16 galaxy clusters, with galaxy groups compared on published masses, testing whether one µ-function and one a0
can explain cluster dynamics. The µ-law alone falls short,
and χ supplies the difference.

1.

Cluster Dynamics in DFD

Rich clusters (M ∼ 1014 –1015 M⊙ ) have characteristic
accelerations:
1014 M⊙ · G
GMbar
2
∼
∼ 10−11 m/s ∼ 0.1 a0 .
acluster ∼
2
r
(1 Mpc)2
(193)
Clusters thus lie in the deep-field regime where µenhancement is significant (Ψ ∼ 3.7–5), not the transition
regime as often assumed.
a. X-ray gas dynamics. In relaxed clusters, X-ray
emitting gas traces the gravitational potential through
hydrostatic equilibrium:
dP
gN (r)
= −ρgas gDFD (r) = −ρgas
.
(194)
dr
µ(xtrue )
Let y ≡ aN /a0 ≈ 0.05–0.1 be the Newtonian baryonic
acceleration for rich clusters. With the self-consistent
closure a = aN Ψ and Ψ = 1/µ(xtrue ) at xtrue = a/a0 =
yΨ, the enhancement Ψ satisfies
1
.
(195)
Ψ=
µ(y Ψ)
For the canonical choice µ(u) = u/(1+u), the closed-form
solution is
p
1 + 1 + 4/y
Ψ = ν(y) ≡
≈ 3.7–5 (y = 0.05–0.1).
2
(196)
We will use ν(y) throughout the cluster analysis below; it
is the inverse function to g µ(g/a0 ) = gN , parameterized
by the Newtonian input y = gN /a0 .
2.

Cluster Sample Analysis

We analyze 16 galaxy clusters, 10 relaxed and 6 merging;
galaxy groups are compared on published group masses
below and in App. I. Data sources include Vikhlinin et
al. (2006), Gonzalez et al. (2013), Clowe et al. (2006), and
Planck Collaboration (2016).
a. Methodology. For each system:
1. Compute characteristic baryonic (Newtonian) accel2
eration: aN = GMbar /r500
, so y ≡ aN /a0 .

2. Calculate DFD enhancement via the inverse-MOND
function:
p
1 + 1 + 4/y
ΨDFD = ν(y),
ν(y) =
.
(197)
2
This is equivalent to ΨDFD = 1/µ(xtrue ) with
xtrue = ν(y) y = aDFD /a0 , but parameterized by
the directly-observable Newtonian input y.
3. Compare predicted dynamical mass MDFD =
Mbar × ΨDFD to observed Mtotal .
4. Evaluate ratio R = Mtotal /MDFD .

TABLE XVI. Cluster analysis with adopted µ(x) = x/(1 + x).
The column y ≡ gN /a0 is the Newtonian baryonic acceleration
in a0 units; ΨDFD = ν(y) is the inverse-MOND boost defined
in Eq. (197). The corresponding DFD acceleration is xtrue =
ν(y) y, larger than y by the boost factor. Computed at a0 =
1.20 × 10−10 m/s2 , as in App. I.
Cluster

Mbar
Mtotal y = gN /a0 Ψobs ΨDFD Obs/DFD
(1014 M⊙ ) (1014 M⊙ )

A1795
A2029
Coma
Perseus
A383

0.79
1.23
1.00
0.65
0.38

Relaxed Clusters
5.50
0.060
8.50
0.070
7.00
0.060
5.80
0.050
2.80
0.050

7.0
6.9
7.0
8.9
7.5

4.6
4.4
4.6
5.1
5.1

1.51
1.58
1.51
1.76
1.47

Bullet
El Gordo
A2744

1.35
2.45
1.52

Merging Clusters
11.50
0.070
21.00
0.080
14.00
0.070

8.5
8.6
9.2

4.3
4.0
4.3

1.97
2.14
2.12

b. Results with adopted µ = x/(1 + x).
c. Systematic pattern. Table XVI reveals a clear pattern (selected subset shown; full analysis in Appendix I):
• Relaxed clusters: Mean Obs/DFD = 1.57 ± 0.08
• Merging clusters: Mean Obs/DFD = 1.99 ± 0.16
The ratios above one are the mass the µ-law does not
supply from the baryons.

3.

Physical Interpretation

a. The missing cluster mass is χ. Clusters are where
DFD’s derived cold component χ (App. AV) shows: the
rest-mass channel lets it gravitate without the µ-screen
(App. GR), so it can supply mass in cluster cores, where
pure MOND cannot. The relaxed masses in Table XVI
are hydrostatic X-ray masses and the merger masses are
lensing masses; a hydrostatic bias b = 0.21 brings the two
classes to the same ratio. With χ adding its Newtonian
pull, all sixteen clusters of App. I need χ = 4.5 ± 0.6 Mb
inside r500 , 0.84 ± 0.12 of the cosmic mix Ωχ /Ωb ≃ 5.4;
under the floor composition of App. AV they carry the
ΛCDM budget, 1.28–1.51 of the cosmic mix inside r500
(computed). The amplitude follows from the derived

62
abundance under the adopted energy (1.21 ± 0.05 of the
cosmic mix predicted inside r500 , 1.28 ± 0.04 measured
with no hydrostatic bias; estimate; App. I).
b. Missing baryons do not close the gap. The literature items for missing cluster baryons (warm-hot gas,
clumping, intracluster light, gas beyond r500 , a bottomheavy IMF) sum to a correction of 1.26, and to 0.96–1.10
inside r500 (App. I), far short of the 1.6–2.0 needed.
c. Galaxy groups. On published X-ray masses the
µ-law falls short of galaxy groups as it does of clusters:
the median Obs/DFD is 1.22–1.36 at r500 for the 23
Chandra groups of Sun et al. (2009) and 1.79 for the 16
groups of Gastaldello et al. (2007), at the bound-matter
a0 with distances computed with H0loc (computed on
published inputs; stellar masses estimated), as a MOND
analysis with the same µ also finds [54]. Under the floor
composition most groups are Newtonian plus χ at 1.2–1.7
of the cosmic mix, as clusters are (App. I).

4.

Nonlinear Averaging and the Correction Stack

Cluster mass budget: the χ component is required
Under the universal µ(x) = x/(1 + x) the sixteen clusters
fall short by Obs/DFD ≃ 1.6 (relaxed) and 2.0 (merging).
The shortfall is the derived χ: 4.5 ± 0.6 Mb , 0.84 ± 0.12
of the cosmic mix, with the relaxed class’s hydrostatic
bias b = 0.21, when χ adds its Newtonian pull, and
1.28–1.51 of the cosmic mix, the ΛCDM budget, under
the floor composition of App. AV (computed). A no-χ
closure by five correction factors, Eq. (200) below, is not
supported: the baryonic items sum to 1.26, nonlinear
substructure averaging gives ≃ 1.04 for the baryonic
substructure the µ-law responds to, the temperature
factor absorbs the hydrostatic bias, and the projection
factor has the wrong sign for lensing masses (App. I).

a. The physics of nonlinear averaging. Clusters are
not smooth distributions—they contain N ∼ 100–1000
galaxy-scale subhalos. The DFD field equation is nonlinear, so the gas-weighted average enhancement Ψ in the diffuse intracluster medium differs from the smooth-cluster
mean-field value. The mechanism is mass redistribution:
when a fraction of cluster mass is concentrated into compact subhalos, the smooth-component contribution to the
local gradient is reduced in the diffuse regions where the
X-ray gas resides. Lower local |∇ψ| means deeper-MOND
enhancement, so ⟨Ψ⟩gas-weighted > Ψsmooth (ȳ).
b. Jensen’s inequality. The function Ψ(x) =
1/µ(x) = (1 + x)/x is strictly convex for µ(x) = x/(1 + x).
By Jensen’s inequality:
⟨Ψ(x)⟩ ≥ Ψ (⟨x⟩) ,

(198)

with equality only for a uniform field. The substructure
boost is therefore real and positive, but its magnitude
depends on the actual statistical and spatial structure
of the field, not on a simple Monte Carlo over assumed
subhalo accelerations.

c. PDE-calibrated Jensen factor. A direct 3D nonlinear AQUAL/DFD solver (Brada-Milgrom 1995 method,
face-centered µ, validated against algebraic-MOND for
spherical NFW within 1%) gives a substructure averaging
factor that depends approximately linearly on the subhalo
mass fraction:
JPDE (fsub ) ≃ 1.00 + 0.39 fsub ,
(robust across cluster mass and subhalo geometry),
(199)
giving JPDE ≃ 1.07–1.12 for subhalo mass fractions fsub =
0.15–0.30. This is the nonlinear AQUAL contribution
(computed). The µ-law responds to the baryons, however,
so the relevant fsub is the member galaxies’ share of the
baryons, ≃ 0.10, and JPDE ≃ 1.04 (computed).
d. The five-factor correction stack. A no-χ closure
would have to come from a product of correction factors,
Ci = Bi × JPDE,i × Ti × Mi × Pi

(200)

for baryonic completeness (B), nonlinear substructure
averaging (JPDE ), an X-ray temperature systematic (T ),
merger nonequilibrium (M ), and lensing/hydrostatic projection (P ), with Ci = 1.5–2.1 needed. It does not close
(computed; App. I): B sums to 1.26 on its own item list,
and to 0.96–1.10 for the items inside r500 ; JPDE ≃ 1.04
for baryonic substructure; T tracks the merger flag rather
than temperature and absorbs the relaxed/merging split,
which is the hydrostatic bias b ≃ 0.21 of the relaxed
clusters’ X-ray masses; M is applied to lensing masses,
which carry no hydrostatic nonequilibrium; and P > 1
has the wrong sign for the lensing masses of mergers,
which simulations find biased low. The clusters need χ:
4.5 ± 0.6 Mb inside r500 , 0.84 ± 0.12 of the cosmic mix,
when χ adds its Newtonian pull (computed). The amplitude follows from the derived abundance under the
adopted energy (1.21 ± 0.05 of the cosmic mix predicted
inside r500 , 1.28 ± 0.04 measured with no hydrostatic
bias; estimate; App. I). A first-principles account awaits
a structure-formation run under the adopted momentumconserving energy and a cluster-by-cluster nonlinear DFD
hydrodynamic and lensing solver that includes χ.
e. Falsifiable prediction: µ-universality. DFD’s µfunction is universal, with n = 1 from galaxies to clusters;
at clusters the missing mass is χ, not a softer µ. Tests:
1. Resolve cluster substructure in weak lensing—
individual subhalos should show n = 1 RAR
2. Measure RAR for cluster member galaxies—should
match field galaxy µ(x) = x/(1 + x)
3. Compare mass-weighted vs. light-weighted cluster
profiles
f. Deep-field lensing: constant p
deflection angle. In
the deep-field regime (b ≫ rm ≡ GM/a0 ), the DFD
deflection
angle asymptotes to a constant: α̂deep =
√
2π GM a0 /c2 , independent of impact parameter. For
an L⋆ galaxy (M = 5 × 1011 M⊙ ), α̂deep ≈ 1.3′′ . This produces a convergence profile κ ∝ 1/R and excess surface

63
density
√
∆ΣDFD (R) =

GMbar a0
4G R

(201)

for R ≫ rm where the galaxy’s χ stays below the µ-law’s
demand, with normalization set entirely by baryonic mass
(zero free halo parameters). Recent KiDS-1000 weaklensing results show approximately flat circular velocities
to several hundred kpc and a smooth RAR across galaxy
types, qualitatively consistent with Eq. (201). They also
show a baryonic Tully–Fisher relation extending well beyond virial radii, which DFD does not reproduce: the
Rest-Mass Channel confines each galaxy’s phantom to its
bound region (App. GR, Remark GR.12), so Eq. (201)
holds out to about the halo radius r200m , and beyond
it DFD’s profile falls below the measured one for the
lighter halo masses, and below the looser-isolation stack
of Brouwer et al. (2021) for every halo mass tried (computed), as the halo model of ΛCDM does; spectroscopically isolated lens samples decide it. The decisive test is
a direct fit of the µn family (n = 1 vs. n = 2) to stacked
ESD profiles from published galaxy-galaxy lensing data,
feasible with existing public datasets.

5.

The Bullet Cluster: Quantitative Analysis

The Bullet Cluster (1E 0657-56) is often cited as strong
evidence for dark matter due to the spatial offset between
the X-ray gas and the gravitational-lensing peaks. In
DFD this offset is reproduced by the derived collisionless
dark-matter field χ (mχ = 32.0 eV, Ωχ h2 ≃ 0.12) , not
by the ψ-MOND enhancement alone.
a. DFD mechanism. During the merger the collisional X-ray gas is shock-stripped and lags at the interaction site, while the collisionless components—the
galaxies and the cold χ field (Ωχ : Ωb ≃ 5.4, born at rest,
pressureless)—pass through and remain at the galaxy
centroid. The χ surface density therefore carries the
dominant lensing convergence (κgalaxy /κgas ≃ 6 for a collisionless fraction ≃ 0.86), placing the lensing peak on
the galaxies and displaced from the gas, in agreement
with the observed ∼ 120–155 kpc offsets. This is the same
collisionless-mass mechanism that operates in ΛCDM; the
DFD distinction is that the collisionless carrier is derived
rather than postulated. Clusters need χ (App. I). For
the Bullet, χ at 0.6–0.9 of the cosmic mix inside the HST
field fits the lensing map when it adds its Newtonian
pull (1.1–1.2 under the floor composition; computed); the
lensing ratio ≃ 6 above is the Newtonian budget of the
cosmic mix.
b. Why ψ-enhancement alone is insufficient. A pure
non-linear ψ (MOND-type) enhancement cannot produce
the offset. At the two brightest cluster galaxies the baryonic field is gN,b /a0 = 0.21–0.34, where the µ-law boost
is ν = 2.3–2.7 (computed); even so, without χ DFD gives
only 24–39% of the lensing mass inside 250 kpc, and it

puts more mass on the gas than on the galaxies (galaxy-togas ratio 0.63–0.76 in 80 kpc apertures on the subcluster,
against 2.05 in the lensing map; computed). This is the
well-known failure of MOND-type gravity at the Bullet,
and it is precisely what makes the derived χ component
necessary.
TABLE XVII. Bullet Cluster lensing budget in DFD: the
collisionless χ carries the offset peak.
Component

Collisional?

X-ray gas (≃ baryons)
yes (lags at shock)
Galaxies + χ (collisionless) no (passes through)

6.

Relative lensing κ
1
≃6

Global Consistency: One Function, Galaxies to Clusters

Table XVIII demonstrates that a single µ-function and
single a0 explain dynamics across four orders of magnitude
in acceleration, with the derived χ supplying the cluster
and group mass; the classical dwarfs, which need χ or
tides as well, are open.
Key Result: clusters need the derived χ
Under the universal µ(x) = x/(1 + x) the clusters
fall short, and the shortfall is the derived cold
component χ.
• Raw Obs/DFD: 1.57 ± 0.08 (relaxed, n = 10),
1.99 ± 0.16 (merging, n = 6) at
a0 = 1.20 × 10−10 m/s2 ; a hydrostatic bias b = 0.21
of the relaxed X-ray masses equalizes the classes
• χ required: 4.5 ± 0.6 Mb inside r500 , 0.84 ± 0.12
of the cosmic mix at the bound-matter
a0 = 1.1966 × 10−10 m/s2 , when χ adds its
Newtonian pull; the ΛCDM budget (1.28–1.51 of
the cosmic mix) under the floor composition
(App. AV); 1.21 ± 0.05 predicted from the derived
abundance (estimate)
• Bullet: without χ DFD gives 24–39% of the
lensing mass inside 250 kpc; χ at 0.6–0.9 of the
cosmic mix fits when it adds its Newtonian pull
(1.1–1.2 under the floor composition)
• Five-factor stack: does not close the gap (App. I)
• Galaxy groups: on published X-ray masses the
µ-law falls short, as for clusters (median Obs/DFD
1.22–1.36 at r500 for 23 groups), and groups need χ
too
Substructure: a direct 3D nonlinear AQUAL/DFD
solver gives JPDE = 1 + 0.39 fsub , about 1.04 for the
baryonic substructure of clusters.
See Appendix I for the complete per-cluster analysis.
Prediction: the µ-function is universal (n = 1) from
galaxies to clusters; the cluster excess is χ.

64
TABLE XVIII. Global consistency: µ(x) = x/(1 + x) and the bound-matter a0 = 1.1966 × 10−10 m/s2 with no galactic retuning.
The column y ≡ gN /a0 is the Newtonian baryonic acceleration in a0 units, and the corresponding boost is Ψ = ν(y) from
Eq. (197).

L.

System

y = gN /a0

DFD Prediction

Galaxy rotation
Galaxy clusters
Classical dSphs
Bullet Cluster
Galaxy groups

0.1–1
Flat curves
Flat curves
✓
0.05–0.1
Ψ ∼ 3.7–5, plus χ
Ψ ∼ 7–9
✓
0.01–0.2 0.2–0.3 dex short (forward law), plus χ or tides
Dispersions
Open
0.01–0.4
Offset to galaxies (χ)
Offset to galaxies ✓
0.02–0.03
Ψ ∼ 6.3–7.6, plus χ
Ψ ∼ 8–14
✓

Summary: Galactic Phenomenology

Summary: Galactic and Cluster Dynamics
DFD reproduces MOND phenomenology at
galactic scales:
• Flat rotation curves: vc = (GM a0 )1/4 = const
in deep-field limit
• Baryonic Tully-Fisher: Mbar ∝ vf4 with correct
normalization
• Radial acceleration relation: no fitted
parameter; 2693 data points
a0 = 1.1966 × 10−10
• No galactic fit parameter:
√
m/s2 follows from 2 α cH0loc on bound-matter
clocks, with no measured coefficient
(Postulate ZS.11), and is tested independently
against SPARC
√
• α-law: a0 /(cH0 ) = 2 α holds exactly; the
physical normalization is consistent with SPARC
once its systematic uncertainty is retained
Quantitative validation:
• SPARC head-to-head: DFD beats Newton in
100% of galaxies tested
• SPARC head-to-head: DFD beats Standard
MOND in 100% of cases
• Wide binaries: γ = 1 exactly near the Sun
(environmental confinement, App. NL), with the
null analyses and against Chae (2023); a boost and
a step beyond R ≃ 13–16 kpc; the orientation null,
no modulation of either parity locally, is the
systematics-robust test
Dwarf spheroidals:
• Classical dSphs: at population-synthesis
mass-to-light ratios the µ-law alone falls
0.2–0.3 dex short on its forward form, nearly all of
it in the four that the Galaxy’s field dominates and
that are not tidally safe; χ subhalos or tidal
disequilibrium can supply the difference
• Ultra-faints: extreme M/L ratios may be inflated
by measurement systematics; χ subhalos are the
alternative
Cluster scales (χ required):
• Raw Obs/DFD under the µ-law: relaxed (n=10)
1.57 ± 0.08; merging (n=6) 1.99 ± 0.16, at
a0 = 1.20 × 10−10 m/s2 and at the bound-matter
a0 = 1.1966 × 10−10 m/s2

Observation

Match

• The shortfall is the derived χ: 4.5 ± 0.6 Mb ,
0.84 ± 0.12 of the cosmic mix (additive
composition, hydrostatic bias b = 0.21), and
1.21 ± 0.05 predicted under the floor composition
and the adopted energy (estimate); a five-factor
correction stack does not close it
• PDE-calibrated nonlinear AQUAL substructure
factor JPDE ≃ 1.04 for baryonic substructure
• Bullet Cluster offset: carried by the collisionless χ,
at 0.6–0.9 of the cosmic mix when χ adds its
Newtonian pull (1.1–1.2 under the floor
composition; §VII)
• Galaxy groups: on published X-ray masses the
µ-law falls short, as for clusters, and groups need χ
(App. I)
• µ-function universal (n = 1) from galaxies to
clusters; substructure boost mechanism verified by
3D AQUAL solver
Key distinction from MOND: DFD provides
falsifiable laboratory predictions (the universal co-located
clock null of the minimal action, Thm. XI.2, and
channel-resolved residuals of the extended non-metric
channel) that MOND does not.

VIII.

THE α-RELATIONS: PARAMETER-FREE
PREDICTIONS

The preceding sections demonstrated that DFD reproduces all established gravitational phenomenology while
providing a natural explanation for galaxy rotation curves.
This section presents DFD’s distinctive theoretical predictions: numerical relations connecting the fine-structure
constant α, the Hubble constant H0 , and the characteristic scales of gravitational phenomenology. These relations
contain no free parameters beyond fundamental constants.
A key result of this section is that the three relations
are derived from Standard Model physics (kα conditional on the one-gauge-vertex axiom, Appendix P)—they
are not arbitrary numerical coincidences but emerge from
gauge structure and QED.

65
A.

c.

The Fundamental Relations

DFD contains two fundamental α-relations plus one
derived relation:
The α-Relations: Two Fundamental + One Derived
Two Fundamental Relations:

Status.

Component Status

Evidence

Ngen = 3
Stated input Identified with χtop (CP 2 ) = 3
Factor 1/α Strong
Only QED at galactic scales
Cloop = 1/8 Derived
Gauge emergence (App. G, App. AP)

1. Self-coupling (from gauge emergence):
ka =

3
≈ 51.4
8α

(202)

2. Clock coupling (from Schwinger correction):
kα = α × ae =

α2
2π

(203)

One Derived Relation:
3. MOND scale (derived from ka + variational
stationarity, Appendix N):
√
a0 = 2 α cH0
(204)

The numerical values are:

B.

Relation I: The Self-Coupling ka = 3/(8α)

a. Statement. The dimensionless coefficient of the Ξ
functional of App. N is:
3
≈ 51.4.
(205)
ka =
8α
b. Decomposition. In generation-normalized form the
coefficient ka is written as three factors:
1
1
1
3
ka = Ngen × Cloop × = 3 × × =
.
(206)
α
8 α
8α
1. Ngen = 3: The number of fermion generations is
one stated input, identified with the Euler number
χtop (CP 2 ) = 3 (App. F, Rem. F.21); no index on the
odd-dimensional CP 2 × S 3 supplies it, since every
elliptic differential operator there has index zero.
The number 3 is not fitted, and a fourth sequential
generation would falsify the identification.

C.

a. Statement. The characteristic scale for speciesdependent clock couplings is:
α2
≈ 8.5 × 10−6 ,
(207)
2π
where ae = α/(2π) is the electron anomalous magnetic
moment (Schwinger’s result).
b. The Schwinger connection. The factor α/(2π) is
one of the most precisely calculated quantities in physics—
the leading-order anomalous magnetic moment of the
electron:
ge − 2
α
ae =
=
+ O(α2 ) ≈ 0.00116.
(208)
2
2π
The clock coupling arises from a two-step process:
kα = α × ae =

1. Step 1: The gravitational potential ψ couples to
the EM vacuum (coupling strength ∼ α)
2. Step 2: The perturbed EM vacuum affects atomic
frequencies through the Schwinger correction (factor
ae = α/2π)
Combined amplitude:
α2
α
=
.
(209)
2π
2π
c. Feynman diagram interpretation. The clock coupling arises from a diagram with two EM vertices:
kα = α × ae = α ×

ψ (gravitational potential)
∼ α (EM-ψ coupling)
γ (virtual photon)

2. Factor 1/α: At galactic scales (a ∼ 10−10 m/s2 ),
only QED contributes to long-range vacuum effects.
QCD is confined, SU(2)L is broken with massive
gauge bosons. The factor 1/α reflects the strength
of QED vacuum polarization effects.
3. Cloop = 1/8: Derived by gauge emergence, not
a heat-kernel estimate: Cloop is the generationnormalized form of the acceleration-channel cou1
3
pling ka = (n3 /n2 ) αM = 32 · 4α
= 8α
, so
−1
ka = Ngen Cloop α with Ngen = 3 gives Cloop =
1/8 (App. G, Thm G.1; independently App. AP,
Thm AP.19 and Cor. AP.20).

Relation II: The Clock Coupling kα = α × ae

∼ α/(2π) (Schwinger)
atom (frequency shift)
d.

Physical meaning.

• First α: How strongly ψ couples to the EM vacuum
• Second α/(2π): The Schwinger anomalous magnetic
moment
• Combined: A two-step process linking gravity to
atomic physics

66
TABLE XIX. Fundamental relations and values.
Relation

Formula Value

Physical Origin

ka (self-coupling) 3/(8α) 51.4
n3 /n2 and 1/(4α) (App. G 4)
kα (clock coupling) α√× ae
8.5 × 10−6
Schwinger correction
a0 (MOND scale) 2 α cH0 1.1966 × 10−10 m/s2 Derived law; bound-matter clock

e. Testable prediction. If kα = α × ae , transitions
more sensitive to the magnetic moment should show larger
gravitational shifts. Hyperfine transitions (sensitive to
ae ) should systematically differ from optical transitions
of similar α-sensitivity.
f. Status. The derivation kα = α × ae places this
relation at theorem grade conditional on the one-gaugevertex axiom (Appendix P), an axiom that is a postulated non-metric operator outside the minimal action
(Thm. XI.2). See Appendix P for the complete theorem chain: Schwinger coefficient (Theorem P.1) + “one
gauge vertex” axiom (Theorem P.2). Observational test:
ESPRESSO α(z) measurement gives (+1.3±1.3)×10−6 at
z ∼ 1; the DFD value is conditional on the undetermined
res
residual ξLPI
(App. P).
D.

Relation III: The MOND Scale a0 (Derived)

√
a. Key result. The MOND scale a0 = 2 α cH0
is not an independent relation. It follows from
ka = 3/(8α) plus the S 3 microsector scaling charge via
variational stationarity (Appendix N, Theorem N.14).
b. Derivation. The crossover point is selected by stationarity of the spacetime functional (Appendix N):

2
Z


3
|a|
Ξ = ka
S[ψ] =
d3 x Ξ(x) − log Ξ(x) ,
.
2
cH0
Ω
(210)
Scaling stationarity gives Ξ∗ = 3/2, the S 3 scaling charge
(Theorem N.12). Then:
3
ka × a20 = (cH0 )2 .
(211)
2
Solving for a0 :
a20 =

3(cH0 )2
3(cH0 )2
= 4α(cH0 )2 ,
=
3
2ka
2 × 8α

(212)

√
a0 = 2 α cH0 .

(213)

There is no fine-tuning; a0 ∼ cH0 follows from topology.
d. Numerical verification. Using α = 1/137.036 and
the bound-matter clock H0loc = 72.09 km/s/Mpc (Postulate ZS.11, Eq. (479)):
ka = 3/(8α) = 51.39
−10

(214)
2

cH0 = 7.00 × 10
m/s
√
2
aderived
= 2 α cH0 = 1.1966 × 10−10 m/s
0

(215)
(216)

2
aobserved
= (1.20 ± 0.02stat ± 0.24sys ) × 10−10 m/s
0

(217)
Agreement: within the systematic error; on the early
clock a0 = 1.1208 × 10−10 , −3.96σ on statistics alone and
−0.33σ on the systematic-dominated comparison.
e. Cross-check. ka × a20 /(cH0 )2
=
51.4 ×
(1.1966/7.00)2 = 1.50 = 3/2 (with the common
powers of ten cancelling). ✓
E.

Consistency and Cross-Checks

The relations satisfy the following identities:
a. I. ka × a20 /(cH0 )2 (variational selection).
3
3
× 4α(cH0 )2 = (cH0 )2 .
(218)
ka × a20 =
8α
2
The α cancels, confirming the variational selection condition is satisfied identically.
b. II. Schwinger check.
α2
α
=
.
(219)
2π
2π
The formula reproduces the known Schwinger coefficient.
c. Summary of consistency.
kα = α × ae = α ×

Check

Expression

Result

ka × a20 /(cH0 )2 (3/8α) × 4α
3/2 (exact)
kα /(α × ae )
[α2 /(2π)]/[α × α/(2π)] 1 (exact)

therefore:

c. The “MOND coincidence” explained. The 40-year
puzzle of why a0 ∼ cH0 is resolved:
• The self-coupling ka is determined by gauge structure (n3 /n2 and 1/(4α), App. G 4)
• The coefficient 3/2 is the S 3 microsector scaling
charge (topologically fixed)
√
• The α coefficient emerges automatically from ka =
3/(8α)

F.

The Three-Scale Hierarchy

Multiplying a0 by α±1 defines two further acceleration
scales:
a−1 : a0 : a+1 = α : 1 :

1
α

(220)

67
Three-Scale Hierarchy

Summary: The α-Relations

a−1 = α · a0 = 2α3/2 cH0 ≈ 9 × 10−13 m/s2
√
a0 = 2 α cH0 ≈ 1.2 × 10−10 m/s2
√
a+1 = a0 /α = 2cH0 / α ≈ 1.6 × 10−8 m/s2

(221)
(222)
(223)

Two fundamental relations:
• ka = 3/(8α) — from n3 /n2 and 1/(4α) (App. G 4)
• kα = α × ae — from Schwinger anomalous
magnetic moment (conditional on the
one-gauge-vertex axiom)

TABLE XX. Characteristic acceleration scales and associated
physical systems.

One derived relation (Theorem N.14):
√
• a0 = 2 α cH0 — follows from ka + S 3 scaling
charge via variational stationarity

Scale Value (m/s2 ) Ratio to a0 Physical Systems

Consistency checks (all exact):

−13

9 × 10
1.2 × 10−10
1.6 × 10−8

a−1
a0
a+1

a.

α ≈ 1/137 Cluster outskirts, cosmic voids
1
Galaxy rotation curves
1/α ≈ 137 Galaxy cores, bulges

• kα = α × ae (Schwinger)
The “MOND coincidence” is explained: a0 ∼ cH0
follows from topology, not fine-tuning.

Physical regimes.

G.

• ka × a20 = 23 (cH0 )2 (the variational selection
condition)

Status Summary
IX.

Summary:
• The three relations are derived from Standard Model
physics and topology (kα conditional on the onegauge-vertex axiom)
• The “MOND coincidence” (a0 ∼ cH0 ) is explained
by gauge structure
• The coefficient Cloop = 1/8 arises from frame stiffness ratios in gauge emergence
a. Falsification criteria.
falsified if:

The α-relations would be

√
1. Precision determination of a0 differs from 2 α cH0loc
by > 15% after accounting for µ-function uncertainty and H0loc resolution.
2. Multi-species clock analysis shows KA inconsistent
α
with kα · SA
pattern at > 3σ.
3. Experimental determination of ka from RAR fits
differs from 3/(8α) by > 25%.

GAUGE COUPLING VARIATION AND
HIGH-ENERGY IMPLICATIONS

Section VIII adopted, conditional on the one-gaugevertex axiom (Appendix P), the coupling kα = α2 /(2π)
of electromagnetic properties to the scalar field ψ. This
section extends the framework to all Standard Model
gauge couplings, derives the modified renormalization
group equations, and explores consequences ranging from
nuclear clock tests to grand unification.
A.

Universal Gauge-ψ Coupling

a. Extension to all gauge sectors. The clock coupling
kα = α2 /(2π) is the coefficient of Appendix P, conditional
on the one-gauge-vertex axiom (a postulated non-metric
operator outside the minimal action, Thm. XI.2). The
parallel construction for non-Abelian gauge fields, under
the same axiom in each sector, yields the universal form:
δαi
α2
= ki ψ,
ki = i ,
(224)
αi
2π
where αi = gi2 /(4π) is the fine-structure constant for
gauge group i.
b. Physical origin. The αi2 dependence is characteristic of one-loop quantum corrections. The gauge emergence
framework (Appendix F) provides a deeper origin for these
couplings through frame stiffness in the internal mode
space.
c. The gauge hierarchy. At laboratory energies:
U(1)EM :
SU(2)L :
SU(3)c :

α ≈ 1/137,

kα ≈ 8.5 × 10−6 ,

(225)

αw ≈ 1/30,

−4

,

(226)

.

(227)

αs ≈ 0.118,

kw ≈ 1.8 × 10
ks ≈ 2.2 × 10

−3

The strong coupling here is the Z-pole reference value
αs (MZ ) = 0.118. At the nuclear scale µ ∼ 1 GeV relevant
to the Th-229 nuclear-clock channel of Sec. XI, αs (µ) ≈

68
TABLE XXI. Status of α-relation derivations.
Relation Formula Physical Origin
ka
kα
a0

3/(8α) n3 /n2 and 1/(4α) (App. G 4)
Derived
α√× ae
Schwinger anomalous magnetic moment Conditional
2 α cH0 Derived from ka
—

0.3–0.5 via standard QCD running, including the Nf =
5 → 4 threshold at µ ∼ mb .
The strong force is most sensitive to gravitational potential:
ks
α2
= s2 ≈ 260.
kα
α

(228)

The Gauge Coupling Hierarchy
Key result (conditional on the one-gauge-vertex axiom
in each sector): All gauge couplings shift with
gravitational potential according to δαi /αi = ki ψ with
ki = αi2 /(2π).
Hierarchy: ks : kw : kα ≈ 260 : 20 : 1
The strong force is ∼ 260× more sensitive to ψ than
electromagnetism.

B.

a. The one-loop β-function. The running of gauge
couplings with energy scale µ is governed by:
bi αi2

dαi
=
,
d ln µ
2π
where bi is the one-loop coefficient:
41
(U(1)Y ),
b1 = +
10
19
b2 = −
(SU(2)L ),
6
b3 = −7 (SU(3)c ).
b. Connection to the beta functions.
Eqs. (224) and (229):
βi
bi

(229)

(230)
(231)
(232)
Comparing

(233)

The ψ-gauge coupling equals the β-function divided by the group-theory coefficient.

C.

Taking the scale derivative at fixed ψ:


dαi (µ, ψ)
dαi (µ, 0)
α2 ψ
2αi dαi
=
1+ i
+ αi ·
ψ.
d ln µ
d ln µ
2π
2π d ln µ
(235)
The modified β-function:


dαi
bi αi2
3α2
(236)
=
1+ i ψ
d ln µ
2π
2π
The ψ-correction is proportional to αi4 —a two-looplike gravitational correction to the running.
a. Laboratory effects. For QCD near confinement
(αs ∼ 1):
δβs
3αs2 ψ
∼ 0.5ψ.
∼
βs
2π

(237)

In laboratory environments (ψ ∼ 10−9 ), this is ∼ 5 ×
10−10 —unmeasurable directly, but the ks coupling itself
has dramatic consequences for nuclear physics.

Connection to the β-Function

ki =

Status

Modified Renormalization Group Equations

In the presence of non-zero ψ, gauge couplings depend
on both energy scale and gravitational potential:


αi2
αi (µ, ψ) = αi (µ, 0) 1 +
ψ .
(234)
2π

D.

Asymptotic Freedom and UV Behavior

a. QCD decoupling. QCD is asymptotically free:
αs (µ) → 0 as µ → ∞. This implies:
αs2 (µ)
→ 0 as µ → ∞.
(238)
2π
The strong sector decouples from ψ in the ultraviolet.
b. Maximum sensitivity at confinement. Conversely,
ks is maximal at the confinement scale where αs ∼ 1:
1
ksmax ∼
≈ 0.16.
(239)
2π
This explains why nuclear physics provides the strongest
low-energy probe of ψ-gauge coupling: the effective coupling ks peaks precisely at the energy scale relevant for
nuclear binding.
c. QED behavior. QED is not asymptotically free; α
increases slowly with energy. The Landau pole occurs at
µ ∼ 10286 eV, far above the Planck scale. For practical
purposes, kα remains approximately constant.
ks (µ) =

E.

Nuclear Clock Prediction: Thorium-229

The ks /kα ≈ 260 hierarchy, combined with the exponential sensitivity of QCD through dimensional transmutation, leads to dramatic predictions for nuclear transitions.

69
a. The thorium-229 isomer. 229 Th has a nuclear isomer with uniquely low transition energy:
Em = 8.3557 eV.

(240)

This arises from near-cancellation between Coulomb
(∼ +300 keV) and nuclear strong-force (∼ −300 keV)
contributions, with a residual of only ∼ 8 eV.
b. Sensitivity coefficients. The isomer energy depends on fundamental constants through:
δEm
δα
δXq
= Kα
+ Kq
,
(241)
Em
α
Xq
where Xq ≡ mq /ΛQCD and from nuclear structure calculations:
Kα ≈ 104 ,

Kq ≈ −104 .

Nuclear Clock Enhancement: Unscreened ExtendedChannel Estimate

(242)

c. The ΛQCD amplification. The QCD scale is determined by dimensional transmutation:


2π
.
(243)
ΛQCD = µ exp −
|b3 |αs (µ)
Differentiating:
δΛQCD
2π δαs
2π
δαs
δαs =
=
≈ 7.6
. (244)
2
ΛQCD
|b3 |αs
|b3 |αs αs
αs
The factor 2π/(|b3 |αs ) ≈ 7.6 represents the exponential
amplification of relative coupling changes through dimensional transmutation. (Note: the coefficient of the absolute
change δαs is the larger number 2π/(|b3 |αs2 ) ≈ 64; these
two bookkeeping conventions must not be mixed.) With
ks ∝ αs2 this shift equals αs (µ) ψ/|b3 | and so depends on
the scale µ at which it is evaluated; the estimates below
take µ = MZ .
d. The DFD enhancement factor. Combining the
above with δXq /Xq ≈ −δΛQCD /ΛQCD and using
δαs /αs = ks ψ:
δEm
2π
= Kα kα ψ − Kq ×
ks ψ
Em
|b3 |αs


= 104 × 8.5 × 10−6 + 7.6 × 104 × 2.2 × 10−3 ψ
≈ (0.085 + 167)ψ ≈ +170 ψ.
(245)
For comparison, an optical atomic clock has
δνopt /νopt ≈ −ψ/2 (ν ∝ e−ψ/2 , Sec. IV H 4).

R≡

(δν/ν)Th-229
≈ −340+600
−240
(δν/ν)optical

(246)

Caveat: This is the unscreened gauge-sector estimate.
The screened treatment in Sec. XI F, incorporating the
µLPI screening function and 2026 Th-229 reproducibility
data, substantially reduces the expected amplitude and
compresses the surviving annual signal window to
26 Hz–O(1 kHz). The unscreened value above serves as
the theoretical ceiling, not the experimental target.
Physical origin:
1. ks ≫ kα (matching coefficients of the extended
channel): Strong force couples to ψ more strongly
2. Dimensional transmutation: ΛQCD exponentially
sensitive to αs
3. Near-cancellation: 8 eV isomer is tiny residual of
∼MeV forces

e. Experimental test protocol. The following estimates use the unscreened enhancement |R| ≈ 340. The
screened predictions, which are the operationally relevant
ones for terrestrial experiments, are given in Sec. XI F.
Height experiment (1 m separation), unscreened:
∆(νTh /νSr )
= 0,
(247)
GR:
νTh /νSr
DFD (unscreened):
Annual
screened:

∆(νTh /νSr )
≈ 3.7 × 10−14 . (248)
νTh /νSr

modulation
GR:

DFD (unscreened):

f. Timeline.
development:

229

(solar

potential),

∆(νTh /νSr )
= 0,
νTh /νSr annual

un(249)

∆(νTh /νSr )
≈ 5 × 10−8 .
νTh /νSr annual
(250)
Th nuclear clocks are under active

• 2024: First laser excitation of nuclear transition
demonstrated
• 2024: Th/Sr frequency ratio at ∼ 10−12
• 2028–30: Improved precision to ∼ 10−15
The DFD prediction is testable within 2–3 years.

F.

Cosmological α(z) Variation

If the cosmological gravitational potential ψ evolves
with redshift, then α evolves accordingly.

70
a. Cosmological potential. In DFD, the cosmological
scalar field tracks the matter density:
ξ res
ψ(z) = LPI Ωm (z),
(251)
2
res
where ξLPI
is the residual screened cavity/clock coupling
scale discussed in Sec. XII and
Ωm,0 (1 + z)3
Ωm (z) =
.
(252)
Ωm,0 (1 + z)3 + ΩΛ
This expression omits radiation, and so do the highredshift entries that use it; the radiation era is not modelled here, so the CMB and BBN entries are matterdominated estimates.
b. The α(z) prediction. Combining with kα =
α2 /(2π):
∆α
ξ res α2
(z) = kα [ψ(z) − ψ0 ] = LPI [Ωm (z) − Ωm,0 ] .
α
4π
(253)
Numerically,
∆α
res
(z) ≈ 4.2 × 10−6 ξLPI
× [Ωm (z) − 0.312] , (254)
α
res
where the physically relevant ξLPI
is a much smaller
screened residual.
c. Numerical predictions.
Epoch
Quasars
CMB
BBN

Redshift Ωm (z)
2
1100
109

0.92
1.00
1.00

∆α/α (DFD)
res
+2.6 ξLPI
× 10−6
res
+2.9 ξLPI × 10−6
res
+2.9 ξLPI
× 10−6

d. Comparison with observational bounds. Laboratory input. In DFD the cosmological α-variation is
res
controlled by the same residual LPI scale ξLPI
discussed
res
for cavity–atom tests (Sec. XII). We treat ξLPI
as an
experimentally determined input, not a cosmology fit parameter. Cosmological bounds therefore constrain the
laboratory value of this residual scale.
TABLE XXII. Observational probes of fine-structure constant
variation.
Probe
ESPRESSO
Quasar dipole
CMB
BBN

z
1.15
1–3
1100
109

DFD pred.
res ppm
+2.1 ξLPI
—
res ppm
+2.9 ξLPI
res ppm
+2.9 ξLPI

Observed
(+1.3 ± 1.3) ppm
∼ 10 ppm
< 2000 ppm
< 20000 ppm

References: ESPRESSO [55]; dipole [56, 57]; CMB [58]; BBN [59].

Using the conservative ppm-level quasar constraints,
res
the scaling ∆α/α ∼ (2.6 × 10−6 ) ξLPI
at z = 2 implies
that an order-unity cosmological residual would already
be uncomfortable.
Status:
res
• BBN and CMB: Satisfied for ξLPI
≤ 1 with > 100×
margin.

res
• Quasars: For ξLPI
of order unity, bounds become
constraining. Current quasar systematics are debated [57].

• The cosmological prediction is only as clean as the
laboratory determination of the residual coupling
scale.
e. Distinctive signatures. DFD predicts specific features distinguishing it from other varying-α models:
1. Functional form: ∆α/α tracks Ωm (z), flat for
3 ≲ z ≲ 100 and falling steeply for z < 1
2. Sign: ∆α/α > 0 (larger α in the past) in the matter
era
3. Spatial correlation: ∆α/α should correlate with
local matter density
f. Future tests. The ELT/ANDES spectrograph will
achieve σ(∆α/α) ∼ 10−7 per quasar system, tightening
constraints on the residual cosmological coupling scale
res
ξLPI
and potentially detecting a ppm-level signal if that
screened residual lies near the upper end allowed by the
clock sector.

G.

Grand Unification

a. Standard unification picture. The SM gauge couplings approximately unify at MGUT ∼ 1015−16 GeV, but
with a mismatch of ∼ 3–5%.
b. DFD corrections. Couplings measured today include ψ-corrections from cosmological evolution:


αitoday = αiGUT 1 + kilow ∆ψ ,
(255)
where ∆ψ = ψtoday − ψGUT is the change in the cosmological scalar field of Eq. (251), assumed here to satisfy
|∆ψ| ≲ 1; the homogeneous background of App. Q is not
included.
c. Differential corrections.
δα1
≲ 5 × 10−5 ,
(256)
α1
δα2
≲ 2 × 10−4 ,
α2

(257)

δα3
≲ 2 × 10−3 .
(258)
α3
d. Effect on unification. The relative shift in the
unification condition:
|δ(α3 − α1 )|
∼ (k3 − k1 )|∆ψ| ≲ 2 × 10−3 .
(259)
αGUT
Since k3 > k2 > k1 , the correction has the sign of ∆ψ,
which the matter-era expression (251) does not fix at the
GUT epoch; its size is at most about 5% of the total
SM mismatch. This is smaller than current theoretical
uncertainties.

71
H.

Vacuum Energy Feedback

TABLE XXIV. Tier 2: Constraining medium-term tests

The ψ-gauge coupling creates a feedback loop connecting vacuum energy, gravitational potential, and gauge
couplings:
source

Observable
∆α/α (z ∼ 2)
α(z) shape
Spatial α corr.

DFD pred.
res ppm
≈ 2.6 ξLPI
∝ Ωm (z)
∝ δm

Current
ppm-level
—
—

Test
ELT
ELT
ELT

loops

shift

ρvac −−−−→ ψ −−−→ αi −−−→ ρvac
a. Self-consistency condition. Let ψ = F (ρvac ) be
the sourcing relation and ρvac = G(αi (ψ)) be the loop
contribution. Fixed points satisfy ψ ∗ = Φ(ψ ∗ ).
b. Naive stability estimate. Linearizing around ψ =
0:
ψ∗ =

ψ0
,
1−λ

TABLE XXV. Tier 3: Theoretical consistency tests
Quantity

a.
λ∼

MP4
ρc

3

×

α
∼ 10113 .
128π 3

(261)

c. Interpretation. That denominator is the source
of the 10113 . The critical density ρc is an output of the
exponent identity GℏH02 /c5 = Cα57 , not an input, and
the Planck-scale quantity that belongs in that denominator is the gradient stiffness c4 /(32πG) ∼ MP2 of the ψ
kinetic term (ψ carries no potential term). In the dimensionless ratio δV /MP4 the one-loop feedback is bounded,
(1)
|δVCW /MP4 | ≤ O(α2 ) (Ext. Deriv. App. AH, vacuum stability, where the dimensional reading in ρvac is recorded
beside it).
Constraint on UV completion: Any UV completion
of DFD must make the ψ-vacuum energy feedback loop
stable. Separately, primed-determinant scaling fixes the
exponent in (H0 tP )2 = Cα57 , with C = 1 on boundmatter clocks (Postulate ZS.11) and C measured on the
early clock (Section XIX). This feedback loop concern is
about UV stability, not the Λ value.
I.

Summary of Falsifiable Predictions

TABLE XXIII. Tier 1: Nuclear clock tests (unscreened gaugesector estimates; see Sec. XI F for screened predictions)
Observable

GR DFD (unscreened) Timeline

Th/Sr ratio (1m height)
0
Th/Sr annual modulation 0
Nuclear vs optical sign
Same

3.7 × 10−14
5 × 10−8
Opposite

2026–27
2026–27
2026–27

Note: These are unscreened estimates. The screened
treatment in Sec. XI F, incorporating µLPI screening and
2026 Ooi reproducibility data, compresses the surviving
signal window to 26 Hz–O(1 kHz). If the measured Th/Sr
enhancement is consistent with unity at 5σ and the crossspecies atomic channels also show persistent nulls, the
DFD gauge-sector coupling structure would be falsified.

Status

GUT shift
≲ 0.2%
Below precision
Modified β
δβ ∝ α4 ψ
Unmeasurable
(1)
CC feedback |δVCW /MP4 | ≤ O(α2 )
Bounded

(260)

where:

DFD prediction

Hierarchy of tests.

1. Nuclear clocks test the core relation ki = αi2 /(2π).
Confirmation validates the entire gauge-ψ framework.
2. Cosmological α(z) tests the ψ-cosmology connection, independent of nuclear physics uncertainties.
3. GUT and CC constraints test high-energy implications, relevant once Tiers 1–2 are confirmed.
Summary: Gauge Coupling Variation
Universal coupling: δαi /αi = ki ψ with ki = αi2 /(2π)
Hierarchy: ks : kw : kα ≈ 260 : 20 : 1
Nuclear clock (unscreened): R ≈ −340; screened
predictions in Sec. XI F
res
at quasar
Cosmological α: ∆α/α ∼ 3 × 10−6 ξLPI
redshifts
Falsification criteria:
• Persistent nulls across all clock channels (same-ion,
cross-species, nuclear) falsifies the gauge-sector
framework
• R ≈ 1 with high precision rules out DFD gauge
coupling

X.

CONVENTION-LOCKED α FROM THE
MICROSECTOR

The preceding sections derived α-relations from gauge
emergence and electroweak physics. This section presents
the microsector completion: a closed form for α−1 =
137.036 from the internal geometry [60], with all conventions locked and no hidden tuning parameters. The closed
form is exact under two stated choices (§X C); its integers
alone give the measured value to 21 ppm, and with the
bracket coefficient c = 1/13, whose provenance is open
(Remark X.4), it lies 4.9 × 10−9 (0.0049 ppm) from the
CODATA 2022 value, 32 times that value’s uncertainty.

72
A.

Design Constraint: No Hidden Tuning
Parameters

We impose a no-knobs policy: once the microsector
geometry, bundle data, and truncation level are fixed, the
predicted α must be stable without invoking subleading
heat-kernel terms as ppm-level tuners. Concretely, α’s
trace is taken in a form whose heat trace terminates, so
no a6 , a8 , . . . term exists to act as a free correction dial
(Sec. X C).
a. Motivation. Any theory that “predicts” a fundamental constant but allows ppm-level adjustments via
regulator moments or trace normalizations is not truly
predictive—it has hidden knobs. The microsector completion must lock all such freedoms.

B.

Operator Choice (Locked)

On the internal microsector X = CP 2 × S 3 , we take a
Laplace-type operator given by the connection Laplacian:
P = −g ij ∇i ∇j ,

(262)

acting on the internal bundle that carries the emergent
gauge degrees of freedom.
a. Bundle structure. The U(1) factor is implemented
via twisting by a line bundle over CP 2 with curvature
proportional to the Kähler form ω, taken trivial over S 3 .
This choice is minimal and convention-stable: the Kähler
form is parallel (∇ω = 0), so derivative terms in higher
Seeley–DeWitt coefficients vanish automatically.
b. Why this is locked. The gauge-kinetic extraction
from a4 is unambiguous with this operator choice. Alternative operators would introduce additional terms proportional to curvature derivatives, creating ppm-level
ambiguities. The connection Laplacian with parallel curvature eliminates this freedom. On the S 3 = SU (2) factor
the operator is taken as the ρ-shifted Casimir ∆ + 1, the
natural Laplacian of the group, whose heat kernel is a
pure Gaussian (§X C).

C.

Regularization/Truncation Rule (Locked)

We define the spectral action with a cutoff function f :
S = Tr f (P/Λ2 ).

(263)

a. The plateau rule holds only in even total dimension. A plateau cutoff, constant near the origin so that
f (n) (0) = 0 for all n ≥ 1, makes the negative moments
f−2 = f−4 = · · · = 0 vanish, and in even total dimension those moments carry a6 , a8 , . . . . On the sevendimensional
CP 2 × S 3 they √
do not: there a6 carries the
R∞
moment Λ 0 u−1/2 f (u) du/ π, which is positive for every cutoff. For the connection Laplacian of Eq. (262) with
its U(1) twist, a6 /a4 = 19/5 (14/5 from CP 2 , 1 from S 3 ),
a correction of 25% (Gaussian) to 38% (sharp) of the a4
term at the cutoff of Eq. (267), and up to 59% for smooth

plateaus (computed). A single joint seven-dimensional
trace therefore does not support the closed form.
b. The exact construction. The closed form of §X I
is reproduced exactly by α’s trace taken in two factors. On CP 2 , which is four-dimensional, the ζ(0) of
the charge trace equals a4 exactly: −1/24 per unit charge
squared, independent of the radius. On S 3 = SU (2)
the ρ-shifted Casimir ∆ + 1 has a heat trace that terP
√
2
minates, m≥1 m2 e−tm = ( π/4) t−3/2 up to exponentially small terms, so that with the heat-kernel
(Gaussian)
√
regulator its contribution is exactly ( π/4)Λ3 . Their
product gives Eq. (276) with no remainder (to 6 × 10−63 ,
computed). The closed form is therefore exact under two
stated choices: the ρ-shifted Casimir as the operator on
S 3 = SU (2), and α’s trace taken as the CP 2 ζ(0) times
the S 3 factor with the heat-kernel regulator, which is
consistent with α having its own trace, separate from
gravity’s (App. WS). Read as one joint seven-dimensional
action the construction would need a CP 2 radius far from
the Einstein product. It does not derive the cutoff of
Eq. (267) or the bracket of Remark X.4, which stay as
printed.
c. Why this is locked. The exact construction has
no subleading heat-kernel term, so none can act as a
ppm-level tuning knob, and no cutoff moment is tuned:
1. The a4 gauge kinetic term is the whole CP 2 factor
2. No subleading heat-kernel contribution exists
3. No moment is tuned

D.

Finite-k Truncation and the (k + 3)/(k + 4) Factor
(Locked)

We implement a finite-k truncation via Toeplitz quantization at level m = k + 3 on CP 1 , where:
d = dim H 0 (CP 1 , O(m)) = m + 1 = k + 4.

(264)

a. Origin of the +3 shift: an exact charge-trace theorem. The shift is not an assumption about a restricted
line bundle. It is the exact finite-level value of the hypercharge second moment on CP 2 itself, and it follows
from the following theorem, which holds on CP n for every
traceless charge.
Theorem X.1 (CP n Toeplitz charge trace). Let
Hk = H 0 (CP n , O(k)) ∼
= Symk (Cn+1 ) be the level-k
Toeplitz (geometric-quantization) Hilbert space, let Q =
diag(q0 , . . . , qn ) be a traceless charge — a U(1) in the
Cartan of su(n + 1) — and let Q̂ be the operator by which
Q acts on Hk . Then




 
k+n+1
k+n
,
dim Hk =
.
TrHk Q̂2 = Tr(Q2 )
n+2
n
(265)
The result is exact at every level: it is a closed-form
identity, not the leading term of a 1/k expansion.

73
Proof. In the monomial weight basis of Symk (Cn+1 ), indexed by exponent vectors m with |m| = k, the operator
P
Q̂ is diagonal with eigenvalue µm = i qi mi . Introduce
the two-variable generating function
n
X
Y
P

−1
G(z, u) =
z |m| u i qi mi =
1 − z uqi
.
m

i=0

P 

q z/(1 −
Tracelessness gives (u∂u ) log G u=1 =
P 2
 i i
2
2
z) = 0, while (u∂u ) log G u=1 =
i qi z/(1 − z) =



2
Tr(Q2 ) z/(1 − z)2 . Since (u∂u )2 G = G (u∂u ) log G +

(u∂u )2 log G and G(z, 1) = (1−z)−(n+1) , the first bracket
vanishes at u = 1 and
 
X
z
.
z k TrHk Q̂2 = Tr(Q2 )
(1 − z) n+3
k≥0


Extracting [z k ] gives Tr(Q2 ) k+n+1
n+2 . The dimension formula is the standard dim Symk (Cn+1 ).
Verification. Exact-integer agreement was checked by
direct enumeration of the weight basis for n = 1, 2, 3, 4 and
k = 2, . . . , 12 over three inequivalent traceless charges at
each n, including the asserted proportionality to Tr(Q2 ).

retained here as a consistency check : the degree of the
determinant line and the shift produced by Theorem X.1
agree because both are the same CP 2 datum n + 1 = 3 —
equivalently, (k + 3)/k is the reciprocal of the level-one
(adjoint) Berezin-transform eigenvalue, the Berezin shift
and the Casimir shift being the same shift. The +3 is thus
computed on CP 2 rather than read off from a restricted
bundle degree.
b. The spectral cutoff. The determinant-channel removal at finite d fixes the spectral cutoff as:
d−1
k+3
Λ3 = k ·
=k·
.
(267)
d
k+4
This is the unique finite-size factor permitted by the
truncation rule; it is not inserted to improve agreement.
The truncation rule also holds the size of the internal
space fixed in this formula: on a continuous CP 2 × S 3
the spectral action of the connection Laplacian lets the
volume run away without end (Prop. FK.3), and α would
drift with it, while the truncated microsector has no size
modulus (Thm. FK.2).

E.

Corollary X.2 (The (k + 3) factor). Specialise Theorem X.1 to the DFD microsector base n = 2 with Q = Y
the Standard Model hypercharge. Then


 
k+3
k(k + 1)(k + 2)(k + 3)
2
2
TrHk Ŷ
= Tr(Y )
= Tr(Y 2 )
,
4
24


k+2
(k + 1)(k + 2)
dim Hk =
=
,
2
2
Ŷ 2 k ≡

TrHk (Ŷ 2 )
k(k + 3)
= Tr(Y 2 )
dim Hk
12

(exact).

(266)
The classical (continuum) mean square of the levelk moment map on CP 2 is Tr(Y 2 ) k 2 /12, the second
moment of a traceless linear functional against the
Dirichlet(1, 1, 1) distribution of Fubini–Study barycentric
coordinates (E[u2i ] = 1/6, E[ui uj ] = 1/12). Hence
k+3
quantum
=
classical
k
exactly, at all orders, with no 1/k expansion.
Three consequences, each replacing an assertion:
1. the numerator (k + 3) of the finite-k factor is the
exact quantum-to-classical enhancement of the hypercharge second moment — a computed quantity,
not a bundle-degree coincidence;
2. the 12 in π 3/2 /24 = π 3/2 /(2 · 12) belongs to the ζ(0)
of the CP 2 charge trace, −1/24 per unit charge
squared (§X C);
3. Tr(Y 2 ) enters at degree exactly 1, because Eq. (265)
is exactly linear in Tr(Q2 ).
Remark X.3 (Consistency with the Spinc reading). The
canonical Spinc structure on CP 2 has KCP 2 = O(−3),
hence Ldet = K −1 = O(3), and restriction to CP 1 ⊂ CP 2
does send O(k) ⊗ Ldet to O(k + 3). That observation is

The Forced Microsector Fork

At this point there is a forced binary fork, determined
solely by what finite Hilbert space carries the microsector
trace.

1.

Branch A: Regular-Module Microsector (Survives)

Take the finite Hilbert space to be the algebra itself:
HF := A = Md (C),

(268)

with Hilbert–Schmidt inner product ⟨X, Y ⟩ = Tr(X † Y ),
and gauge action by inner derivations:
ada (X) = [a, X].

(269)

a. Trace normalization. The UV-normalized trace is
naturally the democratic normalization per matrix degree
of freedom:
trdem (·) :=

1
TrHF (·).
d2

(270)

b. Conversion to physics normalization. When reporting the final gauge kinetic term in canonical generator
normalization on su(d):
1
trsu (·) = 2
Trsu (·),
(271)
d −1
the conversion factor is forced :
(A)

εadj =

d2
d2 − 1

(272)

For k = 60, d = 64:
(A)

εadj =

4096
= 1.000244 . . .
4095

(273)

74
2.

Branch B: Fermion-Representation Microsector
(Falsified)

If instead the kinetic term trace is taken over a ddimensional fermion representation space HF ∼
= Cd (as
in conventional matter spectral triples), unimodularity
literally removes the identity generator channel, yielding
the drop factor:
d2 − 1
4095
(B)
εadj =
=
= 0.999756 . . .
2
d
4096

(274)

Microsector Lock
By Theorem AD.1 (Appendix AD), the regular-module
microsector completion (Branch A) is the unique minimal
faithful real-bimodule trace carrier on the DFD finite
algebra A = Md (C). Branch B fails the bimodule
consistency requirement.
Forced microsector:
• Hilbert space: HF = A = Md (C) (regular module)
• Dimension: dim(HF ) = d2 = 4096
• Gauge action: inner derivations ada (X) = [a, X]
• UV trace: trdem = (1/d2 ) Tr

F.

Branch Selection by Real-Bimodule Consistency

The apparent fork between the regular-module trace
and the fermion-representation trace is resolved by the
finite real-bimodule structure of the DFD microsector,
not merely by numerical comparison.
The finite Hilbert space used in the closed form for α is
not the observed fermion Hilbert space; it is the vacuum
gauge-frame trace carrier. Since gauge variations act by
inner derivations ada (X) = [a, X], the Hilbert space must
carry faithful left and right actions of A = Md (C) and Aop
satisfying the order-zero condition and mutual commutant
closure. By Theorem AD.1 (Appendix AD), the minimal
such module is
HF = L2 (A) ≃ A = Md (C),

dim HF = d2 ,

(275)
d

while the fermion-representation module HF ≃ C has
commutant C and cannot support a faithful right Aop
action. Branch B is therefore not a valid DFD vacuum microsector trace carrier; it fails the consistency requirement
before any numerical comparison is performed.
Holding all other ingredients fixed (geometry, gF , hypercharge trace, and the finite-k rule Λ3 = k(k + 3)/(k + 4)),
we compute α−1 under both microsector trace choices to
demonstrate the magnitude of the consistency selection.
a. Numerical comparison.
b. Branch A: forced. The regular-module microsector
is the unique minimal faithful real-bimodule trace carrier
(Theorem AD.1) and gives the closed form of §X I, which
lies 0.0049 ppm from the measured value with the open
bracket coefficient; the exact construction of §X C has no
higher heat-kernel term.
c. Branch B: structurally excluded. The fermionrep microsector HF ≃ Cd has commutant only C and
therefore cannot support a faithful right Md (C)op action.
Branch B is excluded independently of the 43 ppm numerical mismatch. The mismatch becomes a redundant
consistency check rather than the basis for selection.
Note: this exclusion concerns only the DFD vacuum
gauge-frame trace carrier. It does not imply that Cd is
invalid for ordinary fermion representations elsewhere in
the framework.

• Factor: BOOST = d2 /(d2 − 1) = 4096/4095
The regular-module branch is therefore not chosen by its
numerical value.

G.

The Closed-Form Chain

The closed form for α rests on the following chain:
a. Closure of kmax = 60. The value kmax =
χ(CP 2 , O(Ngen q1 ) ⊕ O⊕5 ) = 55 + 5 = 60 follows from
the generation count, q1 = 3 (derived under the electroweak identification) and the five hypercharged multiplet types, through the conditional determinant-line count
of Lemma F.11 (conditional on the fixed-kernel reading;
the relation of its degree-zero summands to the hypercharge degrees is open, Rem. F.7); since O(3) = K −1 , the
degree 9 is also Ngen c1 (CP 2 ). Within the finite-symmetry
closure framework adopted in this section, the same value
follows from auxiliary structural postulates, which serve
as a cross-check:
1. The microsector channel symmetry G acts faithfully
on a real three-dimensional generation space.
2. G is orientation-preserving, simple (no hidden normal subgroup) and non-abelian.
3. The channel algebra furnishes exactly five conjugacy
classes, matching the five chiral multiplet types in
one SM generation.
4. Choose the minimal such group.
The icosahedral rotation group G ∼
= A5 is the only nonabelian simple finite group of rotations of real threespace, so postulates 1 and 2 already fix it, postulate
4 is not needed and its five conjugacy classes are an output; kmax = |A5 | = 60. Postulate 2 must say non-abelian,
since the cyclic group of order five satisfies the other conditions. The independent Bridge Lemma (Appendix K 4)
and minimal-padding argument then function as nontrivial consistency checks rather than as hidden tuners. Once
kmax is fixed, only discrete choices remain.
b. Unconditional content. What does not depend
on the auxiliary postulates is the following: once any
integer kmax is fixed, the entire microsector output (α−1 ,
CKM structure, neutrino spectrum) follows with zero

75
TABLE XXVI. Microsector fork: numerical comparison at k = 60, each factor ε entering as 1 + w(ε − 1) with the stated weight
w = 7/80 of Table XXVII. Branch B is shown for completeness; it is excluded by the bimodule theorem (Appendix AD).
Branch

Factor

A (regular-module)
B (fermion-rep, excluded)
Experimental (CODATA 2022)

4096
4095
4095
4096
—

α−1

Residual (ppm)

137.03599985

−0.005

137.03014445

+42.7

137.035999177(21)

—

TABLE XXVII. Chain of the closed form for α−1 .
Component

Value

KCP 2 = O(−3)
−3
Ldet = K −1
O(3)
d=k+4
64
(d − 1)/d
63/64
Nspecies
7
Tr(Y 2 )
10
gF
8
w = Nspecies /(gF · Tr(Y 2 )) 7/80
εadj
4096/4095
α−1

Source

Status

Algebraic geometry theorem
Rigorous
Spinc structure
Rigorous
dim H 0 (O(k + 3))
Rigorous
Numerator: exact CP 2 charge trace (Cor. X.2); d: Eq. (264) Derived
SM SU(2) components
SM content
SM hypercharges
SM content
Spectral triple (J × γ × C)
Derived
Hypercharge weighting
Stated; provenance open (Rem. X.4)
Regular-module trace conversion
Forced

137.03599985 Integers × [1 + w(εadj − 1)]

continuous free parameters. The count above fixes kmax =
60 among the integers; the theory’s numerical output is
then falsifiable against >30 independent measurements.

H.

Sharp Falsifier

The microsector choice HF = A is a testable ontological claim:
“The finite Hilbert space of the DFD Toeplitz
microsector is the algebra itself (Md (C)), not
a fermion representation space (Cd ).”
a. If future work derives HF = Cd from first principles:
• DFD fails by 43 ppm
• Cannot be rescued without fine-tuning
• Theory requires fundamental revision
b.

If future work derives HF = A from first principles:

• BOOST factor is forced, not fitted
App. AD goes part of the way: under the standard-form
real-bimodule conditions (order zero, faithful left and right
actions, mutual commutants) and minimality it excludes
Cd and makes A the carrier; those conditions are adopted
as the spectral-triple consistency requirement, not derived
from the field equation.

4.9 ppb (with the open bracket)

I.

The Closed-Form Result

Collecting all locked ingredients, the fine-structure constant is given by a single equation with no continuous free
parameters:
α−1 =

π 3/2
kmax +3
Tr(Y 2 ) kmax
24
kmax +4


7
× 1+
= 137.036
80·4095

(276)

where:
• π 3/2 /24: the normalization of the exact construction
of §X C: its 1/24 is the ζ(0) of the √
CP 2 charge trace
per unit charge squared, and its π is that of the
terminating heat trace of the ρ-shifted Casimir on
S3
• Tr(Y 2 ) = 10: Standard Model hypercharge trace (3
generations of QL , uR , dR , LL , eR )
• kmax = 60: topological cutoff from the Bridge
Lemma (Spinc index on CP 2 , = |A5 |)
• (kmax + 3)/(kmax + 4) = 63/64: the numerator
(kmax + 3) is the exact quantum-to-classical ratio
of the CP 2 hypercharge second moment (Corollary X.2); the denominator (kmax + 4) = d is the
Toeplitz normalisation of Eq. (264)
• [1 + 7/(80 × 4095)]: regular-module microsector
correction (4095 = 642 − 1 = d2 − 1); exactly equal
2
to 1 + 1/(13 kmax
) — see Remark X.4
The closed form gives α−1 = 137.03599985, 0.0049 ppm
from the measured value; the integers alone give 137.03307,

76
21 ppm below it. The trace in it is the hypercharge
trace, and the step that identifies its coefficient with 1/e2
rather than with the hypercharge coupling 1/g ′2 is not
derived; since 1/e2 = 1/g 2 + 1/g ′2 , with sin2 θW = 3/13
(Theorem Z.1) the hypercharge reading would give α−1 =
(13/10) × 137.033 = 178.14.
Remark X.4 (The bracket carries one integer: an exact
identity). The printed microsector correction admits a
second exact reading, in which its three integers collapse
to one. Since 80 · 4095 = 327600 = 7 · 13 · 3600 =
2
2
(7 · 13) kmax
(equivalently 4095/kmax
= 91/80 exactly, and
4095 = 63 · 65 = (kmax + 3)(kmax + 5)),
1
7
1+
= 1+
2
80 · 4095
13 kmax
1
1
= 1+
sin2 θW = 1 +
,
2
Ngen kmax
46800
(277)
using sin2 θW = 3/13 (Theorem Z.1) and Ngen = 3, so
that (3/13)/3 = 1/13. The integer 13 = 3 + Tr(Y 2 ) is the
Weinberg denominator of Theorem Z.1 and appears as
such in the structural-integer table of Appendix AB 5.
This is a second exact reading, not a correction: the
weight w = Nspecies /(gF Tr(Y 2 )) = 7/80 is the stated
weight of Table XXVII, and both readings evaluate to the
same rational 1/46800. The value of α−1 is unchanged at
137.03599985.
What is and is not established. Writing the metaplectic
(half-form) level κ = k + 3/2, Corollary X.2 becomes
Tr(Y 2 )  2 9 
1 2 1
Ŷ 2 k =
κ −4 ,
dim Hk =
κ −4 ,
12
2
both of which are even series in 1/κ that terminate after
one term. A finite-level correction that kept this symmetry
would be even in 1/κ, with no 1/k term; the printed
2
1/(13kmax
) is even in k, not in κ, so the form is not derived.
The coefficient c is a different matter: because Eq. (265)
is exact at every order, the CP 2 Toeplitz sector generates
no such correction at all, so the coefficient’s provenance is
open and is not claimed here. What Eq. (277) establishes
is that a mechanism reproducing the closed form would
2
have to supply 1/13 in the position of a 1/kmax
coefficient.
The two readings of the bracket agree only at kmax = 60,
where 11k 2 − 640k − 1200 vanishes; they are not one
identity in k. The integers alone give α−1 = 137.03307,
21.4 ppm below the measured value, and with c = 1/13 the
closed form lies 4.9 ppb from it, 32 times the uncertainty
of the CODATA 2022 value; the measured value requires
c = 0.0769053(6). The form of the bracket is not fixed at
the part-per-billion level, so the ppb figure is a property
of this closed form, not a test of the bracket and not an
agreement within experimental error.

J.

Summary

Summary: Convention-Locked α
Result:
α−1 = 137.03599985
(4.9 ppb from CODATA with the open bracket;

(278)

21 ppm from the integers)
Locked conventions:
• Operator: connection Laplacian with parallel
curvature on CP 2 ; the ρ-shifted Casimir ∆ + 1 on
S 3 = SU (2)
• Regulator: heat-kernel (Gaussian) cutoff in the
exact construction; the CP 2 factor enters through
ζ(0)
• Finite-k: Toeplitz truncation with d = k + 4 = 64
• Microsector: regular-module (HF = Md (C))
• Trace: democratic UV → per-generator physics
(BOOST forced)
The fermion-rep microsector is falsified:
• 43 ppm deficit cannot be filled
• Under no-knobs policy, only Branch A survives
Falsification criterion: If HF = Cd is derived from
microsector first principles, DFD’s α prediction fails.

XI.

ATOMIC CLOCK TESTS

Atomic clocks remain one of the sharpest laboratory
probes of DFD. One must distinguish channels. Sameion optical comparisons test the pure electromagneticsector coupling; cross-species atomic ratios primarily
test composition-sensitive structure; and nuclear clocks
uniquely access the strong sector. General relativity predicts exact universality for co-located clocks after the common redshift is removed. DFD’s minimal action predicts
the same universality for co-located clocks (Thm. XI.2);
a residual differential response, if present, is channeldependent and environment-dependent and belongs to
the extended non-metric sector (Sec. XV C).

A.

Local Position Invariance Framework

a. LPI in metric gravity. Local position invariance
(LPI) states that non-gravitational physics is independent
of location in a gravitational potential. In GR, all clocks
redshift in the same way:
∆ν
∆Φ
= 2 .
(279)
ν
c
The universal redshift (279) has been verified to 7 × 10−5
by the GP-A rocket experiment and to ∼10−5 in modern
optical clock comparisons. For a clock ratio R = νA /νB ,

77
the universal GR redshift cancels:
∆R
=0
(GR, co-located clocks).
(280)
R
b. Differential coupling language. A convenient way
to parameterize a possible violation is


∆ν
∆Φ
(281)
= (1 + KA ) 2 ,
ν A
c
so that
∆R
∆Φ
= (KA − KB ) 2 .
R
c

(282)

The observable is therefore the difference in effective
couplings, not the absolute redshift of either clock alone.
B.

Common-Factor Cancellation and Observable
Residuals

a. The structural point. Every local transition frequency can be decomposed as
νA (ψ, a) = U (ψ, a) ν̂A (ψ, a),

(283)

where U (ψ, a) is the common electromagnetic scale
factor shared by all clocks (encoding the universal coupling of ψ to the electromagnetic vacuum), and ν̂A is a
dimensionless structure-dependent residual specific
to transition A.
For any co-located clock ratio,
νA
ν̂A
RAB ≡
=
,
(284)
νB
ν̂B
so the common factor U cancels identically.
Theorem XI.1 (Clock-ratio cancellation of the common
sector). For co-located clocks A, B admitting the factorization (283) with the same common factor U (ψ, a), any
differential LPI observable formed from their ratio depends
only on the residual internal-structure response:
 
νA
δ ln
= δ ln ν̂A − δ ln ν̂B .
(285)
νB
Proof. Insert (283) into RAB = νA /νB to obtain (284).
Taking a logarithmic variation, the universal factor U
cancels algebraically.
Theorem XI.2 (Null of the co-located ratio under the
minimal action). Let matter couple to ψ only through
the physical metric (12)—the minimally coupled Smatter
of (20), with Sint at hTT
ij = 0—through a diffeomorphisminvariant action (regulator included), and let ψ = ψ0 be
constant over the co-located comparison. The map t =
e+ψ0 /2 T , x = e−ψ0 /2 X pulls g̃µν back to diag(−c2 , 1, 1, 1)
exactly, so every transition frequency of every species
carries the same factor and
d
νA
ln
=0
(286)
dψ0 νB
obs
identically, to all orders; equivalently KA
= 0 for every
A in (289). A nonzero co-located ratio response therefore

requires an operator outside Smatter , a non-metric term
of the form (352); each λI in (289) is the coefficient of
such a term, and the ψ-linear vertex of the minimally coupled electromagnetic action is the energy-density sourcing
vertex (Thm. II.1, Eq. (R13)), not a varying-α operator.
If ψ varies over the comparison, the gradient and tidal
remainders over a laboratory are ≲ 10−43 and the adiabatic remainder ψ̇0 /ω is ∼10−32 , in fractional frequency.
Special cases already in print: Thm. ZS.3 (homogeneous
cosmology) and the metric-level universality of Sec. XV B
(free fall).
Corollary XI.3 (Kennedy–Thorndike null (minimal sector)). Under the hypotheses of Thm. XI.2, let the laboratory move with any constant velocity V relative to the
preferred (CMB) frame. Every co-located proper frequency
ratio is then independent of V,
∂
νcav
BKT ≡
= 0,
(287)
ln
∂(V 2 /c2 ) νatom ψ0
identically, to all orders in V /c: the Robertson–Mansouri–
Sexl combination βRMS − αRMS − 1 vanishes, and the
same map gives the Michelson–Morley (isotropy) null. A
Kennedy–Thorndike signal therefore requires either a ψ
gradient across the apparatus or an operator coupling matter to the preferred-frame flow uµ , which neither Smatter
nor the non-metric family (352) (scalars in ψ, boost-blind
at fixed ψ by the same map) contains.
Proof. The map t = e+ψ0 /2 T , x = e−ψ0 /2 X of Thm. XI.2
makes the physical metric (12) exactly Minkowskian; the
optical metric (29) is e−ψ times it (App. AN), so the
light cone is shared, and the Maxwell action is conformally invariant. Every minimally coupled matter
and radiation action is therefore its Minkowski form in
(T, X) p
and is Poincaré invariant there, with dτ /dt =
e−ψ0 /2 1 − V 2 /c2 from (12) and V = eψ0 dx/dt =
dX/dT the physical velocity. A Lorentz boost of (T, X)
brings the laboratory to rest, where every proper frequency is the same function of the local physics as at
V = 0; the ratio is thus V-independent, and a Vdependence can enter only through an operator not built
from g̃µν and ψ alone, or through ∇ψ ̸= 0 over the apparatus.
Remark XI.4 (Unequal-arm round trip, remainders, and
what the minimal sector predicts). Explicitly, a cavity of proper length L along V has coordinate length
L/γ and round-trip coordinate time (L/γ)[(c − V )−1 +
(c + V )−1 ] = 2γL/c, i.e. proper time 2L/c; a transverse
arm gives the same, so an unequal-arm interferometer
has ∆τrt = 2(L1 − L2 )/c with ∂∆τrt /∂V = 0. Corollary XI.3 is the matter-sector counterpart of Thm. IV.1:
the preferred foliation enters only the gravitational sector,
through the temporal invariant ∆ of Eq. (Q4), and at
fixed ψ the matter–radiation sector carries no trace of
it. Three remainders survive, all from the corpus’s own
inputs (w ≡ V in Thm. IV.1). (i) The preferred-frame
g00 term of Thm. IV.1(ii), αeff (V 2 /c2 )(U/c2 ), modulates

78
the laboratory potential with the annual change of the
boost, δ(V 2 /c2 ) = 2VCMB v⊕ /c2 = 2.4 × 10−7 , by at
most 3.8 × 10−8 × 2.4 × 10−7 × 9.9 × 10−9 = 9 × 10−23
in fractional frequency (Eq. (85) at 1 AU, unsaturated;
≈ 6×10−33 with the saturation factor µ′ (∆) ∼ 6.6×10−11
of Thm. IV.1(iii)); this modulation is common to all colocated clocks and cancels in the ratio by Thm. XI.2.
(ii) The tidal variation of ψ over a cavity of length L is
∼ 2gL2 /(c2 R⊕ ) = 3 × 10−23 for L = 1 m (the ≲ 10−43
of Thm. XI.2 is this term at L = aB , the Bohr radius).
(iii) The species couplings of Eq. (352) are boost-blind
at fixed ψ and enter only through ∇ψ. All lie far below
the measured null, βRMS − αRMS − 1 = −1.7(4.0) × 10−8
(sidereal; weighted mean −4.8(3.7) × 10−8 ) [61], which
improved on the 2 × 10−13 fractional-frequency null of
Ref. [62]. Two corollaries. (a) In Einstein-equivalenceprinciple language the minimal sector has universal free
fall, exact local Lorentz invariance at fixed ψ, and localposition-invariance residuals only through the species
couplings of Eq. (352): a co-located ratio fitted as
δR/R = KAB ∆Φ/c2 + BAB ∆V 2 /c2 + CAB (n̂ · V/c)
has BAB = CAB = 0 predicted exactly, with KAB the
only admissible residual (zero under the minimal action,
Thm. XI.2). (b) A spatially uniform ψ0 is removed by
the rescaling and is unobservable in the minimal local sector, so local minimal-sector observables depend on ψ only
through its derivatives, through comparison between separated regions, or through the non-metric species couplings
(the local counterpart of Thm. ZS.3).
Theorem XI.2 is the general case of this cancellation and
of the cavity–atom cancellation proven in Sec. XII A: under the minimal action no residual survives in a co-located
ratio, so a structure-dependent residual, if observed, is a
coefficient of the extended non-metric sector of Sec. XV C.
b. Common-sector coupling. The DFD α-relation
kα = α2 /(2π) (Sec. VIII) sets the coupling of ψ to the
common electromagnetic scale:
α2
≈ 8.5 × 10−6 , (288)
2π
where Σ(y) is the screening factor (Sec. XI C). This coupling is not directly observable in ratio experiments because Kcom cancels between numerator and denominator.
c. Observable residual couplings. What ratio experiments measure is the residual structure response:
h
i
(A)
αs
obs
α
KA
(y) = Σ(y) λα SeA
+ λs SA
+ λN CN + λe Ce(A) ,
Kcom (y) = kα Σ(y),

kα =

(289)
where
α
SA
≡

α ∂νA
νA ∂α

(290)

α
α
is the electromagnetic sensitivity, SeA
≡ SA
− S̄ α is the
centered electromagnetic sensitivity (with S̄ α absorbed
αs
into the common sector), SA
is the strong-sector sensi(A)
(A)
tivity, CN and Ce are effective nuclear and electronic
family charges, and the λI are channel coupling strengths.
I
Each λI SA
is the clock-transition form of the non-metric

coefficient βI of (352), the transition analogue of the
bound-state mass shift (353); under the minimal action
(Thm. XI.2) every βI = 0, and the default expectation is
the null of Sec. XV H.
The total clock coefficient is
obs
KA = Kcom + KA
,

(291)

but the observable ratio shift is


∆RAB
obs
obs ∆Φ
.
(292)
= KA
− KB
RAB
c2
Equations (288)–(291) are written per unit ψ, the unit
in which kα is defined (Appendix P). Since Φ = −c2 ψ/2,
a change ∆Φ/c2 moves ψ by 2|∆Φ|/c2 , so in the ∆Φ/c2
slot of Eqs. (281) and (292) each coefficient enters with
twice its magnitude.
d. Why this resolves the pure-α tension. The Yb+
E3/E2 same-ion null (Sec. XI D) constrains λα , the residual pure-α channel coupling — not the common-sector
kα . The conditional value kα = α2 /(2π) (Appendix P,
one-gauge-vertex axiom) survives as the coupling to the
shared electromagnetic scale. Same-ion tests bound only
α
α
the centered sensitivity difference SeE3
− SeE2
, which is a
statement about residual structure, not about the universal ψ–EM coupling.
e. Microsector suppression hierarchy. The residual
channel couplings are set by the microsector class-breaking
parameter
ϵH ≡

Ngen
3
1
=
=
.
kmax
60
20

(293)

The hierarchy follows from class-breaking order: the
common-sector coupling requires zero class insertions;
composition-sensitive residuals require one; and the sameion pure-α residual requires two (because the one-insertion
piece cancels in same-ion ratios). This gives:
1
λα ≈ ϵ2H kαcom ≈
× 8.5 × 10−6 ≈ 2.1 × 10−8 , (294)
400
1
× 8.5 × 10−6 ≈ 4.2 × 10−7 .
20
(295)
The pure-α residual λα ≈ 2.1 × 10−8 is compared against
the Yb+ E3/E2 gravitational-LPI limit |(c2 /α) dα/dΦ| ≲
3 × 10−9 (Filzinger et al. 2023, −2.4(3.0) × 10−9 ), which
supersedes the earlier 3.2 × 10−8 bound. Screened at
the local acceleration (Table XXIX), the residual enters
the same-ion observable as 2λα Σ(⊕) ≈ 1.5 × 10−13 per
unit ∆Φ/c2 , four orders of magnitude below this limit,
which therefore does not test it. The composition/strong
couplings λN,e,s ≈ 4.2 × 10−7 are ∼20× larger.
f. Channel structure. Different experiments project
out different pieces of Eq. (289):
λN,e,s ≈ ϵH kαcom ≈

1. Same-ion comparisons (e.g. Yb+ E3/E2) cancel
composition terms by construction and isolate λα .
2. Cross-species atomic comparisons are dominantly sensitive to λN and λe because ∆CN and
∆Ce are generically nonzero.

79
3. Nuclear clocks add a qualitatively new strongsector contribution through λs .
g. Indicative α sensitivities. The electromagnetic
sensitivity coefficients remain useful bookkeeping quanti(α)
ties. The column KA gives the common-sector pure-α
α
value kα SA ; ratio experiments are sensitive only to the
centered residuals.
TABLE XXVIII. Electromagnetic sensitivities and commonsector coupling values.
Transition

Type

α
SA

(α)

KA

(×10−5 )

133

Cs hyperfine
MW
+2.83
+2.4
Rb hyperfine
MW
+2.34
+2.0
1
H 1S–2S
Opt
≈0
≈0
87
Sr
Opt
+0.06
+0.05
171
Yb
Opt
+0.31
+0.26
171
Yb+ E2
Opt
+1.0
+0.85
171
Yb+ E3
Opt
−5.95
−5.1
199
Hg+
Opt
−3.2
−2.7
27
Al+
Opt
+0.008
+0.007
229
Th nuclear [63] Nucl. 5900 ± 2300 (strong sector)
87

C.

Screening: Derivation from a Response
Functional

The screening factor Σ(y) appearing in Eq. (289) determines how the local gravitational environment suppresses
clock–ψ coupling. Rather than treating this as a heuristic,
we derive it from an explicit coherence-response principle.
a. Response functional. At acceleration a (with y ≡
a/a0 ), the effective noise occupation of the local vacuum
combines the de Sitter background and the Unruh contribution:

b. Connection to earlier notation. Below, µLPI ≡
Σ. The common-sector effective coupling from Eq. (288)
becomes:
α2
p
.
(299)
kαeff (a) = kα Σ(a/a0 ) =
2π 1 + a/a0

TABLE XXIX. Screening factor Σ(y) and common-sector effective coupling across environments.
Environment

a (m/s2 ) y = a/a0

Σ(y)

eff
kα

Galactic outskirts
10−10
0.84
0.738
6.26 × 10−6
−6
3
−2
Outer solar system 10
8.4 × 10 1.09 × 10
9.27 × 10−8
−3
7
−4
Solar orbit (1 AU) 6 × 10
5.01 × 10 1.41 × 10
1.20 × 10−9
Earth surface
9.8
8.19 × 1010 3.49 × 10−6 2.96 × 10−11

c. Implication for experiments. Terrestrial opticalclock tests are therefore much more strongly screened
than a naive solar-orbit estimate would suggest.
d. Falsifiable predictions from µLPI (extended sector).
The y −1/2 scaling makes specific predictions for future
off-Earth experiments:
1. Earth-based clocks: At a ≈ 10 m/s2 , coupling should be ∼40× smaller than at solar orbit—
consistent with null terrestrial LPI tests.
2. Outer solar system: At Jupiter’s orbit (a ≈
2 × 10−4 m/s2 ), coupling should be ∼5× larger
than at 1 AU.
Deviation from the y −1/2 power law would constrain or
falsify the Unruh screening mechanism.

D.

The Same-Ion E3/E2 Constraint

(297)

The PTB Yb+ experiment comparing the E2 and E3
transitions is the cleanest same-ion constraint because
it removes composition differences by design [64]. Both
transitions live in the same ion, so ∆CN = ∆Ce = 0
and any signal primarily probes the pure electromagnetic
channel.
a. Structure of the test. For the same ion,
α
α
∆KE3/E2 = λα (SeE3
− SeE2
) = λα × (−6.95).
(300)

The first term penalizes coherent coupling against the
noise floor; the logarithmic term enforces positivity and
represents the entropic cost of decoupling. Stationarity
gives:

Lange et al. measured the gravitational coupling parameter
 2

c dα
= 14(11) × 10−9 ,
(301)
α dΦ

Neff (y) = 1 + y.

(296)

The coherent response amplitude Σ is determined by
minimizing the free energy of quantum-sector coupling in
this noise background:
Fresp (Σ; y) = 12 (1 + y) Σ2 − ln Σ.

∂Fresp
= (1 + y)Σ − Σ−1 = 0
∂Σ

1
.
1+y
(298)
The screening law is the unique stationary point of this
response functional. The effective coupling of any clock
channel I is then Σ(y) λI .
⇒

Σ(y) = √

consistent with zero, which corresponds to a conservative
one-sided 95% bound
|λα | ≲ 1.6 × 10−8 .

(302)

Filzinger et al. (2023) tighten the underlying limit to
3 × 10−9 , i.e. |λα | ≲ 1.5 × 10−9 (Sec. XI B).

80

• the pure electromagnetic-sector proposal is tightly
bounded;

E.

(i)

Ki ≈ kN CN + ke Ce(i)

(303)

for ordinary atomic clocks once the pure-α piece is
bounded to be subdominant.
a. Indicative scale. The indicative scales carried into
Table XXX are:
• Yb/Sr and Al+ /Yb: ∼ 10−17
• Yb+ (E3)/Sr: ∼ 4 × 10−16 ; Hg+ /Sr: ∼ 10−16
• Cs/Sr: ∼ 10−16 to 10−15 depending on channel
normalization.
These scales are generated by the unscreened pure-α
coefficient kα ∆S α reduced by the ∼40× Earth/solarorbit ratio, not by the composition terms, and the same
generator gives a Yb+ E3/E2 amplitude ∼140× above
the Filzinger limit; under Eq. (289) with Σ(⊕) every row
is ≲ 10−20 . They are indicative scales of the extended
channel, null under the minimal action (Thm. XI.2).
b. Cs/Sr: explicit worked example. This channel is
one of the highest near-term priorities. The pure-α sensitivity difference is
α
α
α
∆SCs-Sr
= SCs
− SSr
= 2.83 − 0.06 = 2.77.

(304)

At the pure-α leading-term level (Table XXVIII), the
predicted differential coupling is
(α)

4

Perihelion

2
0
2
4
6

[×10 15]

8

0

100

0

50

∆KCs-Sr = kα × ∆S α = 8.5 × 10−6 × 2.77 ≈ 2.35 × 10−5 ,
(305)
giving, under Eq. (299) with the annual |∆ψ⊙ | =
2|∆Φ⊙ |/c2 ≈ 3.3 × 10−10 , an annual modulation amplitude of ∼1 × 10−18 at 1 AU and ∼3 × 10−20 at
Earth-surface screening (Table XXIX); Fig. 10 uses a
different screening normalization, not these amplitudes.
Composition-sensitive terms may contribute additional
signal depending on ∆CN and ∆Ce .

200

300

400

500

600

700

100

150

200

250

300

350

Days from Jan 1

5
0
5

Orbital
phase
(degrees
from
Jan 1)
SCHEMATIC:
Predicted
signal based
on DFD
parameters

Cross-Species Atomic Comparisons

For different species A/B, the composition terms in
Eq. (291) generically survive. This is why cross-species
atomic ratios remain important even after the same-ion
E3/E2 null. In the phenomenological “family + clock”
language, one writes

Simulated data
DFD prediction
GR prediction

KCs KSr = 2.35e 05
Amplitude = 3.9×10 15

6

• cross-species atomic comparisons remain open because composition-sensitive terms survive there;
• nuclear clocks remain open because same-ion optical comparisons are essentially blind to the strong
channel.

Predicted Clock Anomaly Signal (Cs/Sr frequency ratio)

8

( Cs/ Sr) [×10 15]

b. What this means for DFD. The same-ion null does
not exclude the channel-resolved DFD clock program. It
excludes the naive claim that one universal pure-α law
controls the whole clock sector. In particular:

FIG. 10. Illustrative Cs/Sr annual-modulation curve (extended
channel). Its screening normalization is not the bound-matter
clock’s, whose screening factors are those in Table XXIX,
and the plotted amplitude is an illustration, not a precision
prediction. GR predicts null (gray dashed).

c. Prior data: Blatt et al. 2008. The 2008 multilaboratory Cs/Sr annual amplitude ySr = (−1.9 ± 3.0) ×
10−15 [65] is consistent with zero.
d. Methodological note. Year-long global fits with
flexible drift models can absorb annual signals into nuisance parameters, while windowed perihelion analyses are
more sensitive to the specific DFD phase signature but
more vulnerable to drift contamination. Both approaches
should be applied to any dedicated campaign and their
results compared.
e. ROCIT and existing hints. The ROCIT ion–
neutral analyses remain interesting because they point
at the very type of cross-sector comparison the channelresolved picture says should be informative. ROCIT-like
results are suggestive rather than definitive: they motivate
focused reanalysis and replication, but they are not the
sole pillar of the clock case.

1.

ROCIT Statistical Detail

For completeness, the full statistical methodology is
recorded here so that independent groups can replicate
the analysis. The complete regression scripts, figures, and
derived outputs are publicly archived [66]; the accompanying analysis paper is Ref. [67].
a. Reported amplitude: Yb+ /Sr. The Yb+ (E3)/Sr
ion–neutral ratio exhibits [67, 68]:
AYb+ /Sr = (−1.045 ± 0.078) × 10−17 ,
Z = 13.5σ (formal error),

6.1σ (leave-one-day-out error).

(306)
The amplitude is phase-locked to Earth’s perihelion (January), corresponding to maximum solar gravitational potential.

81
b.

Regression model.
y(t) = β0 + β1 t + A b(t) + ϵ(t),

(307)

where b(t) is the orthogonalized Kepler driver (solar potential template) with unit RMS, constructed from Earth’s
mean anomaly with perihelion at phase zero.
c. Uncertainty
estimation. Leave-one-day-out
LODO
(LODO) jackknife gives σA
≈ 1.7 × 10−18 ; wild
bootstrap of residuals, sign-permutation, and day-shift
resampling give empirical pemp ≈ 2 × 10−4 .
d. Phase robustness. Regression on alternative phase
hypotheses confirms solar specificity:
Aspring eq. = (−0.18 ± 0.81) × 10−17 , Z = 0.22σ,
Afall eq. = (+0.09 ± 0.76) × 10−17 , Z = 0.12σ.

(308)

All non-perihelion phases are consistent with zero.
Neutral–neutral ratios from independent SYRTE measurements are also null: Aneut-neut = (0.4 ± 7.3) × 10−17 ,
p = 0.58.
e. Channel-resolved interpretation. In the channelresolved language of Eq. (291), the ROCIT signal probes
(A)
(A)
the composition-sensitive terms (kN CN +ke Ce ) rather
than the pure-α sector. Using the unit-RMS Kepler driver
normalization with σ(∆Φ/c2 ) ≈ 1.2×10−10 , the measured
amplitude corresponds to 2|Kion − Kneut | ≈ 9 × 10−8
per unit ∆Φ/c2 . Because this is a distinct compositionsensitive channel, it is not identified with the pure-α kαeff
column in Table XXIX. The same-ion pure-α null does
not test this channel, but the inferred coupling is at least
104 times the screened composition coupling 2λN Σ(⊕) ≈
3.0 × 10−12 per unit ∆Φ/c2 (Eq. (295), Table XXIX), so
the screened extended channel does not account for the
amplitude.

F.

Nuclear Clocks: the Strong-Sector Channel

The 229 Th nuclear isomer is qualitatively different from
ordinary atomic clocks. Its transition energy sits near
a cancellation between Coulomb and hadronic contributions, making it sensitive to the strong sector through
dimensional transmutation.
a. Strong-sector
amplification. A
convenient
parametrization is
δXq
2π δαs
≈−
,
(309)
Xq
b0 αs αs
with
2π
≈ 6.9
b0 αs

(310)

for αs (MZ ) ≈ 0.118 and b0 = 23/3. Combined with
Flambaum-style nuclear sensitivity coefficients of order
|Sq | ∼ 104 , this makes the nuclear clock the natural place
to look for strong-sector scalar couplings.
In the same screened notation used for the electromagnetic channel, with ks a matching coefficient of the

extended strong channel, the strong-sector effective coupling is
kseff (a) = ks µLPI (a/a0 ) =

αs2
µLPI (a/a0 ),
2π

(311)

so that at Earth’s surface2
kseff (⊕) ≈ 7.7 × 10−9 .

(312)
4

Combining Eqs. (309) and (312) with |Sq | ∼ 10 and
the annual half-amplitude of the solar ψ at Earth,
|∆ψ⊙ | = 2|∆Φ⊙ |/c2 ≈ 3.3 × 10−10 (Φ = −c2 ψ/2), gives a
screened Th-229 estimate of about 350 Hz half-amplitude
at αs (MZ ), rising to about 0.9–1.5 kHz at the nuclearscale αs .
b. What the newer data changed. The 2026 Ooi et al.
reproducibility paper [69], together with the measured Th229 electromagnetic sensitivity from Beeks et al. [63] and
the strong-sector amplification logic of Flambaum [70],
materially sharpens the status of the Th-229 channel. At
195 K, with the first-order thermal sensitivity nulled near
196(5) K, they report frequency reproducibility of 220
Hz over 7 months for two differently doped 229 Th:CaF2
crystals. Interpreted conservatively, this means:
1. the unscreened strong-sector prediction (∼ 100
MHz half-amplitude with ks = αs2 /2π, App. G) is
excluded by about five orders of magnitude;
2. the simplest screened strong-sector estimate
(about 350 Hz half-amplitude at αs (MZ ), about 0.9–
1.5 kHz at the nuclear-scale αs ) lies from below the
conservative annual bound derived from the Ooi
data to about twice it3 and is therefore not excluded
at αs (MZ ), while its nuclear-scale range awaits a
formal perihelion-fixed cosine fit;
3. the surviving window for a genuine annual signal is
pushed down into the rough range
26 Hz ≲ δνb ≲ O(1 kHz),

(313)

with the lower end set by the composition/family
floor and the upper end set by the Ooi reproducibility ceiling.

2 This uses α (M ) = 0.118. Running to the nuclear scale relevant
s
Z
to Th-229 gives αs ≈ 0.3–0.5, which would increase kseff ∝ α2s

by a factor of 6–18 and lower the amplification of Eq. (309) by
2.5–4.2, so the estimate below rises by 2.5–4.2 (Sec. IX E). This
scale ambiguity sets the range of the screened estimate below.
3 The Ooi 220 Hz figure is frequency reproducibility (scatter across
measurements over 7 months), not a fitted annual cosine amplitude bound. To map reproducibility to an annual bound:
(i) the 7-month baseline covers ∼60% of one annual cycle, degrading cosine-fit sensitivity by ∼2×; (ii) the scatter includes
systematic contributions (crystal √
dependence, thermal residuals) that do not average down as 1/ N , adding a ∼2–3× floor
factor; (iii) the peak-to-peak range of a cosine is 2A, so the
amplitude A is half the peak-to-peak. Conservative example:
Abound ≈ 220 Hz × 2 × 3/2 ≈ 660 Hz. Moderate: Abound ≈ 1–
2.5 kHz. The 350 Hz estimate lies below all of these bounds; the
0.9–1.5 kHz range lies below the upper end of the moderate bound
and reaches 1.3–2.3× the conservative one.

82
The 2026 result therefore excludes the unscreened amplitude but does not eliminate nuclear clocks.
c. Thermal-systematics control. A sharper thermal
analysis is possible because Higgins et al. measured the
line shifts at three temperatures and identified the nearzero-crossing behavior around T0 = 196(5) K [71]. Near
that operating point, line b is unusually temperatureinsensitive while line c remains much more responsive.
This suggests a co-thermometry diagnostic: any genuine
gravitational annual modulation should appear as a common fractional modulation in the hyperfine-averaged nuclear frequency, whereas a residual thermal drift would
imprint a much larger correlated signal in line c.
d. EFG-free combination. An especially clean observable is the hyperfine-averaged, electric-field-gradient-free
combination of the resolved quadrupole lines,


EFG-free
νTh
= 16 ν3/2→1/2 + 2ν5/2→3/2 + 2ν1/2→1/2 + ν3/2→3/2 ,

(314)
which cancels the leading crystal-field splitting while preserving any true nuclear fractional modulation. A future dedicated annual campaign should analyze both this
EFG-free combination and the line-c co-thermometer in
parallel.
e. Interpretation. The nuclear-clock channel therefore remains decisive for the extended strong channel.
The experiment probes a residual window rather than a
large expected signal.
f. Beyond 229 Th: the 187 Re nuclear sensitivity target.
The Flambaum nuclear sensitivity formalism [70, 72] predicts κq ∝ n/Q for beta decays, placing ultra-low-Q transitions at the top of the hierarchy. 187 Re (Q = 2.47 keV, the
lowest known β-emitter Q-value) achieves κq ≈ 19,000—
roughly 2× the 229 Th sensitivity. A half-life measurement
at fractional precision 10−6 , repeated at different orbital
phases, would constrain kqeff < 0.2. This complements
the 229 Th nuclear clock: 187 Re probes the strong-sector
coupling through a completely different experimental technique (calorimetric or mass-spectrometric rather than
optical frequency comparison), providing independent
confirmation or falsification. A multi-isotope ratio test—
simultaneously monitoring two isotopes with different
κq in the same facility—would eliminate environmental
systematics by design and directly probe composition
dependence.

G.

Channel-Resolved Prediction Table

Table XXX collects the current channel logic in one
place.
The most important conceptual point of Table XXX
is that same-ion nulls and cross-species signals are not
contradictory in the extended channel. They are precisely what a channel-resolved extended sector allows; the
minimal action predicts null in both (Thm. XI.2).

H.

Empirical Checks and Current Status

The status of the clock sector is:
• PTB E3/E2: strong quantitative bound on any
universal pure-α coupling law.
• BACON optical network: extremely stringent
null/near-null behavior in ordinary optical-clock ratios, with direct implications for cavity–atom residuals.
• Ooi 2026: nuclear-clock reproducibility already excludes the unscreened strong-channel amplitude and
does not exclude the screened estimate at αs (MZ )
(about 350 Hz), while the nuclear-scale range (about
0.9–1.5 kHz) reaches 1.3–2.3 times the conservative
annual bound.
• ROCIT ion–neutral analyses (Sec. XI E 1):
a perihelion-locked amplitude in Yb+ /Sr (13.5σ
formal, 6.1σ leave-one-day-out), which neither the
minimal action nor the screened extended channel
predicts. Suggestive rather than definitive pending
replication, but the full statistical methodology is
archived for independent verification.
No single formula is asked to fit every channel at once.

I.

Experimental Priorities

The experimental ordering is:
1. Th-229/Sr and related nuclear-clock reanalyses. This is the unique strong-sector channel and
carries a sharply delimited surviving window.
2. Cross-species atomic comparisons. Hg/Sr,
Yb+ /Sr, Yb/Sr, Al+ /Yb, and Cs/Sr map the
composition-sensitive sector.
3. Same-ion null checks. These continue to pin
down the pure electromagnetic channel and prevent the theory from smearing everything into one
effective constant.
4. Cavity–atom residual tests. Important, but
with geometric cancellation at tree level they are
not the first short-horizon discriminator; their natural role is ultra-clean residual testing at very high
precision.

83
TABLE XXX. Channel-resolved clock comparison guide for the extended non-metric channel (Sec. XV C); the minimal action
predicts a null in every row (Thm. XI.2). Amplitudes are indicative scales with a coefficient the corpus does not derive. “Open”
= live test of the extended channel; “bounded” = simplest version under pressure.
Comparison Dominant channel

Scale (atomic rows: What it tests
unscreened, excluded;
screened by
Eq. (289), ≲ 10−20 )

Yb/Sr
—
∼ 10−17
+
Al /Yb
—
∼ 10−17
Yb+ (E3)/Sr —
∼ 4 × 10−16
+
Hg /Sr
—
∼ 10−16
Cs/Sr
HF
∼ 10−16 –10−15
229
Th/Sr
strong + comp. floor 26 Hz–kHz window
(extended strong
channel)
Yb+ E3/E2 pure α only
null expected

Status

cross-species residual
optical-network null check
ion–neutral response
EM sensitivity contrast
MW/optical cross-check
nuclear strong sector

open
open
open
open
open
decisive/bounded

same-ion bound on the extended α channel bounded

Clock-Sector Summary

a. Step 1: optical metric and constitutive relations.
DFD posits the optical metric

What is solid: same-ion optical clocks strongly
constrain any pure universal kα law; under the minimal
action co-located ratios are null (Thm. XI.2); clock
phenomenology must be channel-resolved; nuclear clocks
are the unique strong-sector probe of the extended
channel.
What is under pressure: the unscreened Th-229
amplitude formula is too large in light of Ooi 2026; the
screened estimate at αs (MZ ) (about 350 Hz) is not
excluded, and the nuclear-scale range (about 0.9–1.5 kHz)
reaches 1.3–2.3 times the conservative annual bound.
What remains decisive for the extended channel:
the surviving Th-229 window, plus cross-species atomic
campaigns that isolate composition-sensitive residuals.

c2 2
dt + dx2 ,
n = eψ .
(315)
n2
Through the Tamm–Plebanski construction, this metric
defines effective vacuum constitutive relations

XII.

CAVITY-ATOM REDSHIFT TESTS

The cavity–atom comparison is part of the DFD laboratory program as a residual test. Slowing light while
holding the cavity spacer fixed would give an order-unity
LPI slope, but that is not the DFD calculation: the
same optical metric that changes photon propagation also
changes Coulomb binding, lattice spacing, and hence the
cavity length, so the leading geometric response of cavity
and atomic sectors cancels at tree level.
What remains is a residual, screened signal. This makes
the cavity–atom channel harder as an experiment but also
cleaner as a precision residual test.

A.

Formal Constitutive Proof of the Cancellation

The cancellation can be organized as a short formal
derivation.

ds̃2 = −

εeff = ε0 e+ψ ,

µeff = µ0 e+ψ .

(316)

The medium is impedance-matched, and the coordinate
phase velocity is
1
vph = √
= ce−ψ .
(317)
εeff µeff
b. Step 2: Coulomb binding changes with the same
constitutive chain. Virtual photons feel the same optical
medium, so the static Coulomb potential scales as
e2
e2 −ψ
=
e .
(318)
4πεeff r
4πε0 r
The local fine-structure constant at tree level is therefore
unchanged:
V (r) =

e2
= α0 ,
(319)
4πεeff ℏ c e−ψ
because the factors from εeff and the coordinate phase
speed c e−ψ cancel.
c. Step 3: the atomic length scale in coordinate units.
With α unchanged at tree level, the Bohr radius is a fixed
proper length; on the physical metric (12) its coordinate
length is
α(ψ) =

(0)

a0 (ψ) = a0 e−ψ/2 ,

(320)

like that of every minimally coupled length standard
(Thm. XI.2).
d. Step 4: the cavity length follows the same electromagnetic scale. A Fabry–Pérot cavity resonance obeys,
in coordinate time,
fcav =

m c e−ψ
.
2L(ψ)

(321)

84
For an electromagnetic solid spacer, the lattice constant
and therefore L scale with the Bohr radius, so L ∝ e−ψ/2
while the coordinate phase speed is c e−ψ . Hence
e−ψ
= e−ψ/2 .
(322)
e−ψ/2
Atomic transition frequencies carry the same factor in
coordinate time, fatom ∝ e−ψ/2 (Thm. XI.2), up to
channel-dependent residual sensitivities of the extended
non-metric sector.
e. Convention note. The e−ψ/2 scaling above is in
coordinate time and is the gravitational redshift factor of the physical metric g00 = −e−ψ quoted in Section IV H 4: in the rescaled coordinates of Thm. XI.2
every minimally coupled standard is Minkowskian, so
the same universal coordinate-to-proper conversion multiplies both cavity and atomic frequencies. Therefore
the ratio R = fcav /fatom is convention-independent, and
the tree-level cancellation holds regardless of which clock
convention is adopted; Cor. XI.3 extends it to boosts.
f. Tree-level result. The leading geometric ratio is
therefore constant:
fcav ∝

R≡

fcav
= const. at tree level,
fatom

ξgeom = 0

(323)
The universal geometric redshift cancels. Any surviving
cavity–atom signal must come from a residual channel,
not from an order-unity tree-level effect.

nario is therefore ruled out by more than seven orders of
magnitude.
b. Check 2: PTB Yb+ E3/E2. The same-ion E3/E2
comparison [64] is exactly the sort of experiment that
would have seen an unscreened cavity-style signal if it
were real. Instead, the observed result is null at a level
that rules out the naive geometric expectation by more
than seven orders of magnitude and forces any viable
theory into a much smaller residual regime.
c. Check 3: BACON optical network. The BACON
collaboration measured Al+ /Sr/Yb frequency ratios with
uncertainties at or below 8×10−18 [73]. A naive geometric
annual signal in Yb/Sr would be of order 8 × 10−11 , far
larger than the observed stability. This is effectively a
ten-million-fold exclusion of the unscreened geometric
cavity/atom picture.
d. Check 4: a direct cavity–atom record. JILA compared a cryogenic silicon cavity with a Sr lattice clock for
205 days, from July 2025 to January 2026 [74], while the
solar potential changed by about 3 × 10−10 in units of c2 .
The cavity frequency shows only a smooth, nearly linear
drift of about 1.4 × 10−12 over the whole record, far below
the ∼ 3 × 10−10 that an order-unity cavity–atom slope
would produce.
These four checks all point the same way: the orderunity tree-level picture is excluded, and only a screened
residual remains.

D.
B.

Once the tree-level cancellation is enforced, the cavity–
atom observable is naturally written as
∆R
res ∆Φ
= ξLPI
,
(324)
R
c2
with
GR
ξLPI
= 0,

DFD
ξLPI
= 0 (minimal action).

(325)

For ordinary terrestrial experiments an extended-sector
residual is small because the local environment sits deep
in the screened regime.
a. Interpretation. The cavity–atom comparison remains valuable because it can isolate a non-metric residual
if the sensitivity frontier is pushed far enough.

C.

BACON and the Screening Regime

What Survives Physically

Four Independent Empirical Checks

Four independent data streams support the geometriccancellation picture.
a. Check 1: fine-structure splitting. If the geometric
unscreened picture were right, the ratio of two transitions
with different α sensitivities inside the same atom would
show an annual modulation of order ∆S α δψannual ∼
10−10 . Precision spectroscopy constrains such effects at
the ≲ 10−17 level. The naive unscreened geometric sce-

Under Eq. (299), evaluated at the local acceleration
(Table XXIX), the Yb/Sr annual signal with ∆S α ≈ 0.25
would be
 
δR
≈ 0.25 × kαeff × 3.3 × 10−10
R Yb/Sr
≈ 1 × 10−19 (1 AU),

2 × 10−21 (⊕),
(326)
with kαeff = 1.20 × 10−9 and 2.96 × 10−11 respectively,
against BACON’s Yb/Sr weighted scatter of about
1.1 × 10−17 . BACON therefore excludes neither evaluation point and does not select between them; the local
evaluation is adopted from Sec. XI C. Under the minimal
action the co-located ratio is null in any case (Thm. XI.2),
and the extended-sector coefficient λα is not derived, so
no numerical amplitude is printed for this channel.

85
E.

Sector-Resolved Parameterization

The cavity–atom channel benefits from a sectoral bookkeeping language. Write

(M )
∆f
(M ) ∆Φ
(327)
= (αw − αL ) 2 ,
f cav
c

(S)
∆f
(S) ∆Φ
(328)
= αatom 2 .
f atom
c
Only differences are observable. With two cavity materials
(for example ULE and Si) and two atomic species (for
example Sr and Yb), the directly identifiable combinations
are
ULE
Sr
δtot ≡ αw − αL
− αatom
,

(329)

Si
ULE
δL ≡ αL
− αL
,

(330)

Yb
Sr
δatom ≡ αatom
− αatom
.

(331)

This remains useful for a future high-precision residual
measurement even after the tree-level cancellation is imposed.
F.

The 4→3 GLS Protocol

The four basic cavity–atom slopes map onto three independent sector combinations:
TABLE XXXI. Mapping of measured cavity–atom ratios to
sector parameters.
Measured slope
ULE/Sr
Si/Sr
ULE/Yb
Si/Yb

Combination
Sr
ULE
− αatom
αw − α L
Sr
Si
αw − αL − αatom
Yb
ULE
− αatom
αw − αL
Yb
Si
αw − αL − αatom

Parameter
δtot
δtot + δL
δtot + δatom
δtot + δL + δatom

The redundancy provides a built-in closure relation and
is valuable even though the target signal is residual rather
than order unity.
G.

Experimental Concept and Controls

The experimental architecture below applies at the
residual level: a residual target changes the amplitude,
not the measurement logic.
a. Hardware.
• two evacuated optical cavities (for example ULE
and cryogenic Si) with PDH-locked lasers;
• co-located Sr and Yb optical lattice clocks;
• a self-referenced frequency comb measuring all four
ratios simultaneously;
• vertical relocation or a dual-station geometry providing a known potential difference.

b. Dispersion control. The dual-wavelength check remains essential. DFD’s optical metric is nondispersive in
the minimal formulation, so any large wavelength dependence would diagnose ordinary optical systematics rather
than a gravitational effect. Causality constrains material
dispersion via the Kramers–Kronig relation:
2 ω α0 Lmat
∂ ln n
≲
,
(332)
∂ ln ω
πΩ F
where F is the cavity finesse, Lmat the material path
length, α0 the absorption coefficient, and Ω the detuning
to the nearest material resonance. For crystalline mirror
coatings and ULE glass near optical-clock frequencies
(α0 < 10−4 , Ω/ω > 10−2 ), this yields |∂ ln n/∂ ln ω| <
10−8 —far below experimental reach.
c. Cavity mechanics. Vertical transport changes
gravitational loading on the cavity spacer. Controls include: elastic modeling to null first-order sag; 180◦ orientation flips at each height (mechanical artifacts change sign,
gravitational effects do not); and a platform tilt budget
maintained at <100 µrad. Gravitational sag contributes
αgrav ∼ 10−9 for ULE, elastic coupling <10−14 for 10−6 g
perturbations, and thermoelastic drift cancels in commonmode ratios. The combined effective length-change bound
M
is |αL
| ≲ 10−8 .
d. Environmental and noise budget. Temperature
stability <10 mK, pressure <10−2 mbar, magnetic field
drift <10 µT with periodic reversal. The ratio Allan
variance is modeled as σy2 (τ ) = h−1 /τ + h0 + h1 τ , with
typical values: white frequency h−1 ∼ 10−32 (300 s windows), flicker h0 ∼ 10−34 , random walk h1 ∼ 10−38 . The
dominant term is white noise.
e. Thermal rejection. Silicon cavities have dn/dT ∼
10−4 /K; with δT < 10 mK the fractional contribution is
<10−6 . ULE has CTE ≈ 0 near 30◦ C; silicon near 124
K has CTE ≈ 0. Operating at these zero-crossings suppresses length changes. Any residual dispersion from coating thermal effects appears differently at two wavelengths,
bounding the dispersion systematic to |ϵdisp | ≲ 10%. Total thermal target: <3 × 10−16 , achievable with demonstrated technology.

H.

Expected Signal and Sensitivity

For a height separation ∆h,


g∆h
∆h
−14
.
(333)
≈
1.1
×
10
c2
100 m
After geometric cancellation, the cavity–atom observable
inherits this factor and a screened residual coefficient. The
terrestrial height-separated signal is therefore extremely
small.
a. Consequence. The practical ranking of experiments is as follows:
• a terrestrial height-separated cavity–atom test is
not a quick binary discriminator;

86
• it is a demanding precision residual experiment,
likely better matched to future long-baseline or
space-based platforms;
• multi-species clock and nuclear-clock programs come
before it in near-term priority.

I.

Current Status and Priority

No existing experiment has yet performed the full sectorresolved cavity–atom residual test at the required precision, so the channel is not experimentally exhausted; its
target is clean but extremely small.
Cavity–Atom Priority
Picture: tree-level geometric cancellation; residual
screened signal only.
Ranking:
1. Th-229/Sr and related nuclear-clock analyses
2. cross-species atomic clock comparisons
3. same-ion null checks that bound the pure α sector
4. height-separated cavity–atom residual tests

J.

Summary: Cavity–Atom as a Precision Residual
Test

The cavity–atom picture is simple to state:
1. the optical metric implies constitutive relations
through Tamm–Plebanski,
2. those constitutive relations alter both light propagation and electromagnetic binding,
3. the cavity spacer’s coordinate length therefore
changes together with the coordinate light speed,
4. the leading geometric cavity/atom response cancels
at tree level,
5. only a residual screened signal remains.

XIII.

MATTER-WAVE INTERFEROMETRY

Atom interferometry provides a complementary test
of DFD in the matter sector. This section examines a
T 3 phase expression proposed to distinguish DFD from
GR, describes concrete experimental designs, and assesses
sensitivity requirements. The signature is not a DFD
prediction: the matter-wave sector is a null prediction of
DFD (status note after Eq. (337)).

A.

The ψ-Coupled Schrödinger Equation

In DFD, the scalar field ψ modifies the dynamics of
massive particles through the physical metric (12). For
nonrelativistic particles in weak fields (|ψ| ≪ 1), the
Schrödinger equation becomes:
iℏ ∂t Ψ = −


ℏ2 2
ℏ2 
∇ Ψ + mΦN Ψ +
ψ ∇2 Ψ + (∇ψ) · ∇Ψ ,
2m
2m

(334)
where ΦN = −c2 ψ/2 is the effective Newtonian potential.
a. The δH perturbation. The Hamiltonian splits as
H = H0 + δH, where:

p2
ℏ2 
H0 =
+ mΦN ,
δH =
ψ ∇2 + (∇ψ) · ∇ .
2m
2m
(335)
The δH term produces a phase shift beyond the standard
gravitational phase. Status: not a DFD result. The
δH above carries the bare kinetic coefficient 1 where the
matter dispersion relation gives 3/2 (App. AN, e−3ψ/2 );
taken at face value it and the phase below would print
a non-GR 1PN residual (γmatter = 1/2, excluded). DFD
equals GR at 1PN for matter waves; see the status note
after Eq. (337).
b. Key phase formula. Evaluating δH along classical
trajectories gives the phase shift:
Z 2T
1
∆ϕ∇ψ = −
(336)
dt (∇ψ) · ∆p(t) ,
2m 0
where ∆p(t) is the momentum difference between interferometer arms.

B.

The T 3 Discriminator (Not a DFD Prediction)

Consider a vertical Mach-Zehnder atom interferometer
with light-pulse beam splitters at t = 0, T, 2T . The effective Raman wavevector is keff ẑ, and the recoil velocity is
vrec = ℏkeff /m.
a. Arm geometry. After the first pulse, the arms have
momentum difference ∆pz = ℏkeff . The spatial separation
grows as ∆z(t) = vrec t until the mirror pulse at t = T .
b. Phase evaluation. In uniform Earth gravity, ∇ψ =
−2g/c2 . The constant part cancels between arms, but the
finite spatial separation produces a residual. Evaluating
Eq. (336) with the arm separation:
∆ϕKC
DFD =

2
ℏkeff
g 3
T .
m c2

(337)

Status: not a DFD prediction. The expression above is
not a phase (its units are s2 /m), and the re-derivation
from the physical-metric dispersion relation, Eq. (AN3),
yields no DFD-specific term: the one-particle Hamiltonian
coincides with the GR test-body Hamiltonian through first
post-Newtonian order for any static potential (γ = β = 1,
§IV C; the rotating-Earth g0i sector is not covered by this
identity and is separately universal, Eq. (82)), so every
T 2 , T 3 and T 4 phase at this order is common to DFD

87
and GR. The static metric is general relativity’s at second
post-Newtonian order as well (App. WS), so no DFDspecific term appears there either, and a species-dependent
coupling Katom (App. B 5) is an Eötvös parameter already
bounded below 10−14 (§XV G). The matter-wave sector is
therefore a null prediction of DFD (cf. §XVIII H); the forecasts, designs and falsification thresholds of this section
and of Apps. B 5, D 3 and W 3 do not apply to DFD.
c. Comparison with GR. The standard GR phase
(after common-mode subtraction) is:
2
∆ϕKC
GR = keff g T .

d.

(338)

The discriminator (not a DFD prediction).
Eq. (337):

∆ϕ ∝ T 3 ,

GR:

∆ϕ ∝ T 2 .

• keff ≃ 1.6 × 107 m−1
• vrec = ℏkeff /m ≈ 1.2 × 10

m/s

Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII. The
absolute GR phase keff gT 2 ∼ 1.6 × 108 rad is removed
by standard common-mode techniques; DFD predicts no
residual beyond the GR phase at this order.

Experimental Designs

Several configurations can search for a T 3 term; since
DFD and GR predict the same phase through 1PN, each
is a null test of DFD:

1.

a. Configuration. Horizontal Bragg interferometer
with baseline direction n̂. Rotate platform by 180◦ about
vertical.
b. Signature.
2
ℏkeff
g · n̂ 3
T .
(340)
m
c2
Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII. The
expression above would flip sign under rotation; many
systematic effects do not.

∆ϕhoriz
DFD =

3.

Design C: Source Mass Modulation

a. Configuration. Place a dense source mass (∼500
kg tungsten) at distance R ∼ 0.25 m. Modulate the mass
position to generate time-varying gs = GM/R2 .
b. Signature.
2
ℏkeff
gs 3
T × G(geometry).
(341)
m c2
Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII. Lockin detection at the modulation frequency; the amplitude
of the expression above scales with T 3 .

∆ϕsrc
DFD =

• g = 9.8 m/s2 , c = 3 × 108 m/s

C.

Design B: Horizontal Rotation

(339)

The contrast does not hold for DFD: through 1PN, DFD
and GR carry the same T 2 , T 3 and T 4 phases (status
note above), so time scaling, orientation dependence and
recoil scaling do not separate them.
e. Numerical estimate. For 87 Rb at 780 nm:

−2

2.

Design A: Vertical Fountain

a. Configuration. 10-meter vertical fountain with
Rb, 780 nm Raman transitions, π/2–π–π/2 pulse sequence.
b. Parameters.
87

4.

Design D: Dual-Species Protocol

a. Configuration. Run Rb and Yb interferometers in
matched geometry. The phase of Eq. (337) would scale
2
as ℏkeff
/m; through 1PN DFD adds nothing to the GR
phases, so the dual-species differential is the GR one,
and a species-dependent coupling is an Eötvös parameter
bounded below 10−14 .
b. Differential signal.
!
2
2
keff,i
keff,j
gT 3
(i−j)
∆ϕDFD = 2 ℏ
−
.
(342)
c
mi
mj
Status: not a DFD prediction (the matter-wave sector
is null); see the status note after Eq. (337) in Sec. XIII.
If both species share the same lattice wavelength, the
expression reduces to ∝ (1/mi − 1/mj ).

• Interrogation time: T = 1–2 s
• Arm apex separation: ∆zmax ≈ vrec T ∼ 1–2 cm
• Expression of Eq. (337): not a phase (units s2 /m)
Status: not a DFD prediction (the matter-wave sector is null); see the status note after Eq. (337) in
Sec. XIII.
c. Existing facilities. Stanford
Wuhan HUST, Hannover VLBAI.

10-m

fountain,

D.

Discriminants and Systematics Control

The table compares the T 3 term of Eq. (337) with
standard systematic effects:
a. Properties of the T 3 term.
1. Time scaling: not a discriminator: both DFD
and GR carry T 2 /T 3 /T 4 identically through 1PN
(§XIII B); the DFD row of Table XXXII is the Newtonian gravity-gradient recoil term, not a DFD signature

88
TABLE XXXII. Systematics overview and discriminants. The row labelled DFD is not a DFD prediction: its T 3 scaling and
even k-parity are those of the standard Newtonian gravity-gradient recoil term, not unique to DFD (§XIII B).
Effect

T -scaling
3

Rotation flip k-reversal parity

DFD (not predicted)
T
Gravity gradient Γ
T 2 /T 3 mix
Wavefront curvature
T2
Vibrations (residual)
≈ T2
AC Stark / Zeeman
pulse-bounded
Laser phase (uncorrelated)
T2

2. Orientation: rotation would flip the term of
Eq. (337) (via g · n̂); many systematics do not
2
3. k-reversal: the term is ∝ keff
(even under keff →
−keff ), like the gravity-gradient recoil term; laserphase systematics are odd and cancel

4. Recoil dependence: the term is ∝ vrec , like the
gravity-gradient recoil term
5. Dual-species: the expression gives a residual ∝
(1/m1 − 1/m2 ); through 1PN DFD adds nothing to
the GR residual, which is null after rejection
b.

Known systematics.

• Gravity gradient noise (GGN): Atmospheric
and seismic mass fluctuations; mitigated by underground siting or subtraction.
• Wavefront aberrations: Dominant accuracy
term; < 3 × 10−10 g equivalent demonstrated.
• Vibration isolation: 102 –103 vertical attenuation
at 30 mHz–10 Hz achieved.
• Coriolis/Sagnac: Separated by rotation protocols.
E.

Sensitivity Forecast

a. Current state of the art. Long-baseline atom interferometers have demonstrated:
• Stanford 10-m fountain: single-shot sensitivity
few×10−9 g, arm separation 1.4 cm.
• Dual-species EP tests: η ∼ 10−12 with 2T = 2 s.
• VLBAI (Hannover): high-flux Rb/Yb, 10-m magnetic shielding.
F.

Existing Analyses and the T 3 Term

Long-baseline atom interferometry experiments routinely suppress or calibrate out cubic-in-T gravitygradient contributions using frequency-shift gravitygradient (FSGG) compensation or k-vector tuning

Yes
Often No
No
No
No
No

2
Even (keff
)
Mixed
Odd
Odd/Even mix
Design-dependent
Odd

schemes [75–77], because within GR such terms are
treated as systematics. As a result, published analyses
typically:
1. Operate at fixed T for the main measurement
2. Do not report a residual-vs-T regression with the
even-in-keff , rotation-odd discriminator
3. Use k-reversal specifically to cancel odd-in-keff
laser/systematic terms
To our knowledge, no experiment has isolated a coefficient beven in ϕres (T ) = aT 2 + beven T 3 that:
(a) Is even under keff → −keff , and
(b) Flips sign under 180◦ rotation of a horizontal baseline
This is the signature the formula of Eq. (337) would give;
DFD predicts none.
a. Practical consequence. A T 3 residual beyond the
modelled gravity-gradient terms would appear as a
“gravity-gradient residual” that was not fully removed by
standard compensation and shows the wrong parity under
k-reversal. DFD predicts none, so reanalysis of archival
data with this discriminator applied is a zero-cost null
test.

G.

MAGIS and AION as Null Tests

The MAGIS (Matter-wave Atomic Gradiometer Interferometric Sensor) and AION (Atom Interferometer
Observatory and Network) programs are next-generation
vertical-baseline interferometers designed for gravitational
wave detection and fundamental physics.
a. MAGIS-100. The 100-meter baseline at Fermilab
will achieve:
• Interrogation times T ∼ 1–2 s
√
• Single-shot strain sensitivity ∼ 10−19 / Hz
Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII.

89
b. AION-10 and AION-100. The UK AION program
plans staged development:

I.

Summary: Matter-Wave Test

• AION-10 (Oxford): 10-m baseline, T ∼ 1 s, demonstration phase

Summary: Matter-Wave T 3 Test (Not a DFD Prediction)

• AION-100 (UK, site pending): 100-m baseline, full
science program

The formula of Eq. (337):

DFD predicts no DFD-specific phase in either.
c. Null-test analysis pass. We recommend that
MAGIS/AION include a dedicated analysis pass:
1. Vary T systematically over the accessible range
2. Fit residual phase to aT 2 + bT 3
3. Apply the even-k, rotation-flip discriminator to b
4. Report b with uncertainty, regardless of whether it
is consistent with zero
This analysis costs nothing beyond what is already
planned and would provide a direct test of the mattersector null prediction of DFD.

H.

Complementarity with Cavity-Atom Test

2
g 3
ℏkeff
T .
m c2
Status: not a DFD prediction. The expression above is
not a phase (its units are s2 /m), and the re-derivation
from the physical-metric dispersion relation, Eq. (AN3),
yields no DFD-specific term: the one-particle
Hamiltonian coincides with the GR test-body Hamiltonian
through first post-Newtonian order for any static potential
(γ = β = 1, §IV C; the rotating-Earth g0i sector is not
covered by this identity and is separately universal,
Eq. (82)), so every T 2 , T 3 and T 4 phase at this order is
common to DFD and GR. The static metric is general
relativity’s at second post-Newtonian order as well
(App. WS), so no DFD-specific term appears there either,
and a species-dependent coupling Katom (App. B 5) is an
Eötvös parameter already bounded below 10−14 (§XV G).
The matter-wave sector is therefore a null prediction of
DFD (cf. §XVIII H); the forecasts, designs and
falsification thresholds of this section and of Apps. B 5,
D 3 and W 3 do not apply to DFD.
Features of that formula (none separates DFD
from GR through 1PN):

∆ϕDFD =

The matter-wave and cavity-atom tests probe different
sectors:

• T 3 scaling, as in the standard gravity-gradient
recoil term

• Cavity-atom: Photon sector (optical metric) vs.
atomic sector

• Rotation sign flip

• Matter-wave: Matter sector (∇ψ coupling to momentum)

• Dual-species mass dependence

Together, they constrain DFD’s sector coefficients:
DFD predicts a null in the matter-wave channel through
1PN (status note after Eq. (337)) and in the co-located
cavity–atom channel (Sec. XII, Thm. XI.2).

2
• Even k-parity (keff
)

Status: a null test of DFD.
A null result is the DFD prediction (the matter-wave
sector is null; status note above).

XIV.

SOLAR CORONA SPECTRAL
ASYMMETRY ANALYSIS

This section presents an analysis of archival
SOHO/UVCS data, whose main result is that the early
and late halves of a day’s Ly-α exposures differ in intensity on about half of the days without a mean velocity
shift, and states what DFD predicts for these data.
A.

Motivation: Intensity Changes Without Velocity
Changes

In the UVCS geometry the disk light arrives nearly
radially and the line of sight is nearly perpendicular to
it, so Doppler dimming by a change in radial outflow,
and a change in density, both alter the scattered Lyα intensity without shifting the observed line centroid.
Intensity variations without velocity shifts therefore have
a standard coronal explanation.

90
B.

At tree level, electromagnetic fields couple to ψ through
their energy, which sources ψ through the (ε+3P ) = 2uEM
channel of the Optical Source Theorem (Theorem II.1;
Eq. (R13)), and through the optical index n = eψ . This
coupling has no threshold and is far too weak to be seen
in the corona. DFD’s prediction for the coronal lines is
therefore the standard one.

C.

D.

Electromagnetic Coupling to ψ

SOHO/UVCS Ly-α Analysis

We analyzed archival data from the Ultraviolet Coronagraph Spectrometer (UVCS) aboard SOHO, examining
334 observation days spanning January 2007 through October 2009 during the minimum phase of Solar Cycle
23/24.

Multi-Species Test: O VI 103.2 nm
1.

Data and Methods

From the UVCS Level-1 archive (2007–2009), we identified 42 observation sequences with wavelength coverage
including O VI 103.2 nm. After quality filtering, 10,995
individual exposures across 25 unique dates were analyzed. For each exposure, the spatially-integrated O VI
spectrum was extracted, the intensity-weighted centroid
computed, and the blue/red wing asymmetry about the
centroid measured. Asymmetries were binned by the
Sun’s ecliptic longitude θ (zero at the March equinox)
and fitted with a sinusoidal model:
∆I
(θ) = A sin(θ + ϕ) + C.
(343)
I
2.

1.

Statistical Methodology: Permutation Tests and FDR
Control

The statistical analysis employs robust nonparametric
methods designed for multiple hypothesis testing across
coronal observation bins [78].
a. Permutation testing protocol. For each day with
≥ 2 frames (the frame table carries no slit position, so all
of a day’s frames form one group), we sorted frames by
observation timestamp, which is the start time of each
frame’s observing sequence, and split at the temporal midpoint (“early” vs. “late”); whichever temporal half has
the higher median intensity is post-hoc labeled “bright”
for sign convention only — the labeling does not affect
the statistical test [79]. Permutation tests (N = 20,000
replicates) generated null distributions by random reassignment of group labels. Two-sided p-values were computed as the fraction of permutation replicates yielding
test statistics as extreme as observed.
b. Multiple testing correction. With 321 testable day
groups, false discovery rate (FDR) control is essential.
We applied the Benjamini–Hochberg procedure [80] at
q = 0.05, ensuring that the expected proportion of false
positives among significant detections is bounded at 5%.
c. Effect size quantification. Cohen’s d provides a
standardized measure of effect magnitude [81]: intensity
contrast d = 0.24 (small–medium), velocity shift d =
−0.03 (null). Of 321 testable groups, 163 (50.8%) passed
the 5% FDR threshold for intensity contrast. Days whose
early and late halves were taken with different instrument
settings (slit width, roll or pointing) reach p < 0.05 far
more often (about 75–98% of them, against 34–47% of
days without a change), so part of the contrast comes
from comparing exposures taken with different settings,
while days without a change still exceed the 5% expected
under the null.

Results

The sinusoid fits treat each O VI exposure, and each Lyα day that passed the FDR threshold, as an independent
point, and the two fitted quantities are different statistics: a per-exposure wing asymmetry for O VI and the
early/late intensity contrast for Ly-α. The 10,995 O VI
exposures come from six observing campaigns. With the
observing campaign as the unit for O VI and the day as
the unit for Ly-α, neither line shows a significant annual
modulation, and neither phase is determined.

E.

Consistency Check: Intensity Without Velocity

a. O VI velocity analysis. Binned by asymmetry
magnitude quartiles, the asymmetry increases by a factor
of about 10 from Q1 to Q4. With the line centre fitted in
each exposure and the observing campaign or the date as
the independent unit, the mean velocity shows no significant rise from Q1 to Q4 (+0.8 ± 0.4 km/s by campaign,
+0.4 ± 0.3 km/s by date).

F.

Summary

The main UVCS result is the Ly-α early/late intensity
contrast, significant at 5% FDR on 163 of 321 days without
a mean velocity shift (Cohen’s d = −0.03); part of the
contrast comes from comparing exposures taken with
different instrument settings. Neither H I nor O VI shows
a significant annual modulation once the day (H I) or the
observing campaign (O VI) is the independent unit.

91
3.

UVCS Analysis Summary
Key Results:
• Ly-α early/late intensity contrast significant on
163 of 321 days (5% FDR), with no mean velocity
shift (d = −0.03)
• Days whose two halves used different instrument
settings pass far more often
• No significant annual modulation in either line
with independent units

G.

Quantitative Multi-Wavelength Test: The
Asymmetry Ratio

The ratio of Ly-α to O VI asymmetry amplitudes compares resonantly scattered chromospheric Ly-α with locally
produced coronal O VI emission.

1.

Thermal Width Analysis

The thermal Doppler width of a spectral line depends
on temperature and atomic mass:
r
kB T
.
(344)
σtherm = λ
mc2

Comparison with Observations

The 2007–2009 data do not measure R, since the annual
fits of Sec. XIV D use different statistics for the two lines
and find no significant annual modulation in either.
The matched dim/bright slope measures Γ. In the
same bins of simultaneous Ly-α and O VI spectra, the
O VI dim/bright intensity ratio plotted against the Ly-α
ratio has slope 1/R, so with σLyα and σOVI measured
in those spectra, rather than taken from Table XXXIII,
Eq. (346) predicts the slope (σLyα /σOVI )2 /Γ, with the
bins, the width measurement and the fit fixed in advance.
With DFD’s standard Γ this slope tests the coronal lineformation model; it does not separate DFD from standard
physics.

XV.

ANTIMATTER GRAVITY TESTS

The recent trapping of more than 1.5×104 antihydrogen
atoms and the first direct measurements of antimatter
free fall [82, 83] open a qualitatively new window on the
Einstein Equivalence Principle (EEP). In pure-metric GR
with minimal coupling, hydrogen (H) and antihydrogen
(H̄) must experience identical gravitational acceleration.
DFD reproduces this prediction at the metric level, but
allows for controlled, testable deviations through nonmetric couplings.

TABLE XXXIII. Thermal line widths at characteristic formation temperatures.
Line
Ly-α (1216 Å)
O VI (1032 Å)

Temperature
104 K
2 × 106 K

Mass
mp
16 mp

Thermal Width
0.037 Å
0.111 Å

The width ratio is σOVI /σLyα = 3.0 at these temperatures; the coronal H I that UVCS observes is at about
1–3 × 106 K, where the Ly-α width is 0.37–0.64 Å.

2.

The Generalized Prediction

For small detuning δ of a Gaussian line profile with
width σ, the fractional intensity change scales as:
 2
∆I
δ
A=
∝
.
(345)
I
σ
We write the asymmetry ratio in the generalized form:

2
σOVI
ALyα
=Γ
R≡
(346)
AOVI
σLyα
where Γ captures any enhancement factor for scattered
versus locally-emitted light. DFD’s optics adds no enhancement, so Γ takes its standard value (App. M).

A.

GR Baseline: Matter–Antimatter Universality

In pure-metric GR, the motion of a test body follows
the geodesic equation
d2 xµ
dxν dxρ
+ Γµνρ
= 0,
(347)
2
dτ
dτ dτ
independent of the body’s internal constitution. In the
weak-field, slow-motion limit relevant to laboratory experiments:
d2 x
≈ −∇Φ(x),
dt2

(348)

where Φ is the Newtonian potential. This implies:
aH = aH̄ = −∇Φ

(GR prediction).

(349)

Definition XV.1 (Matter–antimatter universality in
GR). In pure-metric GR with minimal coupling, hydrogen
and antihydrogen obey:
1. Identical free-fall acceleration: aH (x) = aH̄ (x) =
−∇Φ(x)
2. Identical gravitational redshift for corresponding
clock transitions
Any detected deviation from these equalities falsifies this
minimal framework.

92
B.

2.

DFD Metric-Level Prediction

At the level of the effective metric, DFD reproduces
the GR weak-field limit. The physical metric (12) gives,
in the slow-motion limit:
c2
d2 x
≈ ∇ψ(x) = −∇Φ(x),
(350)
2
dt
2
using Φ = −c2 ψ/2. Thus metric-coupled test bodies—
including antihydrogen—follow the same trajectories as
in GR.
Remark XV.2 (Universal free fall at metric level). At the
effective metric level, DFD reproduces GR’s universal
free fall. Any violation of matter–antimatter universality
must arise from non-metric couplings of physical sectors
to ψ beyond the metric.

Remark XV.3 (C-even vs C-odd couplings). If δL couples only to charge-conjugation-even densities (Fµν F µν ,
Gµν Gµν , Higgs potential), then by CPT symmetry:
σĀ = σA

Non-Metric Couplings and Species-Dependent
Sensitivities

Once Standard Model sectors are embedded as internal
modes in the ψ medium, small non-metric couplings can
arise. At the effective field theory level:
L = Lmetric [gµν [ψ], SM fields] + δL[ψ, sectors],

(351)

where Lmetric represents minimally coupled SM fields and
δL encodes non-metric ψ-dependence.
A generic form for δL is:
X
δL =
βI ψ(x) II (x),
(352)

(C-even couplings only).

B

L

where f˜BH , f˜LH ∼ O(1).
D.

Matter–Antimatter Differential Acceleration
Effective Point-Particle Action

Model bound state A as an effective point particle with
action: Z
Z
q
SA = − mA (ψ) c2 dτ = − mA (ψ) c2 −gµν [ψ]ẋµ ẋν dλ.
(358)
In the weak-field, slow-motion limit with mA (ψ) ≈
(0)
mA (1 + σA ψ), the effective potential becomes:


1
(0)
(0)
VA (x) = mA c2 σA −
ψ(x) = mA Φ(x)(1 − 2σA ).
2
(359)
The effective gravitational mass is:
(0)

mg,A = mA (1 − 2σA ),

I

where:

(356)

However, if δL includes C-odd densities such as baryon
number nB or lepton number nL :
σH̄ − σH ∼ −2(βB f˜H + βL f˜H ),
(357)

1.
C.

CPT Considerations

(360)

(0)
while the inertial mass remains mA .

• I indexes SM sectors (electromagnetic, strong,
baryon number, lepton number, etc.)
• II are scalar invariants (Fµν F µν , Gµν Gµν , nB , nL ,
etc.)
• βI are small dimensionless coupling coefficients
1.

Bound-State Mass Shifts

2.

The free-fall acceleration of species A is:
mg,A
aA = − (0) ∇Φ = −(1 − 2σA )∇Φ = (1 − 2σA )a, (361)
mA
where a = −∇Φ is the GR baseline acceleration.
For hydrogen and antihydrogen:

(0)

For a bound state A with unperturbed mass mA , the
coupling (352) induces a mass shift:
Z
X
2
δmA (ψ) c =
βI ψ ⟨A| d3 x II (x)|A⟩.
(353)
I

Define the dimensionless sensitivity parameter:
Z
1 X
σA ≡ (0)
βI ⟨A| d3 x II (x)|A⟩
(354)
mA c2 I

Free-Fall Acceleration

aH = (1 − 2σH )a,

(362)

aH̄ = (1 − 2σH̄ )a.

(363)

The differential acceleration is:
∆aH H̄ ≡ aH̄ − aH = −2(σH̄ − σH )a,

(364)

giving the fractional difference:
∆aH H̄
|a | − |aH |
≡ H̄
≈ 2|σH̄ − σH |
a
|a|

(365)

Then to first order in ψ:
(0)

mA (ψ) ≈ mA (1 + σA ψ).

(355)

E.

Three Scenarios for σH̄ − σH

a. Scenario 1: Pure energy-density couplings (CPTeven). If δL couples only to CPT-even energy densities

93
and respects charge conjugation:
∆aH H̄
= 0.
(366)
σH̄ = σH ⇒
a
DFD reproduces the pure-metric GR prediction.
b. Scenario 2: Natural C-odd couplings. If ψ couples to baryon/lepton number densities with coefficients
|βB |, |βL | ∼ 10−3 –10−1 (natural, unsuppressed values):
∆aH H̄
∼ 10−3 to 10−1 .
(367)
a
Such couplings are already excluded by ordinary-matter
equivalence-principle tests (Sec. XV G).
c. Scenario 3: Fine-tuned or symmetry-suppressed Codd couplings. If |σH̄ − σH | ≪ 10−3 , this would require
either:
• Accidental cancellation between multiple C-odd couplings, or
• A symmetry mechanism suppressing C-odd couplings relative to CPT-even ones

TABLE XXXIV. Summary of matter–antimatter scenarios.
Scenario
Pure metric (GR)
Natural C-odd
Suppressed C-odd

|σH̄ − σH |
0
10−3 –10−1
≪ 10−3

∆aH H̄ /a
0
2 × 10−3 –2 × 10−1
≪ 10−2

The DFD-induced fractional shift at position x:

(T )
∆ν
∆Φ(x)
(T )
+ κA ψ(x).
(x) ≈ −
ν A
c2

(370)

Comparing H and H̄ 1S–2S frequencies at different
(1S–2S)
(1S–2S)
gravitational potentials probes κH̄
− κH
, which
is independent of σH̄ − σH .
Remark XV.4 (Complementarity of free-fall and spectroscopy). Free-fall measurements probe σA (overall mass
(T )
sensitivity), while spectroscopy probes κA (transitionspecific sensitivity). Together they can disentangle different sectors of the DFD coupling structure.
G.

Relation to Ordinary-Matter EP Tests

Ordinary-matter equivalence-principle tests (torsion
balances, lunar laser ranging, MICROSCOPE) constrain
the Eötvös parameter:
aA − aB
ηAB = 2
= −2(σA − σB )
(371)
aA + aB
to the ∼ 10−14 level for materials with different neutronto-proton ratios [5]. However, these tests involve only
ordinary matter and constrain combinations where baryon
and lepton numbers have the same sign.
For antihydrogen:
fBH̄ = −fBH ,

fLH̄ = −fLH ,

(372)

so that:
F.

σH̄ − σH ∼ −2βB fBH − 2βL fLH .

Experimental Mapping: ALPHA-g and Beyond
1.

Ordinary-matter tests already bound these couplings
through the B/µ and Z/µ differences between materials, to |∆aH H̄ /a| ≲ 10−11 (Ti–Pt, MICROSCOPE).

ALPHA-g Free-Fall Measurements

The ALPHA-g experiment measures the vertical motion
of antihydrogen atoms released from a magnetic trap. The
measured acceleration can be written as:
aH̄ = (1 − 2σH̄ )g,

(368)

where g is Earth’s surface gravity.
Current status (2023): The ALPHA collaboration
reported the first observation of antihydrogen free fall,
showing consistency with downward acceleration at approximately the same rate as ordinary matter [82]. Current precision: ∼20% (aH̄ = (0.75 ± 0.13 ± 0.16) g).
Near-term target: ∼1% precision on ∆aH H̄ /a.
Ultimate target: ∼0.1% precision, probing |σH̄ −
σH | ≲ 5 × 10−4 .
2.

(T,0)

(T )

(1 + κA ψ).

DFD Prediction and Falsification

a. Core DFD prediction. With universal ψ-coupling
(no non-metric sector-specific couplings):
∆aH H̄
σA = 0 for all species ⇒
= 0.
(374)
a
This is the default DFD prediction, matching GR.
b. Extended DFD (with C-odd couplings). If Standard Model sectors couple non-minimally to ψ through
C-odd invariants, ordinary-matter equivalence-principle
tests already bound the deviation to |∆aH H̄ /a| ≲ 10−11 .
Antimatter Falsification Criteria
• Pure-metric GR confirmed in antimatter sector

For a transition T in bound state A, define the tran(T )
sition sensitivity κA analogously to σA . The local
transition frequency is:
(T )

H.

If ∆aH H̄ /a = 0 at 10−3 precision:

Spectroscopy Complement

νA (ψ) ≈ νA

(373)

(369)

If ∆aH H̄ /a ∼ 10−2 :
• Pure-metric GR and DFD falsified
• Requires follow-up with spectroscopy to
disentangle sectors

94
I.

and the (nonrelativistic) acceleration of matter is

Summary

Antimatter gravity experiments provide a unique probe
of gravity-matter coupling:
1. At the metric level, DFD reproduces GR’s universal
free fall.
2. Non-metric couplings to C-odd sector invariants
(nB , nL ) induce species-dependent sensitivities σA .
3. The matter–antimatter differential acceleration is:
∆aH H̄
≈ 2|σH̄ − σH |.
(375)
a
4. Current ALPHA-g precision (∼20%) already constrains gross “antigravity” scenarios.

TABLE XXXV. Experimental targets for antimatter gravity.
Experiment
ALPHA-g (current)
ALPHA-g (near)
ALPHA-g (ultimate)
Spectroscopy

XVI.

Observable
aH̄ /g
∆aH H̄ /a
∆aH H̄ /a
κH̄ − κH

Precision
20%
1%
0.1%
10−12

DFD signal
Gross test
Null
Null
Sector decomp.

COSMOLOGICAL IMPLICATIONS

In this section GR/ΛCDM enters only as an observer
dictionary (how distances/angles are commonly reported),
not as ontology.
A.

ψ-Tomography (ψ-Screen) Cosmology Module

a. Non-negotiable premise. The primary reconstructed object is the “ψ-screen” on the past light cone:
∆ψ(z, n̂) ≡ ψem (z, n̂) − ψobs ,

dimensionless.
(376)
All GR/ΛCDM quantities used in this section (e.g.
dict
obs
DL
, DA
) are reporting-layer variables that serve as
a convenient dictionary for published datasets. The supernova and distance relations of this module, Eqs. (380)–
(381), Estimator A and the closure statements built on
them belong to the optical-distance dressing that the SNlensing sign test retired (App. SL); they are kept as the
record of that test and are not DFD predictions. The duality check (Estimator B), the patchwise, mean-subtracted
CMB estimator (Estimator C) and its cross-correlation
falsifier remain in use.
1.

DFD postulates and sign conventions

• ∆ψ > 0 means ψ (hence n) was higher at emission
than locally (slower c1 at emission).
• ∆ψ < 0 means ψ was lower at emission than locally
(faster c1 at emission).
a. Endpoint vs. observable screen. Equation (376)
is an endpoint definition. Operationally, each dataset
reconstructs an observable screen ∆ψobs defined by the logmultiplicative bias required by the DFD optical relations
below. When needed, one may represent ∆ψobs as a
weighted line-of-sight functional
Z χ(z)
∆ψobs (z, n̂) =
dχ Wobs (χ; z) δψ(χ, n̂),
(379)
0

where χ is a dictionary comoving-distance coordinate and
Wobs is a dataset-specific kernel. The inverse program
reconstructs ∆ψobs directly from data without assuming
a particular Wobs .
2.

Forward model: three optical relations

The module was built around three optical relations.
a. (1) Luminosity-distance bias (SNe Ia). Let
dict
DL
(z, n̂) be the baseline luminosity distance as typically reported under the observer dictionary. The opticaldistance dressing, retired by the SN-lensing sign test
(App. SL), mapped this to an optically biased luminosity
distance:
DFD
dict
DL
(z, n̂) = DL
(z, n̂) e∆ψ(z,n̂) .

(377)

(380)

DFD
dict
Equivalently, ln DL
= ln DL
+ ∆ψ.

b. (1b) Angular-diameter-distance bias. Both DL
and DA are computed from null geodesics of the same optical metric g̃µν . Under that dressing the ψ-field screened
both distances equally:
DFD
dict
DA
(z, n̂) = DA
(z, n̂) e∆ψ(z,n̂) .

(381)

c. (2) Distance duality (Etherington reciprocity).
DFD’s optical metric ds̃2 = −c2 dt2 /n2 + dx2 with
n = eψ > 0 is a smooth, non-degenerate Lorentzian
metric. All three conditions of Etherington’s reciprocity
theorem[84, 85] are satisfied: (i) photons propagate on
null geodesics of a Lorentzian metric; (ii) geodesics are
locally unique (ψ is C 1,α away from sources, Appendix U);
(iii) photon number is conserved (no absorption/emission
mechanism). Therefore
DL (z, n̂) = (1 + z)2 DA (z, n̂).

DFD is formulated on flat R3 with a scalar field ψ and
refractive index n = eψ . The one-way light speed is
c1 (ψ) = c e−ψ ,

c2
∇ψ.
(378)
2
We adopt the gauge choice ψobs ≡ 0, so that ∆ψ = ψem
in this gauge. With this convention:
a =

∆ψ

(382)

This holds exactly; no e
factor appears. The common
screening factor from Eqs. (380) and (381) cancels in the
ratio DL /DA .

95
d. Notation. We distinguish ∆ψscreen (z, n̂) (the distance bias relative to the dictionary baseline, measured
by Estimators A and C below) from ∆ψdual (z, n̂) ≡
ln[DL /(1 + z)2 DA ] = 0 (the DDR violation parameter,
measured by Estimator B; identically zero). Where the
subscript is omitted, ∆ψ refers to ∆ψscreen .
e. (3) CMB acoustic-scale screen (angular anisotropy).
The sky-averaged acoustic scale is fixed by the current
measured-clock no-screen solution, 100θ∗ = 1.04110 and
ℓ1 ≃ 220.4. The following estimator is therefore restricted
to patchwise, mean-subtracted anisotropic lensing, analogous to a convergence map. It may not be used to map
a putative unlensed monopole 297 → 220, to shift the
global θ∗ , or to redistribute H0 and rs . Let ℓ1 (n̂) denote the locally inferred first acoustic peak location from
patchwise CMB power spectra. DFD posits the angular
screen mapping
ℓ1 (n̂) = ℓtrue e−∆ψ(n̂) ,

(383)

where ℓtrue ≃ 220.4 is the current sky-independent
no-screen baseline and cancels out of the normalized
anisotropy reconstruction below.
f. Sign of the ℓ1 mapping. The sign deserves explicit
comment. The distance relations Eqs. (380)–(381) give
DFD
DA
∝ e+∆ψ : objects appear farther, which naively
pushes ℓ1 ∝ DA /rs higher. But that scaling applies
to the sky-averaged monopole, which is absorbed into
ℓtrue . Equation (383) describes the direction-dependent
anisotropy: a foreground sightline with ∆ψ(n̂) > 0 acts
as a convergent screen that magnifies the angular scale
of CMB features in that patch. Larger apparent angular
scale maps to lower ℓ1 , hence the negative exponent. This
is the same sign as standard weak-lensing magnification
of the CMB, where convergence κ > 0 shifts power to
lower ℓ.

3.

Two independent screen estimators and one consistency
check

a. Estimator A: SNe Ia alone (and its degeneracy).
From Eq. (380), an operational estimator on each SN
sightline is
d (zi , n̂i ) ≡ ln Dobs (zi , n̂i ) − ln Ddict (zi ) − M,
∆ψ
SN
L
L
(384)
where M is an unknown constant absorbing absolute
magnitude / distance-ladder calibration. SNe alone cannot fix an additive constant in ∆ψ (monopole), because
∆ψ → ∆ψ + const can be absorbed into M. A robust
SN-only product is therefore the anisotropy field
c (z, n̂) ≡ ∆ψ
d (z, n̂) − ∆ψ
d (z, n̂) . (385)
δψ
SN

SN

SN

n̂

b. Estimator B: SNe + BAO / strong lensing (duality consistency check). Etherington’s reciprocity (382)
implies that the observable ratio


obs
DL
(z, n̂)
d
∆ψ
(z,
n̂)
≡
ln
= 0. (386)
dual
obs (z, n̂)
(1 + z)2 DA

This is not an independent measurement of ∆ψscreen ; it
is a consistency check that the optical metric satisfies
d
Etherington’s conditions. The observed ∆ψ
dual = 0.01 ±
0.02 [86, 87] is consistent with zero.
c. Estimator C: CMB peak anisotropy (screen at last
scattering). From Eq. (383), define the normalized estimator:
d
∆ψ
CMB (n̂) = − ln ℓ1 (n̂) + ⟨ln ℓ1 ⟩,

(387)

which fixes the additive constant by construction
d
(⟨∆ψ
CMB ⟩ = 0). This isolates angular structure in the
screen at last scattering.
d. How to obtain ℓ1 (n̂) without ΛCDM priors.
Choose a sky patching scheme; estimate local pseudoCℓ spectra per patch (beam/mask corrected); fit a local
peak locator template around the first peak (only a smooth
peaked function is required); take the maximizing multipole as ℓ1 for that patch.
4.

Theorem-level internal closure of the reconstructed screen

The two screen estimators and one consistency check
introduced above are more than “three ways of plotting
the same thing”: under the forward optical relations, they
imply overdetermined closure identities that must hold on
the sky (and across redshift bins) if a single scalar screen
∆ψ(z, n̂) is the correct organizing variable.
a. Conventions and hypotheses. Fix a redshift bin
z ∈ [za , zb ] and an analysis mask W (n̂) (common to all
maps in a given test). Assume:
1. (H1) Forward relations. The DFD optical relations (380)–(383) hold on their respective domains
of validity.
2. (H2) Observable identification. The reported
distances used in Eqs. (384)–(386) are the observational reconstructions of the corresponding DFD
obs
distances along that line of sight, i.e. DL
(z, n̂) =
DFD
obs
DFD
DL
(z, n̂) and DA
(z, n̂) = DA
(z, n̂) (up to the
stated measurement errors).
3. (H3) SN calibration constancy. The SN absolute calibration constant M in Eq. (384) is a global
constant (independent of z and n̂), as assumed in
the estimator definition.
No dynamical assumption about µ(x), growth, or a specific dictionary is required for the identities below.
Theorem XVI.1 (Duality consistency). Under (H1)–
(H2), the duality estimator (386) tests Etherington consistency:
d
∆ψ
dual (z, n̂) = 0.

(388)

Proof. From Eqs. (380) and (381), both distances carry
the common screening factor e∆ψ . In the ratio DL /[(1 +

96
z)2 DA ] this factor cancels, leaving the standard Etherdict
dict
ington relation (382): DL
/[(1 + z)2 DA
] = 1. Hence
d
∆ψ
=
ln
1
=
0.
dual
Theorem XVI.2 (SN inversion up to an additive constant). Under (H1)–(H3), the SN estimator (384) satisfies
d (z, n̂) = ∆ψ(z, n̂) − M,
∆ψ
SN

(389)

and therefore its centered field (385) equals the true screen
anisotropy at that redshift:
c (z, n̂) = ∆ψ(z, n̂) − ⟨∆ψ(z, n̂)⟩n̂ .
δψ
(390)
SN

DFD
dict
= ln DL
+ ∆ψ. Using
Proof. From Eq. (380), ln DL
d
(H2) and inserting into (384) gives ∆ψ
SN = ∆ψ − M.
Centering over n̂ cancels M identically, yielding (385) as
the true anisotropy.

Corollary XVI.3 (A–B closure simplification). Und
der (H1)–(H3), since ∆ψ
dual = 0 (Theorem XVI.1) and
d
∆ψ SN = ∆ψscreen − M (Theorem XVI.2), the calibration
constant is directly extractable:
D
E
d (z, n̂)
c
M(z) = − ∆ψ
.
(391)
SN
n̂,W

Equivalently, defining the internal closure residual
field
d +M
c = ∆ψscreen − ⟨∆ψscreen ⟩n̂ , (392)
RAB ≡ ∆ψ
SN
recovers the screen anisotropy (up to measurement noise).
c
Here M(z)
= M−⟨∆ψscreen (z, n̂)⟩n̂,W ; it equals M when
the screen has no monopole, as in the current no-screen
solution, and the residual field RAB is exact in either
case.
d
Proof. Set ∆ψ
dual = 0 in the original A–B difference; the
result follows from Theorem XVI.2.
Corollary XVI.4 (Cross-bin overdetermination: M
must be constant). Under (H3), and with a screen whose
c
monopole does not depend on redshift, the offset M(z)
extracted from Corollary XVI.3 is independent of redshift. In practice, for redshift bins {zj } with overlaps, the
statistic
P c


2
c j) − M 2
X M(z
j M(zj )/σM (zj )
P
χ2M =
,
M
≡
2
2
σM (zj )
j 1/σM (zj )
j
(393)
is an overdetermined consistency test of the SN calibration: large χ2M falsifies at least one of (H1)–(H3) (or
flags unmodeled systematics).
Corollary XVI.5 (Harmonic-space closure for
d
anisotropy: SN vs CMB). Since ∆ψ
dual = 0, the
non-trivial harmonic closure test compares the two independent screen estimators. Let both the centered SN map
(Theorem XVI.2) and the CMB map (Theorem XVI.6)

be defined on a common mask. Then for all multipoles
ℓ ≥ 1:
CMB
aSN
ℓm (z∗ ) = aℓm ,

(394)

where z∗ is the last-scattering redshift, and therefore (after identical smoothing/masking) the pseudo-Cℓ spectra
satisfy
b SN×SN (z∗ ) = C
b CMB×CMB = C
b SN×CMB
C
(ℓ ≥ 1),
ℓ
ℓ
ℓ
(395)
up to the usual mask-coupling and noise-bias corrections.
Proof. Both the SN-centered map and the CMB map reconstruct the same monopole-free screen ∆ψscreen (z∗ , n̂)−
⟨∆ψscreen ⟩n̂ at last scattering (up to measurement noise),
hence equal harmonic coefficients for ℓ ≥ 1.
Theorem XVI.6 (CMB estimator is the centered last-scattering screen). Under (H1)–(H2), the CMB peak estimator (387) reconstructs the monopole-free screen at last
scattering:
d
∆ψ
(n̂) = ∆ψ(z∗ , n̂) − ⟨∆ψ(z∗ , n̂)⟩n̂ .
(396)
CMB

Proof. From Eq. (383), ℓ1 (n̂) = ℓtrue e−∆ψ(n̂) . Taking
− ln ℓ1 + ⟨ln ℓ1 ⟩ cancels ℓtrue and removes the monopole
by construction, yielding Eq. (387).
b. Interpretation. Theorems XVI.1–XVI.6 state “closure” algebraically: a single screen ∆ψscreen (z, n̂) implies
d
(i) Etherington consistency (∆ψ
dual = 0), (ii) an SN reconstruction with only one global degeneracy M, and
(iii) strict agreement of SN and CMB anisotropy maps
on overlapping skies and bins. This makes ∆ψscreen (z, n̂)
an overconstrained observable: independent reconstructions must agree, and persistent mismatch falsifies the
single-screen hypothesis.

5.

A GR-independent falsifier

a. Primary falsifier: cross-correlation with independent structure maps. Let X(n̂) be an independent line-ofsight structure tracer map (e.g. CMB lensing convergence
κ or a projected galaxy density map in a defined redshift
slice). Compute the cross-power spectrum
b ∆ψ×X ≡
C
ℓ

ℓ
X
1
∗
∆ψℓm Xℓm
,
2ℓ + 1

(397)

m=−ℓ

and the dimensionless correlation coefficient
b ∆ψ×X
C
ℓ
rbℓ ≡ q
.
∆ψ×∆ψ b X×X
b
Cℓ
Cℓ
b.

(398)

Null hypothesis (falsifier).

H0 :

Cℓ∆ψ×X = 0

for all analyzed ℓ (or all bins).
(399)
Pre-registered falsification criterion:

97
d
If ∆ψ
CMB (n̂) exhibits no statistically significant cross-correlation with an independent
structure map X(n̂) down to the sensitivity
implied by the measured ∆ψ auto-power and
the map noises, then the ψ-screen mechanism
(as the explanation for the optical biases in
this module) is falsified.
c. Probe scope (which tracers can test the null). The
falsifier above has discriminating power only for a structure map X whose redshift support overlaps that of
the reconstructed screen. In particular, a very local
reconstruction—e.g. a directional Hubble-bias field δH0 (n̂)
built from z < 0.05 galaxies—cannot test the X = κCMB
channel: the CMB-lensing kernel carries only ∼ 6×10−5 of
its line-of-sight weight at z < 0.05 (it peaks at z ≃ 1.9), so
a null CℓδH0 ×κ is expected under both this framework and
ΛCDM and is therefore uninformative (neither support
nor falsification). A powered test of Eq. (399) requires a
tracer sharing κ’s z ∼ 0.5–2 support.
d. The galaxy–lensing cross-correlation is detected.
This cross-correlation, which does not use the screen
map, has been measured. The DESI DR1 LRG galaxy
sample (z ≃ 0.4–1.1, whose redshift support overlaps the
CMB-lensing kernel), cross-correlated with the Planck
PR3 SZ-deprojected lensing convergence κ, yields a positive, lensing-kernel-shaped cross-spectrum detected at
≃ 7σ over ℓ ∈ [30, 400]. The significance is quoted as a
matched-filter amplitude  = (t⊤ C −1 d)/(t⊤ C −1 t) with
S/N = Â (t⊤ C −1 t)1/2 , judged against a null distribution
centered on zero; the detection is validated by three independent null constructions—Gaussian realizations with
matched auto-power, harmonic phase-randomization, and
random rotations of the galaxy field—all of which center
on zero, while a deliberately mis-aligned (equatorial-vs.galactic) control destroys the signal, confirming it is skylocked. It is not, however, a discriminant against ΛCDM:
the galaxy–κ cross-correlation is positive in both frameworks, the recovered amplitude (Â ∼ 0.6–0.7, carrying
galaxy-bias and normalization systematics and computed
without full mask deconvolution) is consistent with the
standard S8 lensing amplitude, and the screen amplitude
itself is left unforced by Prop. XVI.7.
A standard variance model for planning is


h
2
1
b ∆ψ×X ≃
Var C
Cℓ∆ψ×X
ℓ
(2ℓ + 1)fsky



i
+ Cℓ∆ψ∆ψ + Nℓ∆ψ CℓXX + NℓX
(400)
with sky fraction fsky and noise power spectra Nℓ∆ψ and
NℓX .
e. Sharpened alternative: a sign-definite, structurecorrelated screen versus a primordial null. The preregistered null Eq. (399) is two-sided, but DFD opticalbias cosmology makes a one-sided, sign-definite prediction
against it.
Proposition XVI.7 (Forced sign of the screen–lensing

cross-spectrum). Take X = κ, the CMB-lensing convergence (or any nonnegatively weighted low-z structure
tracer). Write both observables as line-of-sight projections
of the same Rrefractive field δψ: the reconstructed screen
χ
∆ψobs (n̂) = 0 ∗ dχ Wobs (χ) δψ(χ, n̂) from Eq. (379), and
R χ∗
κ(n̂) = 0 dχ Wκ (χ) δψ(χ, n̂) with Wobs , Wκ ≥ 0. Then,
in the Limber approximation,
Z χ∗


dχ
ℓ+1/2
∆ψ×κ
W
(χ)
W
(χ)
P
k
=
;
χ
> 0
Cℓ
=
obs
κ
δψ
χ
χ2
0
(401)
strictly, on every multipole where the kernels overlap,
with scale-dependence inherited from the (broad, mid-z–
peaked) lensing efficiency. The amplitude is not fixed
by the framework: only sgn Cℓ∆ψ×κ = +1 and the kernel
shape are forced; the correlation coefficient rbℓ of Eq. (398)
is predicted positive and O(0.1–1).
Justification. The patchwise acoustic-scale shift ℓobs =
ℓtrue e−∆ψ is an achromatic optical bias acquired along
the past light cone, so ∆ψobs is a functional of δψ on 0 <
χ < χ∗ , Eq. (379). Under the optical metric n = eψ , the
convergence κ is a line-of-sight projection of the same δψ
(the identical wells that magnify angular scales also dilate
the acoustic scale): an excess integrated δψ both lowers ℓ1
d = − ln ℓ1 +⟨ln ℓ1 ⟩, Eq. (387) —
— raising the estimator ∆ψ
and raises κ. Two nonnegatively weighted projections of
one field have, by the Limber identity, the cross-spectrum
(401), whose integrand is a product of nonnegative factors;
hence Cℓ∆ψ×κ > 0. A mean dilation ∆ψ = 0.30 (297 →
220) is not a DFD mode or prediction; by construction
the patchwise estimator removes the monopole. The
amplitude depends on the data-reconstructed weight Wobs
and is left order-of-magnitude.
Lemma XVI.8 (Primordial null). Any clustering component generated at recombination (z ≃ 1100) and statistically independent of the low-z field δψ contributes
Cℓ∆ψ×κ prim = 0 for all ℓ, by linearity of (401) and the
absence of a shared line-of-sight source.
Corollary XVI.9 (Third-peak height is primordial). The
Planck third-peak height (H3 /H1 ≈ 0.44, H3 /H2 ≈ 1.0)
is invariant under the achromatic angular dilation ℓ →
ℓ e−∆ψ (rescaling the abscissa cannot change ordinate
ratios), and is unaffected by the acceleration-gated response, for which µ → 1 at recombination and the common 1/µ boost cancels in every height ratio (§XVI C).
Hence the height requires a primordial effective-clustering
component (ωc h2 ∼ 0.10–0.12 in ΛCDM units), which by
Lemma XVI.8 contributes zero to Cℓ∆ψ×κ . Consequently
a positive detection of Cℓ∆ψ×κ confirms the line-of-sight
origin of the patchwise anisotropy, while the height remaining absent from the cross-spectrum identifies it as
primordial cold clustering — exactly the signature of the
derived χ-matter field (App. AV), which sources the peak
height through gravitational wells at recombination, not
through a line-of-sight optical remap.

98
f. Observational
implementation. Reconstruct
d
(n̂)
from
per-patch
acoustic-scale anisotropy
∆ψ
CMB
(Planck PR4, ACT DR6) and cross-correlate against an
independent κ map (Planck/ACT CMB lensing) and/or
a low-z galaxy-density slice (unWISE, DESI), forming
b ∆ψ×κ and rbℓ , Eqs. (397)–(398), over ℓ ∈ [30, 300].
C
ℓ
The decision rule is two-way: (i) a positive detection
b ∆ψ×κ > 0) confirms the line-of-sight origin of the
(C
ℓ
patchwise anisotropy; (ii) consistency with the two-sided
null Eq. (399) falsifies the screen mechanism itself.
The cosmic-variance ceiling from Eq. (400) permits
S/N ∼ O(10–100), but the realistic now-value is depressed by the screen-reconstruction noise Nℓ∆ψ and may
be marginal with Planck/ACT; the Simons Observatory
and SPT-3G are needed for a decisive measurement.
These S/N figures are planning estimates, not predictions.
g. Caveats. The cross-spectrum amplitude (and rbℓ )
depend on the data-reconstructed weight Wobs , not on
DFD first principles: only the sign and kernel shape are
theory-forced, and no amplitude is fit here. The Limber/Gaussian form assumes linear, weakly non-Gaussian
fields with overlapping redshift support (the sign is robust
to these; the precise shape less so). The clean primordial null (Lemma XVI.8) rests on the dust-branch no-go
(§J 5: µ ∈ [0, 1) cannot carry an a−3 clustering charge to
recombination); were a future DFD mechanism to source
primordial clustering optically, the null would weaken.
The quoted ωc h2 ∼ 0.10–0.12 is the ΛCDM-side scale of
what the height needs, not a DFD prediction.

factor asc ∈ [0, 1]:
µbg (asc ) = 1 + η1 (1 − asc ) + η2 (1 − asc )2 ,

(404)

with an explicit prior enforcing µbg (asc ) → 1 for asc ≤ 0.5
(equivalently z ≥ 1) to prevent unphysical early-time drift
in this minimal module.
d. (D) Controlled ψ-regime dependence (test knobs).
Introduce log-linear couplings:
δ ln c1 = γc ∆ψ,
δ ln Geff = γG ∆ψ,
δ ln a0 = γ⋆ ∆ψ,

(405)

δ ln α = γα ∆ψ,
where each γ is dimensionless and constrainable by
combining Estimators A–C. In strict DFD postulates,
c1 = c e−ψ corresponds to γc = −1 when ∆ψ is the relevant propagation screen; allowing γc to float is a controlled
falsification test.

7.

a.

Practical next steps

Required data products (minimum viable).

obs
(z, n̂) (e.g.
• SNe Ia compilation providing DL
Pantheon+).[88, 89]
obs
• BAO and/or strong-lensing products providing DA
(e.g. DESI BAO products).[90]

• CMB maps sufficient to extract patchwise ℓ1 (n̂).[58]
6.

Evolving “constants” as controlled parameters

This module introduces only parameters that (i) have
explicit definitions and (ii) enter at least one observable
channel above.
a. (A) Effective gravity in the quasi-static limit.
DFD often packages nonlinear response via an effective
coupling in the linear growth equation:
G
δ̈ + 2H δ̇ = 4πGeff (asc , k) ρ̄ δ,
Geff (asc , k) =
.
µ(x)
(402)
Clarifying statement: Geff is an effective response factor (a
rescaling by 1/µ in the quasi-static limit), not a claim that
the fundamental constant G varies in the field equation.
b. (B) Acceleration scales: distinguish c H0 from a0 .
The cosmological acceleration scale is c H0 , where H0
is the observer-dictionary Hubble parameter (reporting
layer). Separately define the galactic crossover scale a0
through the DFD relation
√
a0 = 2 α c H0loc ,
(403)
as defined in the α-relations module elsewhere in this
review.
c. (C) Minimal background control: µbg . To keep
the module inverse-first, parameterize any late-time background departure as a minimal polynomial in the scale

• Independent structure maps X(n̂) for the falsifier
(e.g. CMB lensing convergence κ).[91]
b.

Pre-registered reconstruction pipeline.

d via Eq. (384);
1. SN-only anisotropy: compute ∆ψ
SN
c
report δψ SN via Eq. (385).
d
2. Duality screen: compute ∆ψ
dual via Eq. (386) in
matched bins / sightlines.
3. CMB screen map: extract ℓ1 (n̂) patchwise, then
d
compute ∆ψ
CMB via Eq. (387).
b ∆ψ×X and rbℓ ; assess signif4. Falsifier: compute C
ℓ
icance against H0 using phase-scrambled / skyrotated null tests.
c. Organization of this section. The decisive nearterm test of the screen mechanism is the ψ–structure
cross-correlation in Sec. XVI A. The measured-clock noscreen forward value ℓ1 ≃ 220 shows that the first-peak
location is consistent.

B.

The ψ-Universe framework

In DFD language:

99
• Apparent “missing mass” in galaxy kinematics corresponds to the nonlinear response packaged by µ(x),
which is fixed by the DFD stack and constrained
empirically in the galactic sector.
• The CMB is not treated as a pristine “initial condition snapshot”; it is treated as an observation
after propagation through a structured, ψ-varying
universe (the screen).
a. Weak-response µ(x). Throughout this review we
use the weak-response form (App. N, Assumption N.3)
x
µ(x) =
,
(406)
1+x
for (i) consistency with the galactic calibration used in
Sec. VII D, (ii) correct asymptotics (µ → 1 for x ≫
1, µ → x for x ≪ 1), and (iii) convexity of Ψ(x) ≡
1/µ(x) = (1 + x)/x for x > 0, which is the property
needed for Jensen-type averaging arguments used in the
cluster appendix (Appendix I).

C.

CMB observables as ψ-screened measurements

This paper does not claim a full replacement for
CLASS/CAMB. What it does claim is narrower and
sharper:
CMB angular observables admit a direct inverse reconstruction of a screen field ∆ψ(n̂)
from patchwise peak-location estimates, independent of ΛCDM priors (Estimator C),
and that reconstructed field has a clean, GRindependent falsifier via cross-correlation with
independent structure maps (Sec. XVI A 5).

CMB

which is the thing to build and test first.
b. Peak-height ratios. The odd/even peak-height
structure is primarily controlled by baryon-photon microphysics (baryon loading) and projection/visibility effects;
any gravity-sector enhancement that enters as an overall
driving amplitude tends to cancel in ratios. This is why
the analytic estimate R = 2.34 below does not depend on
the gravity theory. The full spectrum at Planck’s ratio
also needs the cold χ-matter: CAMB with baryons alone
gives H2 /H1 = 0.39 (Thm. AV.31).
Asymmetry Factor Decomposition

The odd/even peak asymmetry A factorizes into independent physical contributions:
A = fbaryon × fISW × fvis × fDop ,

Factor Value

Formula
Physical origin
√
fbaryon 0.474 Rb / 1 + Rb Baryon loading (BBN)
fISW
0.50
(integral) SW/ISW cancellation
fvis
0.98 sinc(π∆τ /τ∗ ) Recombination width
fDop
0.90 (projection) Velocity dilution

where each factor has a distinct physical origin:
The product yields:
A = 0.474 × 0.50 × 0.98 × 0.90 = 0.209.

(408)

(409)

The peak ratio follows as:

2 
2
1.209
1+A
=
= 2.34.
(410)
R=
1−A
0.791
Planck’s best fit gives R ≈ 2.21; the analytic value lies
6% above it, while the full spectrum of the derived χmatter field, run with CAMB on the measured clock, gives
H1 /H2 = 2.20 (H2 /H1 = 0.4545, +0.5σ; Thm. AV.31).
The key point is that fbaryon depends only on Rb (fixed
by BBN), and the µ-dependent gravity enhancement cancels completely in the ratio. The cancellation holds for
the analytic estimate; the cold component is still needed
for the ratio itself, since CAMB with baryons alone gives
H2 /H1 = 0.39 against Planck’s 0.45, and the derived χmatter supplies both the ratio and the third-peak height
(Thm. AV.31).

D.

a. Patchwise peak location as a screen effect (core
relation). The operative relation is Eq. (383). Written
as a reconstruction statement:
d
∆ψ
(n̂) = − ln ℓ1 (n̂) + ⟨ln ℓ1 ⟩,
(407)

1.

TABLE XXXVI. Asymmetry factor decomposition for CMB
peak ratio.

The optical illusion principle

DFD uses the same organizing idea across scales: observed inferences can be biased by propagation through a
structured ψ-medium.
• Galaxies: kinematic inferences are affected by local ψ-structure and (in the DFD stack) one-way
propagation effects; galactic “missing mass” is interpreted as mis-modeling of the ψ-medium response
packaged by µ(x).
• CMB: local angular scales inferred from patches
of the sky are biased by foreground ψ-structure,
Eq. (383), and this bias is directly reconstructable
(Estimator C) and falsifiable (Sec. XVI A 5).

E.

Intrinsic anisotropy from ψ-gradients

A distinctive prediction of the ψ-screen program is
that the reconstructed acoustic-scale residual field should
correlate with foreground structure. This is exactly the
falsifier in Sec. XVI A 5. An order-of-magnitude planning
estimate for the expected RMS screen is
σℓ1
∼ σψ ,
(411)
σψ ∼ O(10−5 ) ⇒
ℓ1

100
which should be treated as a planning scale to be replaced
d
b ∆ψ∆ψ once ∆ψ
by the empirically reconstructed C
CMB is
ℓ
built.

F.

Line-of-sight distance bias and apparent
acceleration

The reporting-layer diagnostic
1 d(∆ψ)
,
(412)
weff (z) ≃ −1 −
3 d ln(1 + z)
which translated a reconstructed screen ∆ψscreen into an
apparent acceleration, does not apply: a homogeneous
screen adds nothing beyond the expansion (App. ZS),
and the line-of-sight dressing DL → DL e∆ψ is falsified
(App. SL). DFD’s dark energy is the constant α57 vacuum
term alone (App. AU), w = −1.

G.

Cluster-scale dynamics: Status

Cluster-scale dynamics are treated in detail in Appendix I. Current status:
Under the µ-law alone:
• Relaxed clusters (n=10): ⟨Mobs /MDFD ⟩ =
1.57 ± 0.08, at a0 = 1.20 × 10−10 m/s2 and
at the bound-matter a0
• Merging clusters (n=6):
1.99 ± 0.16

⟨Mobs /MDFD ⟩ =

The shortfall is the derived χ. The relaxed masses are hydrostatic X-ray masses
and the merger masses are lensing masses; a
hydrostatic bias b = 0.21 equalizes the classes.
With χ adding its Newtonian pull, all sixteen clusters need χ = 4.5 ± 0.6 Mb inside
r500 , 0.84 ± 0.12 of the cosmic mix; under the
floor composition of App. AV they carry the
ΛCDM budget, 1.28–1.51 of the cosmic mix
(computed). The amplitude follows from the
derived abundance under the adopted energy
(1.21 ± 0.05 of the cosmic mix predicted inside
r500 , 1.28 ± 0.04 measured with no hydrostatic
bias; estimate; App. I).
Assessment: A no-χ closure by literature correction factors (baryonic completeness, nonlinear substructure averaging, temperature,
merger and projection systematics) does not
hold: the baryonic items sum to 1.26, substructure averaging gives ≃ 1.04 for baryonic
substructure, and the projection correction
has the wrong sign for lensing masses (App. I).
Galaxy groups fall short in the same way on
published X-ray masses and need χ too.

H.

Scope of CMB claims

For clarity:
1. Key observables: Peak ratio R = 2.34 is the analytic baryon-loading result; peak location ℓ1 ≃ 220.4
is the current measured-clock no-screen forward result.
2. Full numerical spectrum: Forward TT/TE/EE
calculations use CLASS/CAMB with the cold χ
carrier; the measured-clock run is App. ZS, the runs
of App. CL are on the one-clock C = 1 background,
and they are a transfer check, not a fit of free ΛCDM
densities.
3. No GR ontology: GR/ΛCDM only appear as
dictionary layers for reported distances/parameters.
4. Limited early-universe claims: For baryogenesis
DFD derives the α4 power of the baryon asymmetry
ηB and its B−L bias; its coefficient is a calibration,
since with the stated carrier the mechanism gives
about (2–3) × 10−9 , three to five times Planck’s
value (App. FK; Ext. Deriv. App. AH). It provides
no inflaton potential and no reheating microphysics;
these remain outside scope. It fixes the primordial
(Wheeler–DeWitt) √
scale on the same α3 rung as the
dark sector, H⋆ = 8π MP α3 = 8πfχ (App. AV 7),
and this scale is not an expansion rate: a de Sitter
stage at H⋆ would wash out the χ relic (App. SM).
The relic caps any stage that freezes perturbations
at H ≃ 4 × 1010 GeV (Eq. (Q13)), so DFD predicts no observable primordial B-modes, r ≲ 10−7
(Cor. SM.4).
5. Falsifiability: The screen mechanism is falsifiable through the ψ-screen cross-correlation test
(Sec. XVI A 5).

I.

ISW Effect: A Falsifiable Prediction

The Integrated Sachs-Wolfe (ISW) effect arises when
CMB photons traverse time-varying gravitational potentials. In ΛCDM, this produces a detectable signal at
ℓ < 30 via CMB × galaxy cross-correlation.
DFD prediction: with the adopted Rest-Mass Channel closure (Qχ = 1 on linear scales, App. GR; no gravitational slip, Cor. NL.17) and the real α57 vacuum term
(App. AU), late-time potentials decay as in ΛCDM at the
same background, so the ISW amplitude is AISW = 1.00
in units of the ΛCDM expectation. A suppression to
∼30% of ΛCDM on the optical-bias reading of dark energy is not a DFD prediction: a homogeneous screen adds
nothing beyond the expansion (App. ZS), and the ∼30%
figure has no derivation.
Current data: Planck 2015 measures AISW = 1.00 ±
0.25 (S/N = 4.0) [92]; the tomographic analysis of Stölzner

101
et al. gives AISW = 1.51 ± 0.30 [93]. DFD’s AISW = 1.00
sits at 0.0σ and −1.7σ respectively. The ISW channel
does not separate DFD from ΛCDM.
ISW Falsification Criterion
If CMB × galaxy cross-correlation yields
AISW < 0.6 or AISW > 1.5 at 3σ → the adopted
Rest-Mass Channel closure (Qχ = 1, no slip) is falsified.
If AISW ≃ 1 → consistent with DFD.

J.

The late-time distance ratio is not a screen

The ratio

∆ψ(z) = ln

ΛCDM
DL
(z)
matter (z)
DL


(413)

(0.274 ± 0.02 at z = 1 for Ωm = 0.3) is not a reconstructed “ψ-screen” accounting for the data conventionally
attributed to dark energy, and Pantheon+ Hubble residuals read against a matter-only baseline do not confirm
one: the ratio is, by construction, what a cosmological
constant produces relative to a matter-only ladder; with
one dark energy — the α57 vacuum term of App. AU —
it is not an independent optical quantity, a homogeneous
screen adds nothing beyond the expansion (App. ZS),
and the line-of-sight dressing DL → DL e∆ψ is falsified
(App. SL).
K.

Late-Time Distance Cross-Consistency

The remaining test of the ψ-screen hypothesis is restricted to late-time, mean-subtracted anisotropy observables. A homogeneous screen adds nothing beyond the
expansion, and the acoustic ratio is exactly conformally
invariant (App. ZS); it is therefore not an estimator in
this block. Using the archived late-time reconstruction:
Estimator

Observable

z range

Value

Measures

2

B (Duality) DL /(1 + z) DA 0.3–2.3 0.01 ± 0.02 ∆ψdual

a.

Interpretation of results.

• Estimator B (duality consistency check): Current constraints show DL /(1 + z)2 DA = 1.01 ± 0.02,
i.e. ∆ψdual ≈ 0.01 ± 0.02, consistent with zero
as predicted. This is expected : Etherington’s
reciprocity holds exactly in DFD’s optical metric
(Sec. XVI A 2 c), so any common screening of DL
and DA cancels in the ratio. Estimator B does not
measure ∆ψscreen . The duality relation carries no
e∆ψ factor: DL = (1 + z)2 DA exactly.
• Acoustic observables (excluded from this estimator set): the measured-clock calculation gives
100θ∗ = 1.04110 with no screen. A ∆ψ ≃ 0.30 CMB
endpoint and its evolving-constants interpretation
are therefore excluded; local and inhomogeneous
lensing tests remain live.

The DDR is satisfied (η = 1).
L.

Matter Power Spectrum from Microsector

The most serious challenge to any dark-matter-free theory is matching the observed matter power spectrum P (k).
ΛCDM’s success relies on cold dark matter providing a
pressureless, clustering component. DFD’s clustering
component is the cold χ relic below; the temporal completion theorem (Appendix Q) governs homogeneous
deviations only (Axiom WS.1).
a. The key result. The same S 3 saturation-union
composition law whose weak-response form is µ(x) =
x/(1 + x) (Theorem N.8) also forces the temporal sector
to depend on deviations from background :
µ(ψ0 + ∆ψ) − µ(ψ0 ) = (1 − µ(ψ0 )) µ(∆ψ).

(414)

This is the temporal External Field Effect—a direct consequence of the saturation-union composition law (Appendix Q, Theorem Q.1).
b. Dust-like cosmology. The unique local temporal
scalar is ∆ = (c/a0 )|ψ̇ − ψ̇0 | (the linear deviation from the
ψ-screen). With K’(∆) = µ(∆), the dust branch emerges:
w → 0,

c2s → 0.

(415)

The homogeneous ψ-deviation behaves as pressureless
dust in the late-time small-deviation regime ∆ ≪ 1; the
temporal functional acts on homogeneous deviations only
(Axiom WS.1). Because a3 a0 µ(∆) = const with µ ∈ [0, 1)
bounded and a3 a0 falling into the past, running back in
time forces µ → 1 (saturation), before recombination for
µ(∆0 ) ≳ 2 × 10−5 when a0 tracks H; at those amplitudes the branch therefore does not extend to recombination, where the scaling breaks and the component goes
stiff/radiation-like (see Appendix J, the third-peak-height
obstruction).
c. Implications for structure formation. DFD admits
a dust-like homogeneous ψ-deviation branch (w → 0,
c2s → 0) derived from the S 3 composition law + deviation
invariance under the adopted K ′ = µ. Being homogeneous, it does not cluster; the clustering carrier is the
cold χ component below.
Theoretical status: graded. The cold, non-thermal
χ carrier is the topologically derived harmonic b3 threeform on S 3 = √
SU (2) (App. AV). Its printed mass coefficient uses the 158 multiplicity, one stated input; conditional on the flux-period identification and the asserted
premise P4′ , Ωχ h2 = 0.11928 is forced-amplitude/statedconvention rather than a fitted density; and Qχ = 1 clustering is conditional on the adopted Rest-Mass Channel.
Given that channel, it supplies pressureless a−3 clustering
while leaving DFD’s absolute-time background untouched.
On the current carrier the linear P (k) is ΛCDM-shaped
with DFD’s inputs (σ8 = 0.806, S8 = 0.822; App. GR).
Numerical status: program. A full transferfunction / survey-pipeline confrontation remains a program item. Published P (k) data are processed through
GR-based fiducial cosmologies (the “GR sandbox”), so

102
direct confrontation requires dictionary translation plus
a forward DFD perturbation solver. The linear operator displayed below provides the mathematical closure at
first perturbative order; full survey-pipeline confrontation
remains a numerical implementation task rather than a
missing theoretical principle.
d. Proof-of-concept: N -body structure formation. A
particle-mesh simulation (643 grid, 200 Mpc/h box) comparing ΛCDM (Ωm = 0.30), Newtonian-baryons (Ωb =
0.049), and DFD-baryons (Ωb = 0.049, µ(x) = x/(1+x))4
on identical initial conditions demonstrates the key point:
Newtonian-baryons produces negligible structure (δrms =
1.5 × 10−4 ), confirming the standard objection; DFD produces 43.8× more structure (δrms = 6.4 × 10−3 ). The
5.4× overshoot relative to ΛCDM is physically expected:
cosmological perturbation accelerations (x ≈ 4 × 10−4 ) lie
deep in the MOND regime where the raw µ-function
enhances gravity by ∼ 2500× without the cosmological External Field Effect (EFE) from the Hubble flow
(aext ∼ cH0 ≈ 6 a0 ). This is a proof-of-concept at 643
resolution; production-quality results require ≥ 2563 with
the EFE implemented.
e. The optical σ8 closure. Appendix AE supplies the
rigorous counterpart of this proof-of-concept simulation.
Two results are established there: (i) on the FRW background of the DFD action, |∇ψ̄| = 0 and hence aFRW
=0
ext
(Theorem AE.1), so the Hubble-EFE estimate aext ∼ cH0
used heuristically in this paragraph is not recovered from
the action for cosmological perturbations; and (ii) the
spherical top-hat closure equation Ωb δb Q(δb ) = Ωm σ8,obs
has the solution δb,R8 = 1.041 × 10−2 , QDFD = 502.24 in
the deep-MOND regime supplied by (i). With the baryons
as the only clustering matter, the apparent σ8 measured
by weak lensing would be a refractive amplification of a
∼ 1% baryon contrast at R8 . That construction is not
DFD’s physical carrier: the cold χ relic carries the matter
power with Qχ = 1, the Rest-Mass Channel confines the
baryons’ polarization to bound systems, and the σ8 and
S8 of record are 0.8063 and 0.822 (App. GR).
f. Forward perturbation skeleton. Linearizing the
DFD field equation around a background ψ̄ in Fourier
space gives
ki Mij kj δψk =

8πG
ρ̄ δk ,
c2

(416)

with the response tensor
Mij = µ0 δij + L0 ĝi ĝj ,

4 The

(417)

baryon density Ωb = 0.049 is Planck’s value used uniformly across all three runs to isolate the gravity-law effect
from the matter-content effect (apples-to-apples comparison).
On the current measured-clock background, h = 0.6752 and
DFD’s conventional-point physical density ωb = 0.0222730 give
Ωb = 0.04886. A rerun with this input would produce qualitatively similar conclusions with a modestly smaller overshoot
ratio; production-quality confrontation at ≥ 2563 with the EFE
implemented is the proper venue for quantitative comparison.

where µ0 ≡ µ(x̄), L0 ≡ d dµ
ln x x̄ , ĝ ≡ ∇ψ̄/|∇ψ̄|. The
linear growth equation is then
δ̈k + 2H δ̇k = 4πGeff (a, k̂) ρ̄ δk ,

(418)

with direction-dependent effective gravitational coupling
Geff (a, k̂) =

G
µ0 + L0 (k̂ · ĝ)2

.

(419)

For µ(x) = x/(1+x): µ0 = x̄/(1+ x̄) and L0 = x̄/(1+ x̄)2 ,
the µln,0 of Appendix AC. At low background acceleration
(x̄ ≪ 1), Geff → G/x̄; at high acceleration (x̄ ≫ 1),
Geff → G. The response acts inside bound systems: the
Rest-Mass Channel confines the polarization to them,
so unbound linear perturbations grow with Geff = G
(App. GR).
g. Background-history input. Equations (416)–(419)
describe the linear response of perturbations once a background history H(a) is supplied. In the present monograph, H(a) is the measured-clock background (H0early =
67.52, Ωm = 0.312). The present closure therefore adds
no new background model.
h. Connection to the reconstructed screen. The ψscreen inferred from the patchwise CMB estimator
(Secs. XVI A 3–XVI A 5) is the line-of-sight integral of
the same perturbation field:
Z χ(z)
∆ψscreen (z, n̂) =
W (χ′ ) δψ(χ′ n̂) dχ′ ,
(420)
0

where W (χ) is the lensing kernel. This means the object
inferred from the inverse optical program and the object
sourced by the forward growth equation are the same field.
Any inconsistency between the reconstructed ∆ψscreen
map and the δψ field implied by the forward growth
operator is a direct falsifier of the cosmological closure.
i. New falsifiers from the perturbation system.
1. If the reconstructed ∆ψscreen map does not match
the δψ field implied by Eq. (418), the forward–
inverse closure fails.
2. If the ISW amplitude does not agree with the sign
and amplitude implied by Geff , the growth law is
wrong.
Dust Branch under K ′ = µ: Not Bolted-On K-Essence
The temporal sector rests on one adopted identification,
K ′ (∆) = µ(∆) (App. Q):
1. Same µ(x) = x/(1 + x), the weak-response form of
the galaxy law
2. Same saturation-union composition law
(Assumption N.5)
3. Deviation invariant ∆ = (c/a0 )|ψ̇ − ψ̇0 | forced by
segment additivity
4. Given that identification, the dust branch (w → 0,
c2s → 0) is a theorem (Appendix Q) √
No-go check: Naive quadratic K ′ (Qt ) = µ( Qt ) gives
w → 1/2 (not dust). The dust branch is not
automatic—it requires the deviation-invariant closure.
See Appendix Q for complete derivation.

103
M.

DFD P(k) Multipole Confrontation

Power Spectrum Multipole Confrontation

DFD/ CDM (±10% bias)
BOSS z1 (zeff = 0.38)
BOSS z3 (zeff = 0.61)
eBOSS QSO (zeff = 1.50)

0.6

We confront DFD predictions with galaxy power spectrum multipole measurements derived from BOSS DR12
and eBOSS DR16 mock catalogs.

1.

In linear theory, the quadrupole-to-monopole ratio is:
4
β + 4 β2
P2
= 3 2 7 1 2
P0
1 + 3β + 5β

(422)

where β = f /b is the ratio of the growth rate f =
d ln δ/d ln a to the galaxy bias b.
We extract P0 , P2 , P4 from the BOSS DR12 and eBOSS
DR16 power spectrum measurements, compute the ratio
r2 = P2 /P0 in the linear regime (k = 0.02–0.15 h/Mpc),
and invert the Kaiser formula to obtain β. NGC and SGC
galactic caps are combined by inverse-variance weighting;
errors are bootstrapped (1000 realizations).

Results

TABLE XXXVII. Archived measured β = f /b extraction from
power-spectrum multipoles; the theory column predates the
measured-clock provenance standard.
Sample

0.4
0.3
0.2
0.1
0.0

0.00

0.25

0.50

0.75

1.00

Redshift z

1.25

1.50

1.75

2.00

Method

The anisotropic galaxy power spectrum P (k, µ) encodes
redshift-space distortions (RSD) through the Kaiser formula. Expanding in Legendre multipoles:
Z
2ℓ + 1 1
Pℓ (k) =
P (k, µ)Lℓ (µ) dµ
(421)
2
−1

2.

0.5

= f/b

Remark XVI.10 (Archived pre-measured-clock RSD extraction). This subsection and Fig. 11 are retained as
pipeline provenance. The source script/run ID and the
H0 , Ωm , h, and bias inputs used to produce the printed
βtheory values are not present in the package, so a 67.5clock rerun cannot be certified. The table is not a current
quantitative confrontation; the current Ωm ≃ 0.312 survey
rerun remains an unresolved program item.

zeff

βmeas

βtheory

BOSS DR12 z1
0.38 0.270 ± 0.009 0.357
BOSS DR12 z3
0.61 0.281 ± 0.007 0.395
eBOSS DR16 QSO 1.50 0.366 ± 0.013 0.404

Figure 11 shows the comparison. The measured β values lie 9–29% below the tabulated theory values. This
pattern is consistent with Finger-of-God (FoG) damping and galaxy bias uncertainty not captured by linear
Kaiser, both of which are stronger at lower redshift where
nonlinear structure is more developed.

FIG. 11. RSD parameter β = f /b versus redshift. Blue band:
DFD/ΛCDM prediction with ±10% bias uncertainty. Points:
measurements from BOSS/eBOSS mocks. The displayed consistency statement is historical; a measured-clock rerun is
required.

3.

Interpretation: EFE-Screened Growth Theorem

In the completed first-order DFD growth closure (Appendix AC), the correction to linear growth is not controlled by the clock-sector coefficient kα . It is controlled
by the EFE-screened linear response tensor of the nonlinear µ operator. For the DFD interpolation function
µ(x) = x/(1 + x), the effective gravitational coupling is
−1

x̄
x̄
2
+
(
k̂
·
ĝ)
, (423)
Geff (x̄, θ) = G
1 + x̄ (1 + x̄)2
where x̄ = |∇ψ̄|/a⋆ is the background gradient ratio and
ĝ is the unit vector along the background gradient.
The cosmological external field, supplied by the Hubble
flow
√ under the epoch-consistency rule that fixes a0 (z) =
2 α cH(z) (Section XIX), gives
1
aext (z)
cH(z)
= √ ≃ 5.85, (424)
= √
a0 (z)
2 α cH(z)
2 α
redshift-independent under the same epoch-consistency
rule. Under the Hubble-EFE hypothesis aext = cH(z) of
Appendix AC, which is not DFD’s cosmological growth
law (App. GR), this gives the quasi-Newtonian envelope
x̄EFE =

1.02 G ≤ Geff ≤ 1.17 G,

⟨Geff ⟩ ≃ 1.12 G,

(425)

with a directional (k̂ · ĝ)2 signature.
a. Why the growth correction is not of order kα . A
statement of the form fDFD = Ωγm [1 + O(kα )] is structurally wrong: kα is the clock-sector coefficient, not the
linear-growth response. DFD predicts near-standard linear growth because the Rest-Mass Channel confines the
polarization to bound systems: unbound linear perturbations grow with Geff = G, and linear cosmological growth
is carried by the cold χ relic with Q = 1 (App. GR).

104
b. Why measured β sits below linear theory. The 9–
29% deficit in measured β relative to either ΛCDM or
DFD theory arises from standard effects common to all
linear-Kaiser analyses:
1. Finger-of-God damping from random velocities
2. Galaxy bias uncertainty (∼10%)
3. Mock calibration systematics
4. Alcock–Paczynski mapping and fiducial-cosmology
dictionary translation
These affect ΛCDM and DFD identically and are not
theory discriminators.

4.

Conclusion and Falsifier

The first-order growth operator is closed (Appendix AC); linear cosmological growth is carried by the
cold χ relic with Q = 1 and its signature is isotropic
(App. GR). DFD’s distinctive signatures appear in the
low-acceleration regime (galaxy rotation curves) rather
than linear-regime RSD.
a. Falsifier. The falsifiers of linear growth are those
of App. GR (GR.8), applied after the standard nuisance
layer of galaxy bias, Finger-of-God damping, Alcock–
Paczynski mapping, and fiducial-cosmology dictionary
translation.
Status: linear-regime growth closure complete;
production-level survey-pipeline likelihood pending. The theorem is separate from survey nuisance modeling: the first-order growth operator is closed in the sense
of Appendix AC.

N.

Observational Status (2024–2025)

Several recent observations provide context for the ψscreen framework. We present these as motivations, not
proofs.

1.

Late-Time Potential Shallowing (DES Y3)

The Dark Energy Survey Year 3 analysis provides a
model-independent, direct measurement of the Weyl gravitational potential from combined weak-lensing and clustering data [94]:
Result: The lowest-z bins are 2–3σ shallower
than ΛCDM+GR expectations, corresponding
to ∼10% weaker potential.
As the universe expands and ρ decreases, the source of
ψ [Eq. (21)] weakens:
∆Φ
∆ρ
∼
⇒ late-time shallowing as ρ ↓ . (426)
Φ
ρ
Status: Not a DFD prediction: DFD’s late-time potentials decay as in ΛCDM (AISW = 1.00, Sec. XVI I).

2.

Dynamical Dark Energy Hints (DESI DR2)

DESI DR2 BAO, combined with SNe and CMB distance
priors, shows dataset-dependent preference for dynamical
dark energy w(z) ̸= −1 [95]:
Result: Some dataset combinations favor
w(z) evolving with redshift rather than a pure
cosmological constant.
Status: DFD’s dark energy is the constant α57 vacuum
term, w = −1 exactly (App. AU, Prop. AU.9), with no
dark-energy clustering: on this test DFD stands with
ΛCDM and inherits the same tension. This is therefore
an open test, not a confirmation: a genuine w(z) ̸= −1,
with or without clustering, would falsify DFD together
with ΛCDM.

3.

Wide Binaries (Active and Contested)

Gaia wide-binary tests probe internal accelerations
down to a ∼ 10−10 m/s2 [96]:
Some analyses: Report ∼20% velocity excess beyond ∼3000 au, consistent with MONDlike phenomenology.
Other analyses: Demonstrate that realistic
triple-population modeling and stricter data
cuts remove the signal.
The µ-crossover radius in DFD is:
r
GM
r× =
≈ 7.04 × 103 au
a0
!

1/2
2 1/2
M
1.1966 × 10−10 m/s
×
,
M⊙
a0

(427)

overlapping the few-103 au range where Gaia analyses
disagree. This crossover applies to pairs in weak host fields.
Near the Sun the Galaxy’s baryonic field exceeds (ln 2)2 a0 ,
so by the environmental-confinement axiom (App. NL)
local pairs are exactly Newtonian at every separation;
DFD sides with the analyses that find no signal, and a
confirmed boost falsifies the axiom.
Status: The data are contested; DFD’s local prediction
is the null.

4.

Counter-Evidence and Null Tests

Any alternative framework must address null tests:
a. EG gravity test (ACT DR6 + BOSS). The
geometry-vs-dynamics ratio EG from ACT DR6 CMBlensing crossed with BOSS galaxies is consistent with
ΛCDM/GR and largely scale-independent within current precision [97]. DFD’s forward prediction is the
GR-form statistic EG = Ωm,0 /f (z) at the measured

105
Ωm ≃ 0.312 (gravity-universal growth; the ψ-screen
cancels in the ratio estimator — a cancellation still
to be verified against the DR6 estimator definition):
EG = 0.402 at z = 0.555 (CMASS) and 0.453 at z = 0.316
(LOWZ), scale-independent, vs the measured 0.31+0.06
−0.05
and 0.49+0.14
−0.11 . At the measured clock this is effectively
the Planck-ΛCDM comparison.
Status: Consistent. See App. AG 7.
b. KiDS-Legacy shear. The current-background forward calculation gives σ8 ≃ 0.8063, Ωm ≃ 0.312,
and hence S8 ≃ 0.822. KiDS-Legacy reports S8 =
0.815+0.016
−0.021 for cosmic shear alone [98]; the KiDSLegacy+DES-Y3+Pantheon++DESI-Y1 joint analysis
gives 0.814+0.011
−0.012 [99]. The DFD residual is about +0.45σ
against the primary KiDS-only upper error, or +0.74σ
against the joint upper error (theory-input uncertainty
not included); DES Year 6, 0.789 ± 0.012, lies 2.8σ below
(survey errors only; App. GR).
Status: Consistent with KiDS-Legacy, 2.8σ above
DES Year 6; no special suppression is claimed.
c. Pairwise kinematic Sunyaev–Zel’dovich (ACT +
SDSS/DESI). The pairwise kSZ momentum between
cluster pairs probes the large-scale velocity field v12 (r)
sourced by linear-regime gravity. Recent measurements
by Calafut et al. [100] and Gallardo et al. [101] find the
pairwise infall signal consistent with Newton/GR plus
ΛCDM on scales 20 Mpc ≲ r ≲ 150 Mpc, and have been
cited as excluding modified-gravity alternatives including
MOND.
DFD reproduces the ΛCDM pairwise velocity prediction at this precision. For the Calafut et al. L61
sample configuration (15 bins, 5–145 Mpc, zeff = 0.5,
⟨M ⟩ ≈ 2.5 × 1013 M⊙ , Tinker 2010 bias b ≈ 1.76), evaluating v12 (r) = −(2/3) a(z)H(z)f (z) r ξ¯h (r)/[1 + ξ h (r)]
via the Sheth–Diaferio–Hui–Scoccimarro linear formalism
gives χ2 /dof ≈ 0.79 for DFD, with ∆χ2 (DFD−ΛCDM) ≈
2.5 × 10−4 : DFD and ΛCDM are indistinguishable at current precision and both fit the data well. The test rules
out pure MOND, which fails at cluster scales independently of kSZ; it does not constrain DFD, whose unbound
linear perturbations grow with Geff = G (growth-regime
note below).
Status: Consistent (χ2 /dof = 0.79). Sharp distinction
from ΛCDM requires per-bin precision better than the
0.1–0.5% growth offset below.
d. Growth-regime note. The pairwise-kSZ consistency above rests on quasi-Newtonian growth on 20–
150 Mpc scales, and that is the growth regime DFD’s
dark sector sets: under the Rest-Mass Channel’s boundsystem confinement (App. GR, Remark GR.12) unbound
linear perturbations do not polarize the medium, so linear velocities grow with Geff = G (Qχ = 1), and the
polarization of bound systems raises σ8 by a factor of
only 1.001–1.005 (computed). Neither the Hubble-EFE
screening envelope of Appendix AC (Geff = 1.02–1.17 G)
nor the suppressed kSZ amplitude of the baryon-only construction of Appendix AE and Appendix AF applies to
linear cosmological velocities, and the consistency verdict

holds.

5.

Observational Summary Table

a. Summary. Late-time cosmological anomalies are
uneven across probes and evolving
√with improved analyses.
The clock constraint H0early = C α57/2 /tP (App. AU)
carries a derived exponent and a measured O(1) coefficient
(C = 0.87784 ± 0.00266, five concordant CMB-anchored
channels; Rem. AP.6): H0early = 67.54 ± 0.10 km/s/Mpc
at that C (adopted cross-channel value 67.52 ± 0.09),
concordant with Planck (+0.2σ) and 6.4σ below the
SH0ES ladder’s full 2025 sample (2–3σ below its JWST
subsample); a ψ-screen-bias reading is excluded by the
homogeneous-mode and conformal-invariance theorems
(App. ZS). Local probes read the bound-matter clock,
H0loc = α57/2 /tP = 72.09, by the two-clock postulate of
Sec. ZS 5: one universe read on two clocks, with the size of
their ratio measured. SH0ES’s full 2025 sample lies 1.5σ
above that clock, and the absolute a0 scale (Euclid/DESI
lensing RAR) tests the postulate once baryonic masses are
known to about 2%. The DESI DR2 evolving-w hint, by
contrast, is not a DFD prediction—DFD has one constant
vacuum term, w = −1, with no dark-energy clustering
(App. AU, Prop. AU.9), so a confirmed evolving w would
falsify it together with ΛCDM. EG and KiDS are consistent at the measured clock; the pairwise kSZ test is
passed because DFD’s linear growth is ΛCDM’s to about
0.5%.

O.

Hierarchy of Astrophysical Scales from α
P.

Summary

Cosmology in DFD is framed as reconstructing
∆ψscreen (z, n̂) from the patchwise CMB acoustic-scale
anisotropy, with distance duality (η = 1) serving as a
metric-consistency check, and testing the single-screen hypothesis with a GR-independent falsifier: cross-correlation
with independent structure maps.
• ∆ψ ≈ 0.30 does not set the global CMB peak location; only mean-subtracted patchwise lensing correlations survive in that channel
This is the shortest path to decisive tests that do not
require adopting GR/ΛCDM priors. The falsification criterion remains: cross-correlation of reconstructed ∆ψ(n̂)
with foreground structure maps (Sec. XVI A 5).

XVII.

QUANTUM AND GAUGE EXTENSIONS

This section describes extensions of DFD connecting
the scalar field ψ to Standard Model gauge structure.
The mathematical foundations are rigorous (Appendix F);

106
TABLE XXXVIII. Observational benchmarks (2024–25 status).
Scale/Probe
Solar System
H0 (CMB/BAO)
H0 (local)
DES (low-z)
DESI DR2
Gal. rotation
Wide binaries (local)
EG (ACT)
KiDS-Legacy
DES Year 6
Pairwise kSZ
Lab (100 m)

DFD Prediction
γ
=1
√= β57/2
Cα
/tP = 67.52(9)
α57/2 /tP = 72.09 (bound-matter clock)
ΛCDM-like potentials
weff = −1, no DE clust.
Flat v; TF scaling
γ = 1, no boost
GR form at Ωm ≃0.312
S8 ≃ 0.822
S8 ≃ 0.822
v12 ≃ ΛCDM
null (Thm. XI.2)

TABLE XXXIX. Acceleration scales from powers of α applied
to cH0loc .
Expression Value (m/s2 ) Interpretation
cH0loc
7.0 × 10−10
√
loc
2 α · cH0 1.2 × 10−10
α · cH0loc
5.1 × 10−12

Vacuum acceleration scale
MOND scale a0
Deep MOND regime

the physical interpretation remains conditional on DFD’s
gravitational predictions being correct.

A.

Status and Conditionality

Mathematical Status
Results (Appendices F–G):
1. The electroweak group U (2) = (SU (2) × U (1))/Z2 is the
isotropy group of CP 2 under one stated identification
(Rem. F.2); colour SU (3) and the family content are
inputs, as in the Standard Model (Rem. F.3).
2. The isotropy charge 6Y fixes q1 = 3 (Rem. F.10);
Lemma F.9 is the necessary integrality condition.
3. Flux-product rule Ngen = |k3 k2 q1 | (Thm. F.18), a
bookkeeping identity; the value 3 enters as a discrete
input (Rem. F.21).
4. Energy minimization selects (k3 , k2 ) = (1, 1) at the
derived q1 = 3, consistent with the discrete input
Ngen = 3 (Thm. F.19; status: App. F, Rems. on q1 and
the generation count).
5. ka = 3/(8α) ≈ 51.4 from frame stiffness × EM duality
(Thm. G.1).
6. θQCD = 0 (derived; tree level, with θbare = 0 fixed by
the spectral action given that nothing is added to it for
the gauge fields; loop-induced ≲ 10−16 ; Thm. G.2).

Physical interpretation: Conditional on DFD gravity being correct.
a. Motivation. If DFD’s scalar field ψ is physically
real and couples to matter’s internal degrees of freedom,
one can ask which gauge structures emerge. The construction below shows that SU (3) × SU (2) × U (1) can arise

Obs.
Consistent
Planck 67.4(5)
SH0ES 73.5(0.9)
2–3σ low
w ̸= −1 hints
Empirical
Contested
0.31/0.49
0.815+0.016
−0.021
0.789(12)
χ2 /dof = 0.79
Not tested

Status
✓
✓
1.5σ
?
?
✓
?
✓
✓
2.8σ
✓
—

from Berry connections in a degenerate internal mode
space.
b. Scope. This section presents the mechanism without claiming it is the unique or correct extension of DFD.
It is a theoretical possibility, not an established feature
of the theory.
c. The quantum framework is an imported input. To
make explicit what is assumed: the quantum framework
used below—a complex Hilbert space, the imaginary
unit i, the action scale ℏ, and unitary evolution—is an
adopted input, not derived from the classical ψ-postulates.
App. QM derives the field-level i, the inner product and
unitary free evolution from a unique forced complex structure J⋆ , given a linearized stability that it argues but does
not prove; ℏ, the interacting operator-ordering map and
the measurement postulate are shared with every quantum theory. This is recorded as a documented import,
not as a DFD prediction or a closed result.

B.

Internal Mode Bundle and Berry Connections

a. Setup. Assume the ψ-medium supports degenerate internal mode subspaces at each point:
Hint (x) ≃ C3 ⊕ C2 ⊕ C,
with local orthonormal frames:

E
E
(2)
Ξ(x) = χ(3)
, χb
a
a=1,2,3

(428)
(1)

, χ

E

.

(429)

b=1,2

b. Frame transformations. Under local changes of
basis U (x) ∈ U (3) × U (2) × U (1), the frames transform as
Ξ → ΞU . The resulting non-Abelian Berry connections:
(3)

= i U3† ∂i U3 ∈ su(3),

(430)

(2)

= i U2† ∂i U2 ∈ su(2),

(431)

Ai
Ai

(1)
Ai = ∂i θ ∈ u(1),

(432)

transform as gauge fields.
c. Structure group. The structure group is S(U (3) ×
U (2) × U (1)), with one abelian factor more than the
Standard Model’s; the hypercharge is fixed by the isotropy

107
The gauge couplings are determined by the frame stiffnesses κr . The Stückelberg stiffnesses ηr are a separate
set of parameters for which this construction supplies
no value; neglecting them above is an assumption, not a
result.

reading (Rem. F.2).

C.

The C3 ⊕ C2 ⊕ C partition

The (3, 2, 1) partition enters as a dictionary entry: it
is the minimal partition carrying the factors SU (3) and
SU (2), which the proposition below assumes:
Proposition XVII.1 (Proved in Appendix F 1). Among
all block partitions whose stabilizer contains exactly two
simple non-Abelian factors SU (3) and SU (2) and a singlet sector, the unique minimal partition is (3, 2, 1) with
N = 6. Its stabilizer has two abelian factors modulo
the overall phase; the single hypercharge is fixed by the
isotropy reading (Rem. F.2).
a. Physical requirements. The Standard Model requires:
• SU (3)c for color (3-dimensional fundamental)
• SU (2)L for weak isospin (2-dimensional fundamental)

E.

Generation Counting

The construction relates the generation count to its
flux data by a bookkeeping identity; the count itself is a
stated input.
Theorem XVII.2 (Proved in Appendix F 5). For M =
CP 2 × S 3 with flux configuration (k3 , k2 , q1 ):
Ngen = |k3 · k2 · q1 |.
a.

(436)

The logical chain.

1. Flux quantum: The isotropy charge 6Y fixes q1 =
3; the integrality condition of Lemma F.9 admits
q1 ∈ {3, 6}, and q1 = 6 would give the weak doublets
an even charge (App. F, Rem. on q1 ).
2. Energy minimization: Yang-Mills energy is minimized at (k3 , k2 ) = (1, 1) (Theorem F.19).

• U (1)Y for hypercharge
• A singlet sector for right-handed leptons
b. Minimality argument. A two-block partition
(na , nb ) cannot provide a singlet sector—every vector
transforms non-trivially under at least one SU factor.
Hence three blocks are required. The minimal choice
satisfying all requirements is (3, 2, 1), giving N = 6.
c. Uniqueness. Explicit enumeration (Table in Appendix F 1) shows that no other partition with N ≤ 6
satisfies all requirements.

3. Generation count: Ngen = |1 · 1 · 3| = 3, a bookkeeping identity that reproduces the discrete input
Ngen = 3 (App. F).
b.

Mathematical foundation.

The proof combines:

• Künneth factorization for product manifolds [102]
• Atiyah-Patodi-Singer index theorem on S 3 [103]
• Hirzebruch-Riemann-Roch on CP 2
• Gravitational-U (1)Y anomaly cancellation

D.

Yang-Mills Kinetic Terms from Frame Stiffness

a. Gradient penalty. Twisting the internal frames
costs energy:
X
Lstiff =
ηa ∥∂i |χa ⟩ ∥2 .
(433)
a

b. Hidden local symmetry. This admits a
Stückelberg/hidden-local-symmetry form:
2 
X  κr
ηr  (r)
(r) (r)ij
(r)
,
L=
− Tr Fij F
+ Tr Ai − Ωi
2
2
r=3,2,1

c. Significance. The electroweak group follows under one stated identification and the flux quantum is
derived from it; the generation count, colour and the
family content are inputs (App. F status remarks); the
count assembles from:
• The partition (3, 2, 1), recorded as a dictionary entry
• The flux quantum q1 = 3, derived from the isotropy
charge
• Energy minimization selecting (k3 , k2 ) = (1, 1)

(434)
(r)
where Ωi = iUr† ∂i Ur .

F.

c. Yang-Mills limit. Retaining only the first term of
the preceding Lagrangian, i.e. neglecting the Stückelberg
contribution, leaves the Yang-Mills kinetic term:
X κr
(r)
Lgauge = −
Tr Fij F (r)ij ,
gr ∼ κ−1/2
.
r
2
r=3,2,1
(435)

CP Structure

a. CP violation pattern. The construction predicts
that CP violation enters through complex phases in the
Yukawa sector, with:
• Strong CP violation suppressed (θ̄ = 0 at tree level
with θbare = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields; Thm. L.3)

108
• Weak CP violation supplied by Postulate Y.10
(App. AO)
• CKM-like mixing matrix structure from fermion
mass generation
b. Strong CP suppression. With θbare = 0 fixed by
the spectral action given that nothing is added to it for
the gauge fields, the det-orthogonal tree-level protection
together with the η-invariant argument gives θQCD = 0 at
tree level (Theorem L.3), with loop corrections bounded
at the Standard-Model level (≲ 10−16 ; Sec. XVII M).

G.

Higgs and Mass Spectrum

The gauge emergence framework also addresses the
Higgs sector and fermion mass hierarchy (full derivations
in Appendix H).
a. Higgs emergence. The Higgs doublet (1, 2, +1/2)
emerges as the off-diagonal connector between the C2
(SU(2)) and C1 (singlet) sectors of the (3, 2, 1) partition.
The Mexican-hat potential arises from frame stiffness
energy.
b. Yukawa hierarchy. The three generations correspond to zero modes localized at different “vertices” of
CP 2 . Yukawa couplings are overlap integrals:
Z
Y (n) = gY
ψ̄ (n) · ϕH · ψ (n) dµF S .
(437)
CP 2

If the Higgs ϕH is localized near one vertex (third generation), the hierarchy follows:
Y (1) : Y (2) : Y (3) ≈ ϵ2 : ϵ : 1,

H.

ϵ ∼ 0.05.

(438)

The Fine-Structure Constant from
Chern-Simons Theory

function
2
π
sin2
, k = 0, 1, . . . , kmax − 1, (440)
k+2
k+2
the effective coupling βU (1) = ⟨k + 2⟩ determines α.
The value of kmax is given by the conditional
determinant-line count, a closed Spinc index on CP 2
(§X):
w(k) =

kmax = χ(CP 2 , E) = χ(O(9)) + 5χ(O) = 55 + 5 = 60.
(441)
Here E = O(9) ⊕ O⊕5 is the twist bundle, and the computation uses Hirzebruch–Riemann–Roch for the canonical
Spinc structure.
a. Refined microsector completion. Section X
presents a convention-locked closed form that lies
4.9 × 10−9 from the measured value with the open
bracket coefficient (21 ppm from its integers alone):
α−1 = 137.03599985 (residual −0.005 ppm). This
involves a forced binary fork between regular-module and
fermion-rep microsectors, with only the regular-module
branch surviving under a no-hidden-knobs policy.

3.

Lattice Comparison

This analytical result has been compared with lattice
Monte Carlo simulations (Appendix K 2). The lattice
parameters are fixed before comparison to α:
a. Inputs:
• kmax = χ(CP 2 , E) = 60 (from Spinc index)
• βU (1) = ⟨k + 2⟩ = 3.797 (from CS weight function
at kmax = 60)
• Wilson ratio = (n2 /n1 ) × Ngen = 2 × 3 = 6 (on the
frame-stiffness assumption)
• βSU (2) = 6 × 3.80 = 22.80 (derived)

A central result of the DFD microsector is the closed
form for α = 1/137 from topological quantization on S 3 .

1.

Chern-Simons Quantization

On a compact 3-manifold M3 , the Chern-Simons level
k is quantized:


Z
k
2
SCS =
Tr A ∧ dA + A ∧ A ∧ A , k ∈ Z.
4π M3
3
(439)
For M3 = S 3 with gauge group U(1), the allowed
values are k = 0, ±1, ±2, . . .

2.

The Maximum Level: Topological Derivation

The effective coupling is computed from a weighted
sum over Chern-Simons levels. With the SU(2) weight

b.

Lattice results (L = 6–16, 25+ independent runs):

• At βU (1) = 3.80: within 0.9% of 1/137 for L ≤ 12
(βSU (2) = 23.0 for L ≤ 10)
• L = 16 with 40k thermalization: mean deviation
+1.13 ± 0.50% over nine of ten runs (2.3 standard
errors)
• Converged value (kmax → ∞, βU (1) = 3.94): the
run at βU (1) = 3.95 gives α = 1/147 (−6.8%)
• Wilson ratio: of ten ratios from 3 to 9 only 6 fits;
two further runs at 5.75 give −0.20% and +2.20%
(mean +1.0%; Table LXXXVI)
The lattice agrees to within 0.9% for L ≤ 12 and lies
+1.1 ± 0.5% high at L = 16. In the code’s normalization
1/αW follows from the inputs (Appendix K 2), so the
comparison checks the input chain and does not determine
α independently. The theory would have failed if topology
gave no cutoff (kmax → ∞).

109
I.

1.

The Bridge Lemma: kmax = 60 from Closed Index

The Bridge Lemma identifies kmax = 60 as a closed
Spinc index on CP 2 .
1.

The Mass Formula

v
mf = Af · αnf · √
2

(446)

Statement

where:
c

2

For the canonical Spin structure on CP with twist
bundle E = O(9) ⊕ O⊕5 :
kmax := Index(DCP 2 ⊗ E) = χ(CP 2 , E) = 60.
2.

(442)

• nf = sector-dependent exponent from CP 2 coupling path

Proof

√
For the canonical Spinc structure, D ∼ 2(∂¯ + ∂¯∗ ),
so Index(D ⊗ E) = χ(CP 2 , E) by Hirzebruch–Riemann–
Roch. The holomorphic
Euler characteristic satisfies


χ(O(m)) = m+2
for
m
≥
0. Therefore:
2

χ(E) = χ(O(9)) + 5χ(O) =

• Af = prefactor, data-selected except for the top
and the charm (App. FM)

2.

Sector-Dependent Exponents



11
+ 5 = 55 + 5 = 60.
2

(443)

3.

• α = 1/137.036 (fine-structure constant)
√
• v/ 2 = 174.1 GeV (Yukawa normalization)

The exponents depend on the sector (leptons, upquarks, down-quarks) due to the different Yukawa coupling paths: up-quarks couple to H̃, down-quarks to H
directly, and leptons through a different gauge path.

Physical Selection

The value kmax = 60 is consistent with the microsector
physics. The effective coupling βU (1) = ⟨k + 2⟩, computed
from the SU(2) Chern–Simons weights
2
π
sin2
,
(444)
w(k) =
k+2
k+2
is the lattice input βU (1) ≈ 3.80 for kmax = 60. Here levels
run k = 0, 1, . . . , kmax − 1 (standard SU(2) WZW/CS
convention), giving:
P59
(k + 2) w(k)
⟨k + 2⟩kmax =60 = k=0
= 3.7969 ≈ 3.80.
P59
k=0 w(k)
(445)
Bridge Lemma
Index: kmax = χ(CP 2 , E) = 55 + 5 = 60 [Spinc HRR]
Physics: βU (1) = ⟨k + 2⟩ = 3.797 at kmax = 60; lattice
comparison with α−1 = 137 to about 1%
Icosahedral: kmax = 60 = |A5 |
E8 echo: roots(E8 )/4 = 240/4 = 60 ✓

TABLE XL. Charged fermion mass predictions.
Fermion
Electron
Muon
Tau
Up
Charm
Top
Down
Strange
Bottom

a.

nf
2.5
1.5
1.0
2.5
1.0
0
2.5
1.5
0

Af
2/3
√1
2
8/3
1
1
6
6/7
1/42

Predicted
0.528 MeV
108.5 MeV
1.797 GeV
2.11 MeV
1.270 GeV
174.1 GeV
4.75 MeV
93.0 MeV
4.15 GeV

Observed
0.511 MeV
105.66 MeV
1.777 GeV
2.16 MeV
1.27 GeV
172.76 GeV
4.67 MeV
93 MeV
4.18 GeV

Error
+3.32%
+2.72%
+1.12%
−2.23%
+0.04%
+0.78%
+1.75%
+0.03%
−0.83%

Statistics.

• Mean absolute error: 1.42% at leading order (a
refinement to 0.61% after generation-1 priming is
claimed in the companion volume; the 0.082% after
two-loop running does not stand, since the running
factors it uses are not Standard-Model running)
• Maximum error: 3.32% (electron, leading order)

J.

Nine Charged Fermion Masses

One formula covers the nine charged fermion masses;
the prefactors other than those of the top and the charm
are data-selected (App. FM).

• All nine predictions within 3.4% of observed central
values
• One universal normalization for all 9 fermions
3.

Structural Ratios

The selected prefactors
√ have the ratios Ad /Au = 2.25,
At /Ab = 42, Aτ /Aµ = 2.

110
K.

CKM Matrix from CP 2 Geometry

a.

Numerical verification.
MP = 1.221 × 1019 GeV

The quark mixing matrix emerges from overlap integrals between quark generations localized at different CP 2
positions.

√

−18

19

−18

α = (1/137.036) = 8.04 × 10

Wolfenstein Parameterization

(451)

2π = 2.507

(452)

vpred = 1.221 × 10
1.

(450)

8

8

× 8.04 × 10

× 2.507

= 246.09 GeV

(453)
(454)

Observed: v = 246.22 GeV. Agreement: 0.05%.
The CKM matrix has the standard Wolfenstein form:


1 − λ2 /2
λ
Aλ3 (ρ − iη)

−λ
1 − λ2 /2
Aλ2
VCKM ≈ 
3
2
Aλ (1 − ρ − iη) −Aλ
1
(447)

2.

2.

• Factor α8 : The exponent 8 = dim(CP 2 × S 3 ) + 1
is the dimension count plus the Higgs radial mode
(Thm. Z.3).
√
• Factor 2π: The prefactor is fitted: loop measures
carry (2π)−n , so the loop-normalization motivation
does not hold, and no printed rule fixes it (App. Z).

Geometric Derivation

The Cabibbo angle λ is determined by the ratio of
vertex separations:
λ = e−d12 /σH ≈ 0.225,

3.

Predictions

a. Key prediction within the localization model.
Within the chosen CP 2 localization scheme, the Cabibbo
angle is set by the ratio d12 /σH , which is not computed here. The apex-dependent ratios use the standard
Wolfenstein combinations with the p
Appendix AO apex
(ρ̄, η̄) = (0.157, 0.358): |Vub /Vcb | = λ ρ̄2 + η̄ 2 = 0.088 at
leading order; the exact standard-parameterization matrix
of Appendix AO gives 0.091, vs. observed 0.086 ± 0.006
(0.8σ).

L.

Electroweak-Scale Relation

The “hierarchy problem” asks why v ≪ MP (17 orders of magnitude). In the Standard Model, this requires
fine-tuning. In the present DFD microsector, the relation below is best treated as a numerically successful
structural benchmark rather than a finished theorem
of the core postulates.

1.

The Relation

v = MP × α 8 ×

√

2π

These motivations are structurally suggestive, but
they are not yet a substitute for a referee-proof
first-principles derivation.

(448)

where d12 is the CP 2 geodesic distance between first and
second generation vertices, and σH is the Higgs localization width.

(449)

Physical Origin

Electroweak-Scale Benchmark
√
In this manuscript the relation v = MP α8 2π is best
read as a microsector benchmark supported by the
proposed topological structure, not as a closed hierarchy
theorem independent of the rest of that construction.

M.

Strong CP: Tree-Level Closure, Loop-Bounded

The strong CP problem asks why |θQCD | < 10−10 . In
the Standard Model, this is unexplained. In DFD, θ̄ = 0
at tree level is derived, with θbare = 0 fixed by the spectral
action given that nothing is added to it for the gauge
fields (Appendix L), and the loop-induced value is at the
Standard-Model level (≲ 10−16 ): det-orthogonal mass
matrices, positive on the branch of Postulate Y.10, keep
arg det(Mu Md ) = 0, and the fermion Clifford module
on the even-dimensional CP mapping torus has spectral
pairing and no zero modes, forcing the η-invariant to
vanish.

1.

Tree Level

At tree level, with θbare = 0 fixed by the spectral action
given that nothing is added to it for the gauge fields,
θ = 0 follows from the det-orthogonal tree-level protection
together with the η-invariant argument (Thm. L.3); CP 2
topology quantizes the instanton number:
R
• The θ-term ∝ Tr(F ∧ F ) requires a 4-form
• On CP 2 : H 4 (CP 2 ) = Z, generated by ω 2

111
TABLE XLI. CKM parameters: localization-model values vs. observation (PDG 2024). This geometric model motivates the
structure; the canonical integer-pattern treatment (λ = 31α, A = 108α, with per-channel σ) is in Appendix AO and Table CV.
Parameter
λ
A
|Vub /Vcb |
|Vtd /Vts |

Predicted

R
CP 2

Tr(F ∧ F ) =

Loop Level

Potential loop corrections to θ:
a. (a) Quark mass phases. δθ = arg(det Mu ×
det Md ). In gauge emergence:
Z
Yij = gY
ψ̄i ϕH ψj dµFS
(455)
CP 2

b. (b) Instanton contributions. π3 (SU(3))
H 4 (CP 2 × S 3 ). The cohomology is:

→

H 4 (CP 2 × S 3 ) = H 4 (CP 2 ) ⊕ H 1 (CP 2 ) ⊗ H 3 (S 3 ) = Z ⊕ 0 = Z

(456)
The only 4-cycles are in CP 2 .
c. (c) Electroweak contributions. CKM phase δCP ̸=
0 (weak CP violation exists), but this feeds into θQCD
only at the Standard-Model level:

Tree level: θbare = 0, fixed by the spectral action given
that nothing is added to it for the gauge fields (App. L),
and arg det(Mu Md ) = 0 for det-orthogonal mass matrices
positive on the branch of Postulate Y.10 (App. AO).
Anomaly (Theorem L.3): The fermion Clifford
module on the 8-dimensional (even) CP mapping torus
has symmetric Dirac spectrum and no zero modes, so
η = h = 0, ACP = 1, and the anomaly generates no
θ-term. Loops: the induced angle is at the
Standard-Model level, ≲ 10−16 (App. AO).
Mechanism: The 8-dimensional mapping torus (from
M = CP 2 × S 3 ) carries a graded Clifford module, so
η = 0 by spectral symmetry without an explicit
computation; the product-background Dirac gap removes
zero modes.
Prediction: No QCD axion. Detection at ADMX,
ABRACADABRA, or CASPEr falsifies DFD.

N.

• SU(3)c lives on C3 (the 3-dim block)
• The (3, 2, 1) partition topologically separates
these sectors at tree level
• CKM phases arise from misalignment of fermion
localization with gauge eigenstates—this is a weaksector effect that reaches the QCD vacuum angle
only through the degree-thirteen Jarlskog structure
(a multi-loop effect), at the Standard-Model level
(≲ 10−16 )
Comparison to known solutions.

Mechanism

θ = 0 enforced by

✓
✓
✓
✓

Strong CP: Tree-Level Closure, Loop-Bounded

• SU(2)L lives on C2 (the 2-dim block)

d.

Status

0.225
0.22501 ± 0.00068
0.81
0.826+0.016
−0.015
p
2
2
λ ρ̄ + η̄ ≈ 0.088 (exact matrix: 0.091) 0.086 ± 0.006
0.211 (App. AO)
0.211 ± 0.007

• The instanton density is exact:
8π 2 k3
2.

Observed

PMNS Matrix from CP 2 Geometry

The PMNS matrix has large mixing angles, unlike the
hierarchical CKM. DFD explains this through different
localization patterns.

1.

Observed Mixing

Angle

PMNS (observed)

θ12
θ23
θ13

33.4◦ ± 0.8◦
43–49◦ (octant open)
8.6◦ ± 0.1◦

CKM (observed) Ratio
13.0◦
2.4◦
0.2◦

2.6
18–20
43

DFD analog

Peccei-Quinn
Dynamical (axion)
Not needed
Nelson-Barr
Spont. CP breaking
N/A
Massless u
θ unphysical
N/A
DFD
spectral action + det-orthogonality + η-invariant Thm. L.3

2.

Physical Mechanism

• CKM (quarks): Both up-type and down-type
quarks localized at vertices → small overlaps →
small mixing
• PMNS (leptons): Charged leptons at vertices, but
neutrino R-H sector at the center → large overlaps
→ large mixing

112
3.

Tribimaximal Base

When neutrinos are centered, they have equal overlap
with all three vertices:
p

p
p2/3 p1/3 p0
(457)
UTBM = −p 1/6 p1/3 p1/2
1/6 − 1/3
1/2
giving θ12 = 35.3◦ , θ23 = 45◦ , θ13 = 0◦ .

consequence of the Weitzenböck identity on curved spatial
slices.

1.

Setup: DFD Spatial Geometry

The deep-field scalar profile ψ(r) = ψ0 − B ln(r/r0 )
with
B=

4.

Corrections

The deviations from TBM are those of App. X:

2p
GM a0
c2

induces a conformally flat spatial metric hij = e2αψ δij .
In the deep-field annulus (galactic outskirts), this metric
has strictly positive Ricci curvature in angular directions:
Ricθθ = Bα(2 − Bα),

TABLE XLII. PMNS angles: tribimaximal + corrections.
Angle TBM
θ12
35.3◦
45.0◦
0◦

θ23
θ13

Correction Source
Predicted Observed
trimaximal TM1,
34.4◦
33.4◦
2
sin θ12 = 0.3184
(App. X)
µ-τ reflection (maximal)
45.0◦
43–49◦
2
◦
sin θ13 = 3α
8.5
8.6◦
(stated conjecture)

PMNS Matrix: Trimaximal Structure
Large neutrino mixing arises because:
• Charged leptons at CP 2 vertices (hierarchical, like
quarks)
• Neutrino R-H sector at the center (democratic)
• Tribimaximal mixing as leading order
This explains why PMNS ̸= CKM.

a. CKM mixing. The CKM matrix has Wolfenstein
structure:


1 λ λ3
VCKM ∼  λ 1 λ2  , λ = e−d/σ ≈ 0.22,
(458)
λ3 λ2 1
where d/σ is the ratio of vertex separation to Higgs width.
The CP-violating phase is supplied by Postulate Y.10
(App. AO).
b. Neutrino masses. Lepton number L is not topologically protected. Right-handed Majorana masses
MR = MP α3 ≈ 5 × 1012 GeV (Appendix P) give the
see-saw formula:
M2
mν ∼ D ∼ 0.1 eV.
(459)
MR
Large PMNS mixing arises from different localization
patterns for charged leptons vs. neutrinos.

O.

Infrared Scale for Yang-Mills from DFD
Geometry

The DFD deep-field geometry adds a curvature term
to the infrared spectrum of Yang-Mills fluctuations—a

(460)

Ricrr = 0.

(461)

For 0 < αB < 2, the angular Ricci components are
positive.

2.

Weitzenböck Identity

For 1-forms on a Riemannian 3-manifold:
∆Hodge A = ∇∗ ∇A + Rich (A).

(462)

The Ricci tensor enters as an effective positive potential
for Yang-Mills fluctuations.

3.

The DFD-Induced Infrared Bound

Proposition XVII.3 (DFD-induced infrared scale). On
a bounded domain Ω containing a deep-field annulus with
Rich (v, v) ≥ Λ h(v, v) for some Λ > 0, the smallest
nonzero eigenvalue λ1 of the spatial Yang-Mills operator satisfies:
p
(GM a0 )1/4
λ1 ≥ C1 Λ,
meff ≡ λ1 ∼
.
(463)
cR
a. Numerical scale. For Milky Way parameters
(M ∼ 1012 M⊙ , R ∼ 10 kpc):
meff ∼ 10−30 eV,

(464)

far below the QCD mass gap.

4.

Clarification: What This Does Not Claim

Important Clarification
This mechanism does not solve the Clay Yang-Mills mass
gap problem:
• The Clay problem is formulated for pure SU(N )
Yang-Mills on flat R4
• The DFD mechanism requires curvature of spatial
slices
• The induced scale ∼ 10−30 eV is irrelevant for
hadron physics

113
a. What is established. In any realistic DFD cosmology, Yang-Mills fields never live on exactly flat spatial
backgrounds. The same deep-field parameter a0 that controls galactic dynamics also adds a tiny curvature term
to the gauge fluctuation spectrum through background
geometry. This is a structural result, not a solution to
the mass gap problem.

P.

Testable Predictions

The gauge extension makes structural statements and
model-dependent predictions:
a. Structural statements (consistent with observation).
• Ngen = 3 — a stated input, identified with
χtop (CP 2 ) (Rem. F.21)
• Gauge group SU (3) × SU (2) × U (1) — electroweak
factor under one stated identification, colour an
input (App. F)
• Chiral fermion spectrum — an input, as in the
Standard Model (Rem. F.3)
b.
c.

Model-dependent predictions (testable).
Current status.

• ka ≈ 51.4: Consistent with SPARC RAR fits
• Nuclear clock: null under the minimal action
(Thm. XI.2); extended-channel window 26 Hz to
kHz (Sec. XI F)
• Fermion masses: 1.42% leading-order mean error,
far outside the measured uncertainties
• CKM matrix: the four Wolfenstein values give
χ2 = 10.1 for four parameters against the global fit
(App. AO)

Q.

a.

Caveats and Required Verification

What is claimed (each is graded where it is treated).

• The electroweak group is the isotropy group U (2)
of CP 2 under one stated identification; colour and
the family content are inputs, and (3, 2, 1) is the
minimal partition carrying them
• q1 = 3 derived from the isotropy charge 6Y (App. F)
• Ngen = 3: a discrete input, numerically |k3 k2 q1 |
(bookkeeping identity, not an index theorem —
App. F)

• θQCD = 0 (tree level, with θbare = 0 fixed by the
spectral action given that nothing is added to it
for the gauge fields; loop-induced ≲ 10−16 ; Theorem G.2)
• τp = ∞ if baryon number is violated only by the
electroweak anomaly (Thm. F.26)
• UV stability of all topological results (Theorem F.28)
• ka = 3/(8α) from frame stiffness ratio × EM duality
(Theorem G.1)
• α−1 : closed form exact under two stated choices;
last 21 ppm on the open bracket (§X)
• Bridge Lemma: kmax = χ(CP 2 , E) = 60 for E =
O(9) ⊕ O⊕5
• CKM matrix with λ = 0.225 (localization model;
canonical 31α = 0.2262, App. AO)
• PMNS matrix (trimaximal TM1, App. X)
√
• Higgs scale: exponent 8 of v = MP α8 2π (prefactor
fitted)
• Strong CP: θ̄ = 0 at tree level, loop-bounded, with
θbare = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields (Theorem L.3; no axion)
b.

Experimental status.

• ka ≈ 51.4: Consistent with SPARC RAR fits
• Nuclear clock: null under the minimal action
(Thm. XI.2); extended-channel window 26 Hz to
kHz (Sec. XI F)
• Fermion masses: 1.42% leading-order mean error
(not within the measured uncertainties)
• CKM parameters: χ2 = 10.1 for four against the
global fit, λ at 1.8σ (App. AO)
• PMNS angles: trimaximal, θ23 = 45◦ , in soft ∼2σ
tension with NuFIT 6.0 (octant unresolved)
• Higgs scale: v = 246.09 GeV vs 246.22 GeV observed (exponent derived, prefactor fitted)
c. Falsification criteria for topological results. The
gauge emergence framework makes three hard predictions:
1. 4th generation detection → falsifies Ngen = 3

• Energy minimization selects (1, 1, 3) flux configuration

2. QCD axion detection (KSVZ/DFSZ range) →
falsifies θ = 0

• κr = nr κ0 from Ricci curvature of CP nr −1 (Theorem F.25; a motivated assumption, §XVIII K)

3. Proton decay observation (any rate τp < 1040
yr) → falsifies τp = ∞

114
TABLE XLIII. Predictions from the gauge extension.
Prediction

Value

Test

Status

ka (self-coupling)
3/(8α) ≈ 51.4
RAR normalization
✓
Strong CP suppression θQCD ≈ 0
|dn | < 10−26 e · cm
Pending
ψ-coupled running
δg/g ∝ ki ψ
Nuclear clock ratio
2026–27
α = 1/137
From kmax = 60
21 ppm (integers; bracket open) ✓
9 fermion masses
1.42% LO; seven prefactors data-selected PDG comparison
✓
CKM λ
31α = 0.2262
PDG: 0.22501(68)
✓

d. What is currently claimed. The gauge emergence
framework is proposed to organize the following from
CP 2 × S 3 topology:
• Standard Model gauge group SU (3) × SU (2) × U (1)
• Three fermion generations, a stated input identified
with χtop (CP 2 ) (App. F)
• Fine-structure constant α = 1/137 from ChernSimons
√
• Electroweak-scale benchmark v ∼ MP α8 2π
• All 9 charged fermion masses (1.42% leading-order
mean error; seven prefactors data-selected)
• CKM and PMNS mixing matrices
• Strong CP: θ̄ = 0 at tree level, loop-bounded, with
θbare = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields (Theorem L.3)
• Proton stability: τp = ∞ (if baryon number is
violated only by the electroweak anomaly)
e.

What remains.

1. Experimental confirmation: LPI test, clock
anomalies, T 3 phase (a null prediction)
2. Community verification: Independent review of
derivations
Note: the gravity sector can stand independently of the
microsector. The microsector itself remains a live development program: several results are strong, but others still
rely on structural assumptions that deserve independent
mathematical closure.

Summary: Gauge Extension and Microsector
Conditional: the electroweak U (2) as the isotropy
group of CP 2 under one stated identification (colour an
input); τp = ∞ (if baryon number is violated only by the
electroweak anomaly). (Ngen = 3 enters as a discrete
input — App. F.)
Derived:
• Fine-structure constant: closed form of §X, last
21 ppm on the open bracket; identification with
1/e2 not derived
√
• Higgs scale: exponent 8 of v = MP α8 2π
(prefactor fitted)
• Strong CP: θ̄ = 0 at tree level, loop-bounded, with
θbare = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields
(Theorem L.3)
• Bridge Lemma: kmax = χ(CP 2 , E) = 60 connects
α to mass tower
• Top and charm masses; the other seven rest on
data-selected prefactors (App. FM)
• CKM matrix: λ = 31α = 0.2262 (integer pattern,
App. AO); CP 2 vertex-separation motivation
• PMNS matrix: trimaximal TM1 (App. X);
sin2 θ13 = 3α a stated conjecture
• Koide relation: Qℓ = 2/3 is not derived (DFD’s
α-power lepton spectrum gives Qℓ ≈ 0.665;
measured 2/3 holds to ∼0.001%, beyond DFD’s
∼1% mass precision)
Coupling constants: ka = 3/(8α) (Theorem G.1).
Status: Partially closed microsector program with
several strong results and several still-open structural
selections. Awaiting both experimental and mathematical
verification.
Full proofs: Appendices F–H and K.

XVIII.

OPEN PROBLEMS AND LIMITATIONS

This section names the main open problems and limitations of DFD, grouped in §XVIII A, and separates
theoretical gaps from scope boundaries. The list is not
complete: further open steps are graded where they arise
in the text, and the standing review of §XVIII K grades
the inherited proofs flagged by a proof-pattern scan.
a. Axiomatic status of the frontier completion. The
structural upgrades in Secs. XI B, XI C, V A 3, and the
forward perturbation skeleton (Sec. XVI L, Eqs. (416)–
(419)) are stated as an axiomatic extension of the core

115
DFD postulates. Every derived result (clock ratio cancellation, screening law, Geff , trace–TT decoupling) is a
theorem of the enlarged system. The additional axioms —
common-scale factorization, response functional, microsector hierarchy, dust branch, parent strain field — are
explicitly labeled throughout.
We distinguish four claim-status levels:
T0: Theorem from the core DFD postulates: exact RAR
inversion for µ(x) = x/(1 + x).
T1: Theorem from the enlarged frontier-axiom system:
clock-ratio cancellation, variational screening law,
A5 finite-symmetry closure (kmax = 60), speciesassignment canonicality, linearized perturbation operator, Geff growth law, trace–TT principal decoupling, luminal TT wave equation.
E: Empirical benchmark or auxiliary modeling input:
residual channel hierarchy λα ∼ ϵ2H α2 /(2π),
λN,e,s ∼ ϵH α2 /(2π).
F: Open program item: first-principles derivation of the
species–class map from CP 2 × S 3 , full production
P (k)/Boltzmann-level cosmology pipeline, narrowing of the nuclear-clock prediction band beyond the
stated benchmark, the interacting operator-ordering
map, which DFD shares with every quantum field
theory (App. QM). The cosmological growth regime
is not on this list: the Rest-Mass Channel confines
the polarization to bound systems, so unbound linear perturbations grow with Geff = G (App. GR).
The main open items are listed first (§XVIII A); the subsections after them treat specific problems, several of
which DFD resolves.
A.

Main Open Items

The items below are the main open problems of the
theory, one line each and grouped by sector. Each is
graded where it is treated, and the list is not complete.
The inputs the theory adopts or measures are listed in
Tables CXXVII and CXXVIII; an item leaves this list
when it is derived, reduced to a stated input there, or
handed to a named observation, and each line says which.
The main open problem is the origin of the primordial perturbations: the printed theory contains no mechanism for
them, so they enter through one stated starting condition,
the smooth beginning of App. Q, with their amplitude
As a calibrated number (App. FK).
a. Gravity in bound systems and the dark sector.
• The Rest-Mass Channel, that the µ-law acts only inside
bound systems, is adopted, not derived; its Poisson
term is the Einstein–Hilbert term’s own static scalar
energy. The floor law in energy form is adopted too; in
spherical symmetry it fixes the momentum-conserving
rule of motion, which keeps action and reaction and
gives up χ’s equivalence principle only where χ falls
short of the medium’s demand (App. GR, App. AV).

• The energy’s three-dimensional filling rule is a stated
clause, χ counting by what it adds to the size of the
total rest-channel field; its mode-by-mode form has no
rule of motion of its own, and which system’s demand a
χ halo fills is open where two systems’ fields blend, as for
the Milky Way and the Large Magellanic Cloud. Two
named observations test the energy: the Local Group’s
matter barycentre, which it predicts at rest where the
total-ψ rule predicts a drift, and the disc’s vertical
force at the Sun, which it puts at 66–69 M⊙ pc−2 with
χ at the rotation-curve density, against the measured
68.8 ± 1.9 and the static µ-law’s 84 ± 4, 3.1–3.2σ high
(App. AV).
• The floor composition does not follow from an action
of the printed type; in energy form it is one adopted
input, and total-field equilibrium states the same law
for the medium in spherical symmetry (App. AV).
• At estimate grade galaxy χ halos assemble as in ΛCDM,
and SPARC then needs cored inner halos, which fit it
with no change to the rotation-curve law. In spherical
simulations under the adopted energy neither slow disc
formation, which contracts χ as Newtonian gravity does
at z = 0 (computed on 120 SPARC galaxies), nor, in
dwarf and low-surface-brightness galaxies, the bursty
feedback from stars that cores halos under Newtonian
gravity makes these cores (computed); their origin is
open, and in discs above about 1011 M⊙ of baryons the
halo’s mass is the open item: the abundance-matched
halo fails SPARC and the lensing of isolated late-type
galaxies there, and lighter halos pass both (App. GR,
App. AV). Dwarf satellites share the question: cuspy
χ subhalos at abundance-matched masses exceed the
dark-matter densities measured at 150 pc in several
M31 dwarfs (estimate).
• The flat weak-lensing profile of isolated galaxies is not
kept beyond their bound radius, as the halo model of
ΛCDM does not keep it; spectroscopically isolated lens
samples decide whether the measured far signal is the
galaxies’ own (App. GR).
• Environmental confinement is an adopted axiom: the
Bose sector caps the boost but does not confine it. The
axiom names the even-handed (median) readout of the
host mode, which fixes where the switch sits, and states
that the readout is sharp; the Oort comets need it at
least as sharp as a Hill switch with exponent about 4
to 5, not an exact step. Local wide binaries, which it
predicts Newtonian, and the edge step at 13–16 kpc
test it; the outer-halo globular clusters beyond the step
show about a third of the predicted boost (App. NL).
• The nonlocal galaxy law has no free function on its
depth of record, FB-OUT: the detector form and the
width belong to the accumulator reading only. The completion’s reciprocal terms are computed for discs (the
fitted RAR knee is a0 , and the Freeman surface density
is the local law’s, a0 /2πG = 137 M⊙ pc−2 ), isolated
dwarfs follow the local law, and the law’s relativistic
content is the optical metric on the preferred foliation.
Satellite dwarfs are handed to observation: on the data

116
published in 2020–2026 the seven M31 satellites that
are tidally safe at the new distances sit about 2–3σ
above the forward law’s own prediction, which the ceiling ν(τhost ) caps, and about 2σ above the completed
law (estimate), and the four selected as tidally safe on
earlier distances, only two of which are tidally safe at
the new ones, 5.7σ and 4.4σ; the ceiling omits the χ
branch of the floor law, and a combined 5σ excess over
the completed law in deeper dispersions of externalfield-dominated M31 satellites falsifies it in satellites
that pass an equilibrium check and whose stars the host
pushes, relative to their own χ, harder than their own
gravity (App. NL).
• The exact (Bose) shape of the MOND response rests
on one stated input, the ground-state principle µ =
P(n=0), which gives ν = Z1 (App. BS).
• The domain-local screen is an optional extension that
no claimed result uses; the domain that sets its local MOND scale is not derived, and per-galaxy a0
set against measured environment densities decides it
(App. ZT).
• The epoch law a0 (z) is contested: massive discs at z ≃ 2
sit where the predicted rise should show and do not
show it on the published baryon census, on which they
rotate 0.1–0.17 dex slower than DFD predicts, 1.5–2.2σ
when the census’s calibration is counted; fits that leave
the baryonic mass free neither show nor exclude the
rise; on the published census the survey’s amplitude
lies 0.094–0.103 dex above DFD’s a0 on either clock, a
coherent baryonic-mass shift (App. A0Z).
b. Cosmology.
• The origin of the primordial perturbations and the horizon problem are open: both enter through one stated
premise, the smooth beginning of App. Q, a smooth
background at the bang with small regular ripples of
the calibrated amplitude As and the stated tilt, and the
theory states no mechanism for either: no field in the
printed theory produces the perturbations, the derived
routes close (the spectral action, the Higgs, the χ and
Chern–Simons sector, thermal and holographic fluctuations, the trace anomaly, anisotropic scaling, relaxation
of the internal spectrum, the E-transport clock and the
Majorana sector; Prop. FK.3), and the fast-light past
is the ordinary expansion read on the arena clock, too
slow to open the horizon (App. Q). Closing either needs
an accelerating stage that reheats, which the printed
theory does not contain, so As is one calibrated input,
which the formula 32πα5 matches without a derivation
(App. FK).
• The vacuum coefficient R = ΩΛ C = 0.6042 ± 0.0027
behind H0early and the dark-energy density is one measured input, as the cosmological constant is in general
relativity; no printed principle fixes the shape of the
spectral cutoff on which it sits, and it depends on the
internal spectrum (App. FK).
• The Hubble tension is read as one universe on two
clocks (Sec. ZS 5): the CMB-anchored channels read the
early clock and local probes the bound-matter clock, so

SH0ES’s high local value is the outcome the postulate
expects; the postulate fails if bright sirens or masssheet-robust time delays settle at 67.5, and the absolute
a0 scale tests it once baryonic masses are known to
about 2% (App. ZS). The size of the clock ratio, λ =
H0loc /H0early (Eq. (ZS4)), is measured and its derivation
is open.
• The early universe rests on two stated inputs, the noboundary state with nothing inserted and the smooth
beginning, which carries the arrow of time; the optical
engine needs a negative-pressure source and reheating
that the printed theory does not supply, and the χ relic
bounds it, so r ≲ 10−7 is a bound, not a predicted
number (App. Q, App. SM).
• The χ abundance is derived conditional on one identification, of fχ = M̄P α3 as the flux period, on the asserted
premise P4′ that the 2π/α winding instanton changes
the flux sector (Lemma AV.27), on the stated count
158 in mχ = 32.005 eV and on two stated conventions;
the uniform count, 171, would raise Ωχ h2 by 2%, which
the CMB can separate (App. AV).
• The baryon asymmetry is one calibrated input: the
printed mechanism derives its α4 power and its B−L
current, but its first-principles value is about (2–3) ×
10−9 , three to five times Planck’s value, and the sign
rests on an adopted axiom (App. FK).
• Structure formation on the current χ carrier is ΛCDMlike on linear scales (σ8 = 0.806, S8 = 0.822) and in
the halo abundance at z = 6–15 (computed); the open
parts are numerical: the small-scale lensing boost of
the confined polarization, which depends on the boundbaryon profiles of haloes, and the high-redshift efficiency
lever of a0 (z) (App. PK, App. GR).
c. The tensor sector and the field equation.
• In the mathematics, smooth solutions break down in
finite time on every non-zero background, and whether
weak solutions exist for all time and are unique is open,
as it is for the equations of gas dynamics; the horizon
conditions apply at the event horizons of DFD’s black
holes, and their uniqueness and regularity are open
(§III F).
• Through second post-Newtonian order the one- and
two-body dynamics and the flux are general relativity’s,
with the parent strain’s σ kept (ϵ = 1, deflection 15π/4;
§IV, App. WS), and ψ loops and scalar memory are
bounded far below observability (App. ZU, App. AT).
d. Particle physics and the inner space.
• The seven vacuum axioms that single out CP 2 × S 3 are
adopted, not derived (App. AB, §XVIII C).
• The electroweak group is the isotropy group of CP 2
under one stated identification (the hypercharge as the
direction diag(1, 1, −2), normalised as 6Y ), which is
adopted, not derived; colour and the family content are
inputs, as in the Standard Model (App. F).
• The generation count Ngen = 3 is one stated input,
used once; no index on the internal space supplies it
(Rem. F.21).
• The closed form for α is exact under two stated choices,

117
the ρ-shifted Casimir on S 3 = SU (2) and α’s own
trace with the heat-kernel regulator, and q1 = 3 and
kmax = 60 follow given the electroweak identification,
kmax through the conditional determinant-line count;
the last 21 ppm rests on the bracket coefficient c = 1/13,
whose provenance is open (§X); the trace in the closed
form is the hypercharge trace, and its identification
with 1/e2 rather than with the hypercharge coupling is
not derived (§X I).
• The charged-fermion masses other than the top and the
charm rest on seven data-selected prefactors, one per
species
√ (App. FM).
• The 2π prefactor of the Higgs scale is fitted, with
a no-go on deriving it, and λH = 1/8 is a conjecture
whose matching scale is not derived: imposed at the
top-quark mass it gives mH = 124.6 GeV, 0.40% (2.7σ)
below the measured value (App. Z).
• The CP phase of quark mixing rests on a stated postulate, the channel space C[A5 ] is a stated postulate, and
sin2 θ13 = 3α is a stated conjecture that JUNO tests
through θ12 (App. AO, App. Z).
• The neutrino branch is one of four equally natural
fractions, chosen by oscillation data, and the Diracneutrino Yukawa rung is a stated input (App. X).
√
(5)
• The strong scale ΛMS = 4π MP α19/2 is one stated
discrete input: its exponent is the only half-integer rung
in the empirical window, and it gives αs (MZ ) = 0.1187,
0.8σ from the world average; the string tension and
the glueball mass then follow from QCD, their ratios
to Λ being lattice numbers, as in the Standard Model
(App. YM).
• The soliton reading of the proton is posited: it needs a
quartic term, which the printed action does not contain
(Rem. F.27). Proton stability does not use it and holds
if baryon number is violated only by the electroweak
anomaly (Thm. F.26).
• Why the vacuum energy of matter does not gravitate
is open in DFD as in every quantum field theory with
gravity: the ψ source is linear in the renormalized stress
tensor, a uniform vacuum energy acts on the background
expansion and, as Λ does, on bound systems, and the
lock of the lapse to the scale factor does not stop it
gravitating. The dark-energy relation assumes that
the matter piece, of order m4t /64π 2 ≈ 1053 ρΛ , and the
spectral action’s own Planckian vacuum term, about
10120 ρΛ (App. WS), each cancel (App. AU).
e. Foundations.
• The interacting quantum dynamics and the measurement postulate are those of every quantum field theory;
ℏ is the unit of action, and the Dirac β is derived
(App. QM).
The inherited proofs flagged by the proof-pattern scan are
graded in §XVIII K: twenty-two are re-derived and twentytwo are cited, not re-derived; they form a verification list,
not open problems. Mach’s principle is not a claim of
DFD: a uniform ψ background is equivalent to a change
of units, which leaves every mass ratio, clock ratio and
G unchanged, so it cannot generate inertia. The applied

conjectures of App. V, among them the Cooper-pair and
Casimir coefficients, lie outside DFD’s claims and are not
open items of the theory.

B.

Quantum Superpositions and the Penrose
Paradox

a. The Penrose paradox. In GR-based approaches to
gravity-quantum coupling, spatial superposition of masses
appears to create branched geometries. If a mass M is in
superposition at locations A and B, does spacetime curve
“both ways”?
b. Why DFD resolves this paradox. In DFD, there is
one flat R3 with one scalar field ψ. The resolution follows
from there being a single ψ sourced by the quantum
expectation of the density—the source equation is linear
in ρ:
∇ · [µ(|∇ψ|/a⋆ )∇ψ] = −

8πG
ρ.
c2

(465)

For a quantum superposition |Ψ⟩ = cA |A⟩ + cB |B⟩:
1. The source density is ρ = |cA |2 ρA + |cB |2 ρB (quantum expectation value)
2. The ψ field responds to this weighted average
3. No “branched geometry” exists; there is one ψ field
for the system
c. The semiclassical Schrödinger–Newton reading and
the operator completion (App. ZU). The source equation
above is linear in the density ρ—which is what resolves the
branched-geometry paradox—but this does not by itself
decide linear versus nonlinear quantum evolution. Under
the semiclassical reading—the classical-ψ-by-design axiom
(Sψ ∼ (MPlanck /a⋆ )2 ≫ ℏ) taken as fundamental, with
ψ sourced by the c-number expectation ρ = ⟨Ψ|ρ̂|Ψ⟩ =
m|Ψ|2 —the source is quadratic in Ψ, and substituting the
resulting ψ[Ψ] back into the matter evolution gives a Ψdependent generator: the nonlinear Schrödinger–Newton
(semiclassical-gravity) equation
h
i
ℏ2 2 1 2
iℏ ∂t Ψ = −
∇ − 2 mc ψ[Ψ] Ψ,
2m
(466)


8πG
∇ · µ(|∇ψ|/a⋆ ) ∇ψ = − 2 m|Ψ|2 .
c
The semiclassical side is not DFD’s. The operator completion of the same printed action (App. ZU) replaces it:
the c-number source ρ = m⟨Ψ|ρ̂|Ψ⟩ is the Hartree factorization of the derived two-body vertex, the one-particle
sector is exactly free, and the completion predicts exactly
zero gravitational self-shift of a trapped crystal’s centerof-mass mode (the Kohn null, Thm. ZU.2) to all orders
in αψ with zero parameters. The Schrödinger–Newton
frequency shift implied by the equation above is therefore
not a DFD prediction but the falsifier of the operator completion: a measured positive sideband at the predicted

118
size (∆ω/ω ∼ 10−13 , osmium; see App. ZU) would falsify the completion and support the semiclassical reading.
Exact linearity and unitarity of the test-particle core remain as derived by the one-H/J⋆ theorem (Thm. QM.10,
Cor. QM.12), given the linearized stability it assumes;
what changes is the self-gravitating sector’s status: linear
under the operator completion, nonlinear only under the
semiclassical commitment.
d. Discrimination from Diósi–Penrose: the deepMOND self-energy and the sustained-fringe prediction.
The Diósi–Penrose (DP) mechanism predicts collapse
when the gravitational self-energy difference exceeds ℏ/τ ,
with a Newtonian self-energy EG ∼ Gm2 /d. DFD’s selfenergy differs structurally: for a delocalized mass the
internal self-acceleration sits far below the MOND scale—
e.g. a ∼ 1010 -amu, 100 nm superposition has gself /a0 ∼
9 × 10−4 ≪ 1—so it lies in the deep-MOND (µ → x,
AQUAL) regime, where the self-binding energy is enhanced over the Newtonian Gm2 /d, but every laboratory
and Solar-System mass sits in hosts (the Earth, the Sun,
the Galaxy) whose baryonic field exceeds (ln 2)2 a0 , so by
the environmental-confinement axiom (App. NL) its selffield is exactly Newtonian; the enhancement would arise
only in a weak host. However, an enhanced self-energy
is not the same as self-decoherence. DFD’s no-branching
theorem (Appendix AH) settles this: the classical ψ is
sourced by the c-number expectation ρ = ⟨Ψ|ρ̂|Ψ⟩, giving one averaged optical metric for the whole superposition with no per-branch self-force. Even where such a
self-energy arises, it does not decohere under c-number
sourcing, no matter how strong the binding. Hence DFD
predicts sustained fringes at MAQRO (distinct from
Diósi–Penrose collapse), and this is not a free choice: the
no-branching property follows from the two-body character of the derived vertex itself—the mean field of a
two-body operator is a functional of the one-body density
matrix of the full state alone (App. ZU)—so per-branch
self-force is excluded without invoking the classical-ψ
axiom. If one nevertheless entertained per-branch selfgravity, the deep-MOND enhancement would speed the
Newtonian Schrödinger–Newton collapse by ∼ 20× (collapse window ∼ 7 × 10−5 s for a 10−13 kg / 250 nm mass)
and would operate precisely in the deep-MOND regime
gext < a0 —but that is the excluded alternative, not the
DFD prediction.
Space-based matter-wave interferometry (MAQROclass) is the natural discriminator: DFD predicts sustained interference where Diósi–Penrose predicts collapse.
The deep-MOND-enhanced collapse window above is the
per-branch self-gravity alternative excluded by the nobranching theorem, retained here only to delimit what
DFD is not. The DFD prediction (sustained fringes) is
already testable at the MAQRO 10−13 kg frontier.

C.

UV Completion: the Topological Proposal

a. The traditional UV problem. In General Relativity, the UV completion problem is acute: spacetime curvature diverges at singularities, and the theory is nonrenormalizable when quantized. This requires unknown
“quantum gravity” physics at the Planck scale.
b. How DFD addresses this problem. DFD has a different structure that addresses the traditional UV problem:
1. Flat arena, absolute time: DFD postulates
flat R3 with absolute time. Where µ → 1 the
physical metric has general relativity’s curvature,
but DFD’s time is the uniform-expansion slicing,
maximal around isolated bodies (App. WS), whose
slices through a collapsing star stop at a limiting
surface inside the horizon, R = 3GM/(2c2 ) for
Schwarzschild, with the lapse decaying exponentially in absolute time, and never reach the singularity. For maximal slices this is proved for spherical
collapse and for crushing singularities and argued
for generic rotating collapse; that DFD’s slices stay
maximal during collapse rests on the argument of
App. WS.
2. Classical ψ by design: The action scales as
Sψ ∼ (MPlanck /a⋆ )2 ≫ ℏ, ensuring quantum fluctuations of ψ are negligible. Its quantized (operator)
completion is App. ZU.
3. Gauge structure from topology: The gauge
fields are Berry connections on the internal mode
bundle over spacetime (§XVII B); the electroweak
group is the isotropy group of CP 2 under one stated
identification, and colour and the family content are
inputs, as in the Standard Model (App. F)—this is
the proposed UV physics.
4. Constants from topology: the closed-form value
of α, v and the leading fermion-mass and mixing
patterns are computed from the topology, not from
unknown high-energy physics; the charged-fermion
masses other than the top and the charm, the Higgs
prefactor and several flavour inputs are not yet
derived (§XVIII A).

TABLE XLIV. Comparison of theoretical frameworks and
their UV statuses.
Theory
Gen. Relativity
Fermi Theory
Chiral PT
BCS
DFD

Low-Energy
Curved spacetime
4-fermion contact
Pion/kaon dynamics
Cooper pairs
Scalar-optical

UV Completion
Unknown
Electroweak
QCD
e-phonon
CP 2 × S 3 (proposed)

119
c. The proposed completion. The CP 2 × S 3 gauge
emergence framework is proposed as DFD’s UV completion, as QCD is for chiral perturbation theory; its
ultraviolet finiteness is graded conjectural (§XVIII K).
Specifically:

• Of finite lifespan for fully nonlinear dynamical
evolution on a non-zero uniform gradient, where
smooth solutions reach a gradient singularity in
finite time (Sec. III F); open on the degenerate background ∇ψ → 0

• The α-relations are organised by this topology, each
at its own grade (App. F)

b. Open question. Does the coupled system (DFD
scalar, TT tensor and the constrained pieces of the parent
strain) admit a well-posed initial value formulation for
arbitrary strong-field, dynamical configurations? Where
µ → 1 it is the Einstein–Hilbert system in DFD’s slicing
(App. WS).
c. Partial results. The low-energy EFT preserves hyperbolicity under small perturbations. The perturbation
metric:

• Strong CP: with θYM = 0 fixed by the spectral
action given that nothing is added to it for the
gauge fields, θ̄ = 0 at tree level, loop-bounded (Theorem L.3; no axion required)
d. What remains. The origin of the CP 2 × S 3 topology is the deepest open question, and it is not the only
one: the gauge group is selected rather than forced, the
generation count is an input, and several constants are not
yet derived (§XVIII A). For physics purposes the topology serves as the foundational postulate from which the
derived results follow.

G µν = W ′ (X)η µν + 2W ′′ (X)∂ µ ψ∂ ν ψ
00

(467)
ij

satisfies hyperbolicity conditions (G < 0, det G > 0)
for the constrained µ-family, except at ∂ψ = 0, where
both conditions fail since W ′ (0) = µ(0) = 0.
d. Required work. Full numerical relativity codes for
DFD would need:
1. ADM-like decomposition of the coupled system

D.

Hyperbolicity and Numerical Evolution

a. Current status. The DFD field equation with constrained µ-function is:
• Elliptic in the static limit (well-posed boundary
value problem)
• Hyperbolic for small perturbations about smooth
backgrounds

2. Gauge conditions ensuring constraint propagation
3. Boundary conditions for the µ-crossover regime
4. Stability analysis for black hole merger configurations
This is deferred to future work.

E.

Cluster-Scale Phenomenology: the χ Budget

Clusters: the χ component is required
Under the universal µ(x) = x/(1 + x) alone the sixteen clusters of App. I fall short of their measured masses (Obs/DFD
= 1.57 ± 0.08 relaxed, 1.99 ± 0.16 merging, at a0 = 1.20 × 10−10 m/s2 and at the bound-matter a0 = 1.1966 × 10−10 m/s2 ).
The shortfall is the derived χ: with χ adding its Newtonian pull and a hydrostatic bias b = 0.21 on the relaxed X-ray
masses, all sixteen need χ = 4.5 ± 0.6 Mb inside r500 , 0.84 ± 0.12 of the cosmic mix; under the floor composition of App. AV
they carry the ΛCDM budget, 1.28–1.51 of the cosmic mix (computed). The amplitude follows from the derived abundance:
under the floor composition and the adopted momentum-conserving energy of App. AV, cluster halos assemble as in ΛCDM,
and 1.21 ± 0.05 of the cosmic mix is predicted inside r500 , against 1.28 ± 0.04 measured with no hydrostatic bias and
1.51 ± 0.04 with b = 0.21 (estimate; App. I), while χ boosted like the baryons would over-predict the cluster masses
1.43-fold. Open: a structure-formation run under the adopted energy and a cluster-by-cluster nonlinear DFD
hydrodynamic and lensing solver that includes χ.

a. The correction stack does not close the gap. A
no-χ closure by five correction factors (baryonic completeness, PDE-calibrated substructure averaging JPDE =
1 + 0.39fsub , temperature, merger and projection systematics) does not hold: the baryonic items (WHIM, intracluster light, gas beyond r500 , IMF, clumping) sum to
1.26 and to 0.96–1.10 inside r500 ; JPDE ≃ 1.04 for the
baryonic substructure the µ-law responds to; the tem-

perature factor tracks the merger flag and absorbs the
hydrostatic bias; and the projection factor has the wrong
sign for lensing masses.
b. Per-cluster results.
• Relaxed clusters (n=10): Obs/DFD = 1.57 ±
0.08 under the µ-law alone, at a0 = 1.20 ×
10−10 m/s2 and at the bound-matter a0

120
• Merging clusters (n=6): Obs/DFD = 1.99±0.16
under the µ-law alone
• χ required: 4.5 ± 0.6 Mb , 0.84 ± 0.12 of the cosmic
mix, for all sixteen with b = 0.21 when χ adds its
Newtonian pull (1.28–1.51 under the floor composition; 1.21 ± 0.05 predicted, an estimate)
See Appendix I for complete analysis.
c. Galaxy groups. On published X-ray masses the
µ-law falls short of galaxy groups as it does of clusters (median Obs/DFD 1.22–1.36 at r500 for 23 Chandra groups),
and under the floor composition groups need χ at about
the clusters’ share of the cosmic mix (App. I).
d. Universality of µ. µ is universal from galaxies to
clusters; the cluster excess is χ, not a scale-dependent µ.

F.

Cosmological Constant: the Hierarchy from
Topology

a. The traditional problem. In ΛCDM, the cosmological constant “problem” has two aspects:
1. Fine-tuning: ρΛ ∼ (10−3 eV)4 while QFT predicts
4
ρvac ∼ MPlanck
—a 10122 discrepancy
2. Coincidence: Why is ΩΛ ≈ 0.7 today, comparable
to Ωm ?
b. Topological determination. Section XIX links H0
and MP through topology. A corollary is:

2
H0
= Cαkmax −Ngen = Cα57 ≈ 1.39 × 10−122 (468)
MP
This isolates the cosmological hierarchy: the exponent
is not fine-tuned, while the residual O(1) normalization

C is measured. The exponent 57 = kmax − Ngen = 60 − 3
follows from:
• kmax = 60: the Spinc index χ(CP 2 , E)
• Ngen = 3: the generation count (discrete input; all
derivation routes exhausted, Rem. F.21)
c. One dark energy. The α57 vacuum term is the
only dark energy in DFD, and no ψ-screen on inferred distances is read as a second, apparent acceleration (App. ZS;
App. SL). A further statement holds:
• Distance duality holds exactly: DL = (1 + z)2 DA
(Etherington reciprocity in the optical metric,
∆ψdual = 0). There is no e∆ψ factor in the luminosity distance (Abstract item E1).
d. Status. The size of the cosmological constant is
set by topology, not tuned: its exponent is fixed, its O(1)
coefficient C is measured on the early clock and is 1 on
bound-matter clocks (Postulate ZS.11), where the closure
scale (3/8π)α57 carries no coefficient; the observed density
in laboratory units still carries ΩΛ and the measured clock
ratio (App. FK), and the cancellation of the vacuum
energy of matter and of the spectral action’s Planckian
vacuum term is open (App. AU; §XVIII A). The 10−122
is:
α

G.

57


=

1
137

57

≈ 10−122

(469)

Full Cosmological Treatment

CMB and cosmology: the analytic framework and its named open inputs
The cosmological observables are derived within ψ-physics (§XVI C):
• Peak ratio R = 2.34 from baryon loading (Planck: 2.21; 6% high; the χ-matter spectrum, run on the measured clock,
gives 2.20, within 0.6σ)
• Peak location ℓ1 ≃ 220.4 from the measured-clock no-screen forward solution, 100θ∗ = 1.04110 (with no global
∆ψ ≈ 0.30 map)
• No postulated dark sector: the dark-matter and dark-energy carriers are identified within the spectrum — dark
matter is the χ-matter field (App. AV, the harmonic b3 three-form), dark energy is the α57 geometric vacuum energy
• Relic abundance — conditional-derived (Finite SU (2)60 CS Vacuum Relic): the χ abundance is derived (no
inflatonP
⇒ no chosen angle) as the finite SU (2)60 CS/WZW vacuum Casimir expectation,
⟨ϑ2 ⟩ = j |S0j |2 C2 (j)/[k(k + 2)] = 0.073 ⇒ Ωχ h2 = 1.635 ⟨ϑ2 ⟩ = 0.1193 (−0.6σ from Planck; App. AV, Thm AV.13).
The operator (Casimir) is forced, the measure is conditional on the asserted premise P4′ (Lemma AV.27), and the
normalization (k(k + 2): Sugawara k+2 × bare k, with ϑmax = 12 turn from χ’s derived Z2 ) rests on two stated
1
conventions; in consistent (turns2 ) units the finite vacuum lies 12.45% below the uniform value 12
: the continuum
2
sin measure lowers it by 15%, the Sugawara factor K/k raises it by 3.3%, and finiteness itself moves it
√ by 0.09%
(App. AV). The radiation-era ODE gives the bridge exactly, with no non-DFD input: κrad = Γ( 14 )2 /(6 3π). The
named open inputs are the two stated conventions, the stated count 158, the premise P4′ and the identification of
fχ = M̄P α3 as the flux period (the flux-period identification, Lemma AV.6; charged as one binary trials bit in
App. CE).

121
a. Boltzmann codes. CLASS and CAMB are GRbased numerical tools that solve the coupled BoltzmannEinstein hierarchy. With the parent strain’s σ kept, DFD’s
linear perturbation equations are general relativity’s, in
the uniform-expansion slicing and for the curvature constant k = 0 (App. WS, Proposition WS.6(f)), so they may
legitimately be run forward on DFD’s background with
the χ cold component — Appendix CL has done this on
the one-clock C = 1 background, and Appendix ZS on
the measured clock. What is not legitimate is treating
DFD as ΛCDM and refitting that model’s free densities:
1. The background is not a fitting freedom: DFD’s
expansion history rests on the measured vacuum
coefficient ΩΛ C and is not refitted
2. The cold component is not a free density but the
derived χ field (Ωχ h2 = 0.1193, App. AV)
3. Λ is not a free vacuum-energy knob but the α57 geometric vacuum term, whose coefficient is measured
The semi-analytic DFD derivation of R = 2.34 and the
measured-clock no-screen forward result ℓ1 ≃ 220 form
the current backbone; the forward Boltzmann runs of
Appendix CL are on the one-clock C = 1 background;
the measured-clock plik-lite fit is in Appendix ZS.
b. Scope boundaries. DFD does not address:
• Nucleosynthesis: BBN proceeds the same way
This is not a “problem” for DFD any more than it is
for electromagnetism—it is simply outside the theory’s
domain. Two early-universe questions are not on this
list. The early expansion is DFD’s own: DFD has no
inflaton, its fast-light past is the ordinary expansion read
on the arena clock, and its χ relic forbids any stage
that freezes perturbations above H ≃ 4 × 1010 GeV, so
primordial B-modes are absent (r ≲ 10−7 ; App. SM); the
pre-BBN optical engine would inflate the optical metric
only with a negative-pressure source and a reheating
stage that the printed theory does not contain, and the
primordial perturbations and the horizon problem are
open: they enter through one stated starting condition,
with no mechanism (§XVIII A). Baryogenesis is the other:
DFD derives the α4 power of the baryon asymmetry ηB
and its B−L bias, and its printed mechanism gives about
(2–3) × 10−9 , three to five times Planck’s value (§XVIII A;
App. FK).
H.

atomic-spectroscopy fits is at or below this level, vastly
below any conceivable experimental sensitivity. The observational convergence of CODATA 2018 and PDG 2024
on rp ≈ 0.841 fm is therefore fully consistent with DFD;
DFD has no proton-radius “puzzle” to explain.
b. Wide-binary sky orientation. DFD predicts exactly zero even (cos2 θ) dependence of the wide-binary
force boost on the angle θ between a pair’s separation vector and the Galactic-centre direction, at every separation
and mass ratio (App. NL, Thm NL.10); the only allowed
term is odd, ∝ (m1 − m2 ) cos θ, and vanishes for equalmass pairs. The zero is structural, not small: the depth
never forms the vector sum gint + gext and the energy is
linear in |∇ΦN |2 . MOND/AQUAL predicts 6.9% peakto-peak at 104 AU and 46.6% at the gint = gext resonance.
It applies to pairs in weak host fields, beyond 13–16 kpc;
local Gaia pairs [53], which DFD predicts Newtonian,
cannot test it; the test is robust to the mass-to-light and
contamination systematics that dominate the contested
amplitude measurements.
c. Matter-wave phase at first post-Newtonian order.
The matter-wave T 3 phase of §XIII is a null prediction,
not an open forecast: the DFD and GR test-body Hamiltonians coincide identically through 1PN (γ = β = 1,
§IV C), so no DFD-specific T 2 , T 3 or T 4 phase exists at
that order in any atom interferometer; the static metric
is general relativity’s at 2PN as well (App. WS), and the
extended species channel Katom is an Eötvös parameter
already bounded below 10−14 (§XV G).

I.

Experimental Verification Timeline

The decisive tests of DFD have different timescales:
a. Priority ordering. Nuclear clocks and crossspecies atomic campaigns come first, because geometric cancellation reduces the cavity–atom channel to a
screened residual test rather than a near-term binary
discriminator.

J.

Closed Routes

The routes below were built, evaluated and closed;
each entry names the quantity that closed it. Nothing
in this monograph rests on any of them. Where a
formal proof exists it is cited; the remainder are numerical
closures at the stated size.

Null Predictions: Where DFD Says “No Effect”

DFD has several sharp null predictions where the absence of any DFD-induced shift is itself a verifiable consequence.
a. Proton charge radius. The proton charge radius
is a null prediction of DFD. At hadronic scales, the refractive field ψ evaluated at the nuclear potential is of
order 10−39 ; any DFD correction to electromagnetic scattering cross-sections, muonic-hydrogen Lamb shifts, or

K.

Standing Internal Review of Inherited Results

The monograph’s proofs were passed through a systematic proof-pattern scan over all ⟨theorem, corollary,
lemma, proposition⟩ environments in both volumes — 582
such environments in total, of which 419 are immediately
followed by a proof. The scan searches for argument shapes
that are known to fail silently, not for errors: it cannot

122
TABLE XLV. Experimental verification timeline.
Timeframe

Test

Decision

Near-term (1–3 yr)
Nuclear clocks (Th-229/Sr)
Strong-sector window: 26 Hz to ∼kHz
Near-term (1–3 yr)
Cross-species clock campaigns
Map composition-sensitive channels
Medium-term (3–7 yr) Same-ion null checks
Bound the pure-α sector
Medium-term (3–7 yr) Matter-wave T 3 (null prediction) —
Long-term (> 7 yr)
Cavity–atom / space missions
Ultimate residual tests

TABLE XLVI. Closed routes and the quantity that closes each. Displacements are against the observed radial-accelerationrelation scatter floor of 0.057 dex [104].
Route

Closing quantity
Reference
√
11
Per-source composition read literally over a τ → N τ ; vflat wrong by 562× at N = 10
Thm NL.21
partition
p
Enclosed-mass composition (direction
ambient τ ≥ πGΣ/a0 ln(R/h) = 1.53 at R0 ;
Prop. NL.22
discarded)
0.184pdex
Field-based law, root inside the scale
τ ≥ gbar /a0 by Jensen; 0.139 dex
Prop. NL.24
integral
cannot chain; 0.157 dex, window-shape spread only App. NL, NL-F
Fixed-width logarithmic window
5.7%
Tidal-tensor class
1.631 dex; no external-field effect at all
—
Local Lagrangian modified inertia
baryonic Tully–Fisher normalisation off by a factor —
4
Local non-Lagrangian modified inertia
galaxies evaporate on radial orbits
—
Every tested memory kernel
anti-dissipative: energy gain per orbit > 0
—
p
1/3 2/3
—
τ = gobs /a0 (total-acceleration branch)
gobs = a0 gbar : declining curves, no BTFR
−108
Thermal bath as the dark-matter density
2.6 × 10
vs ρc = 7.6 × 10−10 J m−3 : 98 orders —
Thermal bath as ρΛ
the same 98 orders
—
Gravitational decoherence, tabletop
∼ 60 orders below reach
—
Gravitational decoherence, nHz background ∼ 80 orders below reach
—
Binning rotation-curve data on measured ν +0.47 selection bias; bin on gbar
—
Response sign changes read as a physical
23 zeros at π/rs (the acoustic phase); 62–124%
Rem. NL.33
scale
window-dependent
Any causal linear response for the CMB
zero DC power from an oscillating source, at any Thm NL.28
third-peak height
amplitude
Rectification as the escape from that no-go rectified DC ≤ 0.0166 of the baryon potential vs Prop. NL.29
νreq = 6.364; amplitude, not symmetry
ψ itself as the standing pre-recombination mψ = 0 exactly, protected by an exact symmetry Thm NL.30
well

decide whether any individual proof is sound. Forty-five
statements matched; those that the corpus prints are
graded below by the outcome of their re-examination.

the flux for the approximant and both branches of the
exact law; Thm. U.13, as the flow of a convex energy,
with (A2) in the form of §III A and p = 2.

How to read this list. A statement in
the first group has a proof re-derived under
the current standard, and it appears where
it stands in its re-derived form. A statement
in the second group is cited, not re-derived:
its printed proof asserts a step that the monograph does not supply, or it is a summary, a
tally or a classification rather than a theorem.
Its grade is given with it, and it should not
be cited elsewhere at a higher grade. Entries
marked (C) live in the companion volume.

2. Galactic sector: Thm. NL.1, whose four limits follow from Eq. (NL5), the scale integrals converging for w0 > 0; Thm. BS.19, whose remainder is
−e−2X /48 + O(e−4X ), from the pole of the Mellin
transform at s = −2; Prop. AT.20.

a.

Re-derived (twenty-two).

1. Well-posedness: Thm. U.4, by strict monotonicity of

3. The Hubble–Planck exponent and a0 : Cor. AP.11,
Cor. O.3 and Thm. AP.24, whose stationary
point
√
Ξ∗ = 3/2 is a minimum and gives a0 = 2 α cH0 with
ka = 3/(8α). The generation-kernel axiom and the
potential U (Ξ) remain the premises stated there.
4. Internal geometry: Lemma AB.11, Lemma AB.6,
Thm. AH.9 (C) and Thm. AH.14 (C). The index count
55 + 5 = 60 is re-derived.

123
5. Strong CP: Thm. L.1 and Thm. AH.66 (C). The termby-term cancellation holds for Re s > dim X, where
Weyl’s law makes the η series absolutely convergent;
η(s) therefore vanishes identically, and so does its continuation to s = 0. The loop bound is the StandardModel estimate and is cited.
6. Screen closure: Cor. XVI.3 and Cor. XVI.4. The exc
tracted offset is M(z)
= M − ⟨∆ψ(z, n̂)⟩n̂ . It equals
the calibration constant, and is independent of redshift,
when the screen has no monopole, as in the current
no-screen solution.
7. Bookkeeping in the particle sector: Cor. AH.46 (C);
Thm. AH.50 (C), whose rate, with the B+L multiplicity, is below a third of H at the reheating temperature, with equilibrium reached later, which is all the
conversion factor requires; Thm. AH.183 (C), whose
printed pieces reproduce while the value of δB-T stays
open; Thm. AH.28 (C), where the µ–τ reflection forces
θ23 = π/4 and | sin δCP | = 1, JPMNS = −0.0337 at the
trimaximal sin2 θ12 = 0.3184, and normal ordering follows on the adopted neutrino branch, which the axioms
do not force; Thm. QM.3, where the level-9 sections
span 55 dimensions and the 60-dimensional microsector
is the index of O(9) ⊕ O⊕5 ; Thm. AH.44 (C).
b.

Cited, not re-derived (twenty-two).

1. Ultraviolet finiteness from the microsector
trace, graded conjectural:
Thm. AH.37 (C),
Thm. AH.192 (C), Thm. AH.16 (C), Thm. AH.1 (C),
Thm. AH.159 (C) and Thm. AH.160 (C). The trace
over the 60-dimensional microsector is finite, but
the four-dimensional loop integral over external
momenta is not cut off by it (status note after
Thm. AH.1). The value α = 1/137.036 at the cutoff in
Thm. AH.16 also conflicts with the measured running,
α−1 (MZ ) = 127.95.
2. Strong-sector normalisations, graded as targets:
Thm. AH.47 (C), Thm. AH.48 (C), Thm. AH.115 (C)
and Thm. AH.132 (C); only the proportionality to

ΛQCD follows, and the numerical factors are calibrated.
Thm. AH.15 (C) holds as the one-loop Standard-Model
result for the given group and matter; its frame-stiffness
mechanism is not derived.
3. Thm. F.25, graded as a motivated assumption: the
step from Ricci curvature to the energy cost of a unit
rotation is asserted, and the rank-one block, whose
projective space is a point, is not covered. The Wilson
ratio built on it inherits that grade.
4. Selections presented as consequences: Thm. AH.179 (C)
gives a single vacuum given the selected bundle;
Thm. Z.3 is a relation√ whose exponent 8√is derived, whose prefactor 2π is fitted (MP α8 2π =
246.09 GeV, while the measured v needs 2.5080)
and whose quartic is the conjectured λH = 1/8;
Thm. AH.140 (C) is the conjunction of its parts and carries their grades, including the charged-fermion masses
other than the top and the charm, which are not derived (App. FM).
5. Summaries, tallies, classifications and rules, not
theorems: Thm. AH.198 (C), Thm. AH.208 (C),
Thm. AH.216 (C), Thm. AH.218 (C) and Thm. AP.43.
6. Others: Thm. AH.33 (C), whose printed mechanism
gives about (2–3) × 10−9 , three to five times Planck’s
ηB , as its statement records; Prop. AH.204 (C), a
conjecture as printed.
The list is not a closed set: a pattern scan finds the shapes
it was told to look for, so the absence of a statement from
this list is not a certificate. The list is published because a
corpus of this size that claims to be checkable should show
which parts of it have been checked, to what standard,
and when.

L.

Summary: Resolved and Remaining Items

124
TABLE XLVII. Summary of “open problems” — resolutions.
“Problem”

Kind

Resolution

Status

UV completion
Cosmological Λ

Fundamental
Fundamental

Conjectural
Dict.

Higgs hierarchy
Clock coupling kα
Majorana scale MR
Dust branch (w → 0)
P (k) full match

Fundamental
Technical
Technical
Technical
Program

Boltzmann code

Technical

Strong CP (tree level,
loop-bounded)

Technical

Topology proposed as completion
(H0 tP )2 = Cα57 : exponent derived, C measured
(Appendix √
O)
v = MP α8 2π, prefactor fitted
kα = α2 /(2π) (Appendix P)
MR = MP α3 (Appendix P)
K ′ (∆) = µ(∆) (Appendix Q)
Linear growth carried by χ with Q = 1 (App. GR);
production pipeline (App. PK)
Forward runs of App. CL on the one-clock C = 1
background; the measured-clock fit is App. ZS
θ̄ = 0 at tree level, loop-induced ≲ 10−16
(Theorem L.3)

MOND µ(x)

MOND a0
Neutrino hierarchy
PMNS matrix
CMB peaks

Fermion masses

Exponent
Thm. (cond.)
Identification
Thm. (cond.)
Mechanism
Open

Derived (θYM = 0
fixed by the
spectral action
given that nothing
is added to it for
the gauge fields)
Proved (given the
Phenomenological exponential response from S 3 composition;
µ = x/(1 + x) its weak-response form (Theorem N.8) ground-state
principle and the
assumptions of
App. N, including
the deep-field
slope)
√
Derived
Free parameter
a0 = 2 αcH0loc (Theorem N.14)
Significant
m3 /m2 = α−7/20 (Appendix X)
< 0.2σ
Significant
Trimaximal TM1; sin2 θ13 = 3α conjectured
Partial
Significant
R = 2.34, ℓ1 = 220
analytic R 6% high;
χ-matter spectrum
within 0.6σ
(measured clock)
√
Significant
mf = Af αnf v/ 2, seven prefactors data-selected
Open

125
DFD: Unified Framework + Falsifiable Predictions
Theorem-grade and derived results:
1. MOND function derived (given the ground-state principle and the assumptions of App. N, including the
deep-field slope): exponential response fixed by S 3 saturation-union composition (Lemma N.6, App. BS);
µ(x) = x/(1 + x) is its weak-response
form (Thm. N.8).
√
2. MOND scale derived: a0 = 2 α cH0loc from topological constraint (Thm. N.14).
3. Dust branch: K ′ (∆) = µ(∆) gives w → 0, c2s → 0 (Thm. Q.7). A no-go lemma proves that the quadratic identification
fails.
4. Strong CP (derived): θ̄ = 0 at tree level, loop-bounded, with θYM = 0 fixed by the spectral action given that nothing is
added to it for the gauge fields (Thm. L.3). No axion.
5. G–H0 invariant: (H0 tP )2 = Cα57 ; exponent forced, C measured on the early clock and C = 1 on bound-matter clocks
(Postulate ZS.11; Appendix O).
6. Clock coupling: kα = α2 /(2π) from Schwinger + no-hidden-knobs (Appendix P) — conditional on that axiom.
Quantitative matches:
• α−1 = 137.036 (21√ppm from the integers; the bracket coefficient is open)
• Higgs: v = MP α8 2π (exponent derived, prefactor fitted)
• Fermion masses: 1.42% leading-order mean error (9 particles; seven prefactors data-selected)
• CKM: λ = 31α = 0.2262 (integer pattern; CP 2 overlap motivation)
• PMNS: trimaximal TM1 (App. X); sin2 θ13 = 3α a stated conjecture
• CMB: the analytic R = 2.34 (6% above Planck’s 2.21; the χ-matter spectrum, run on the measured clock, gives 2.20,
within 0.6σ), ℓ1 = 220 (third-peak height carried by the derived χ-matter field, App. AV; abundance normalization
conditional-derived, Ωχ h2 = 0.1193, −0.6σ; see below)
res
(App. P)
• ESPRESSO: conditional on ξLPI
Named open items: the main ones are listed, grouped, in §XVIII A, among them the origin of the primordial perturbations
and the horizon problem, which enter through one stated starting condition, the Rest-Mass Channel postulate, the
mode-by-mode rule of motion of the momentum-conserving energy, the adopted floor law and the environmental confinement
of the MOND response, the inner cores and masses of the χ halos, the flux-period identification, the stated count 158, the
asserted premise P4′ and two stated conventions behind the χ abundance, the vacuum coefficient R, and the baryon
asymmetry, a calibrated input whose printed mechanism gives about (2–3) × 10−9 , three to five times Planck’s ηB ; the
standing review of §XVIII K grades the inherited proofs.
Scale structure: kmax = 60, Ngen = 3 (discrete input; Rem. F.21) + H0 (observed) together with the measured clock
coefficient C ⇒ all dimensionful constants (per-input grades: App. FK, Rem. FK.8).

XIX.

A TOPOLOGICAL LINK BETWEEN H0
AND MP

The preceding sections treated MP (equivalently G)
as an input parameter. Here we present a dimensionless
constraint linking G, ℏ, H0 , c, and α, such that given
one scale measurement, the others follow from topology
on bound-matter clocks, and with the measured coefficient
C on the early clock.

A.

The Dimensionless Invariant

The primary claim is a purely dimensionless relation,
derived to theorem status via Gaussian mode integration
on the finite-dimensional microsector (Appendix O):
Proposition XIX.1 (Topological Invariant — Spectral-Action-Derived). DFD predicts the following dimensionless constraint:
GℏH02
= C αkmax −Ngen = C α57
c5
c

and C is an O(1) normalization coefficient — measured,
C = 0.87784 ± 0.00266 (Rem. AP.6).
Theorem-grade status (Appendix O):
• The exponent 57 = kmax − Ngen is forced by primeddeterminant scaling on the finite Toeplitz state space
(Lemma O.1, Corollary O.3).
• The identification with the observed invariant I =
GℏH02 /c5 is derived via Gaussian mode integration
on the finite-dimensional microsector (Lemmas O.4–
O.6, Theorem O.7).
• The O(1) coefficient C is not derivable from
the printed corpus: an eight-route, enumerationcomplete no-go proves every printed route to the normalization is exponent-only, convention-importing,
or C-covariant. C is a measured input — exactly as
ΛCDM measures H0 — concordant across five CMBanchored channels on the early clock (Rem. AP.6);
C = 1 is the clock of bound matter (Postulate ZS.11,
Eq. (479)).

(470)

where kmax = 60 (Spin index from Lemma F.11), Ngen =
3 (generation count), α is the fine-structure constant,

This formulation has several advantages:
• Dimensionless: No unit conventions or hidden
factors

126
• Symmetric: Predicts G from H0 or H0 from G
• Falsifiable: A single testable constraint
a. Bidirectionality. Given (α, ℏ, c) and a measured
G, the invariant predicts H0loc with no coefficient, and
H0early with the measured C. Equivalently, given H0loc
it predicts G. Neither is privileged as “input”—the constraint is symmetric. This prevents any accusation that
one quantity was “chosen” to match the other.
b. Error propagation. Taking logarithms and differentiating:
δH0
δG
= −2
G
H0

(471)

The precision of any G prediction is limited by H0 uncertainty. With current H0 uncertainties of ∼1–2%, the
constraint tests G at the ∼2–4% level.
pc. Equivalent form (Planck mass). Defining MP =
ℏc/G, the invariant becomes:
MP = C −1/2 α−(kmax −Ngen )/2 ×

ℏH0
ℏH0
= C −1/2 α−28.5 × 2
2
c
c

(472)
d. Numerical verification. Using CODATA values for
G, ℏ, c, α:
GℏH02
= 1.393 × 10−122 (at H0 = 67.54 km/s/Mpc)
c5
RHS: α57 = 1.586 × 10−122 ,
C α57 = 1.393 × 10−122

LHS:

(473)
The derived exponent lands the 10−122 hierarchy with
no tuning — one unit of the exponent is a factor 137, so
the residual O(1) ratio defines the measured coefficient
C = 0.8778 rather than admitting any alternative integer.

B.

Implication for the Cosmological Constant
Problem

The cosmological constant problem asks: why is
ρΛ /ρPlanck ≈ 10−123 ? This is often called “the worst
fine-tuning in physics” because naive quantum field theory predicts ρΛ ∼ ρPlanck .
If Eq. (470) holds, the ratio is topologically constrained :
Proposition XIX.2 (Cosmological Constant Scaling).
The critical density satisfies:
3
GℏH02
3
×
=
C α57 ≈ 1.7 × 10−123
ρPlanck
8π
c5
8π
(474)
(on bound-matter clocks, C = 1 under Postulate ZS.11,
it is 1.9 × 10−123 , with no coefficient). With ΩΛ ≈ 0.69:
ρΛ /ρPlanck ≈ 1.1 × 10−123 .
ρc

=

Scope. This identity fixes the exponent of the vacuum scale; the O(1) coefficient enters through the clockdictionary normalization (Rem. AP.6), and the statement is microsector-scoped — the Standard-Model mattersector vacuum energy and the spectral action’s Planckian

vacuum term are not part of this identity and remain
unaccounted (Rem. AU.3).
Derivation. The critical density is ρc = 3H02 /(8πG). The
Planck density is ρPlanck = c5 /(ℏG2 ). Thus:
3
3H02
ℏG2
GℏH02
× 5 =
×
ρPlanck
8πG
c
8π
c5
Substituting Eq. (470) gives the result.
ρc

=

(475)

The exponent 57 = kmax − Ngen = 60 − 3 has two
sources:
• kmax = 60: the Spinc index χ(CP 2 , E) for twist
bundle E = O(9) ⊕ O⊕5 (Lemma F.11)
• Ngen = 3: the generation count (discrete input; the
flux-product expression is a bookkeeping identity —
App. F)
Cosmological Constant: Spectral-Action-Derived Exponent (Appendix O)
The “fine-tuning” of 10−123 is derived via Gaussian mode
integration, given the cancellation of the matter vacuum
and of the spectral action’s Planckian vacuum term
(Rem. AU.3):
ρc
3
3
=
Cαkmax −Ngen =
Cα57 ≈ 1.7 × 10−123
ρPlanck
8π
8π
(476)
The exponent 57 is forced by primed-determinant scaling
(Corollary O.3). The identification with the physical
hierarchy is derived via Gaussian mode integration over
the 57 nonzero KK modes (Lemmas O.4–O.6). The
residual O(1) normalization is the measured clock
coefficient C = 0.878 on the early clock (Rem. AP.6); on
bound-matter clocks C = 1 (Postulate ZS.11).

C.

Testable Consequence: The Hubble Constant

Interpreted as an H0 constraint from (G, α), the invariant Eq. (470) yields:
r
√
C α57 c5
C α28.5
H0 =
=
(477)
Gℏ
tP
p
where tP = ℏG/c5 is the Planck time.
Using CODATA values for G, ℏ, c, and α, the BAO+θ∗
coefficient C = 0.87784 ± 0.00266 gives H0 = 67.543 ±
0.103 km/s/Mpc. The adopted cross-channel value uses
the independent plik-lite central/error channel, rounded
as
H0early = 67.52 ± 0.09 km/s/Mpc

(478)

The exponent is the zero-parameter content: the
10−122 hierarchy follows entirely from the microsector
closed form for α and the exponent 57 = kmax − Ngen ,
and no neighbouring integer is reachable (one exponent
unit moves H0 by a factor 11.7). The O(1) coefficient

127
C is measured once — the eight-route no-go proves it
underivable from the printed corpus (Rem. AP.6) — and
is concordant across five CMB-anchored channels (Planck
plik-lite 67.520 ± 0.094, BAO+θ∗ 67.543 ± 0.103, ACT
DR6 67.481 ± 0.094, SPT-3G 67.417 ± 0.232, plus θ∗ only); DESI DR2 BAO alone sits 3.4σ higher (App. ZS).
On bound-matter clocks and rulers C = 1 holds (Postulate ZS.11), and the identity then contains no measured
coefficient:
H0loc =

α28.5
= 72.09 km/s/Mpc
tP

(479)

from the measured G, with zero parameters. Read the
other way, G = α57 c5 /[ℏ (H0loc )2 ] is derived from the local
Hubble rate: the Cepheid-free local mean 71.87 ± 0.96
(Sec. ZS 5) gives a G 0.6% ± 2.7% above the laboratory
value, 0.2σ (Eq. (471)). The early-clock value of Eq. (478)
is H0early = H0loc /λ (Eq. (ZS4)), with λ the measured size
of the postulate.
a. Comparison with observations. Recent JWST observations provide high-precision tests of this prediction.
Two major collaborations have released results:
b. Assessment. On the early clock, DFD’s H0early =
67.52 ± 0.09 km/s/Mpc lies +0.2σ from Planck ΛCDM
and 2–3σ below SH0ES’s JWST-subsample estimates,
6.4σ below its full 2025 sample, while agreeing with
CCHP JAGB at −0.1σ. On the bound-matter clock,
H0loc = 72.09 lies within 0.6σ of the SH0ES JWST subsamples, 1.5σ and 1.7σ below the full 2025 sample and
the Local Distance Network, 0.9σ and 1.6σ above CCHP
TRGB and JAGB, and 0.2σ above the Cepheid-free mean
71.87 ± 0.96 of Sec. ZS 5. The two JWST teams obtain
systematically different results, with the disagreement not
yet resolved [105, 108].
Key observations:
• The five CMB-anchored DFD clock channels (pliklite, BAO+θ∗ , ACT DR6, SPT-3G, θ∗ -only) are
mutually concordant and agree with Planck; DESI
DR2 BAO alone lies 3.4σ higher
• SH0ES’s JWST-subsample measurements lie 2–3σ
above and its full 2025 sample 6.4σ above; CCHP
results lie 0.1–1.5σ above the early-clock value
• The C=1 value is the bound-matter clock H0loc (Postulate ZS.11), 0.2σ from the Cepheid-free local mean
c. The Hubble tension in DFD. Reading the CMB
inference as biased by a ψ-screen (∆ψ ≈ 0.30 from ψtomography) while local ladders read the true clock does
not resolve the tension. Two exact theorems exclude
that interpretation (App. ZS): the homogeneous background ψ̄(t) carries no screen beyond the expansion it
describes — at fixed scale factor it cancels identically
in every CMB observable, so no homogeneous screen of
any size can bias the inference — and the optical/matter
metrics are conformally related, so a ψ-screen cannot
be probe-selective; the surviving inhomogeneous channel is bounded 3.5 orders of magnitude too small. The

CMB-anchored channels therefore read the early clock.
Both measurements are robust and about 5σ apart (4.5σ
for the Cepheid-free local mean, 6.4σ for SH0ES’s full
2025 sample), and ΛCDM has to assign the difference to
an error in one of them. DFD reads them as one universe on two clocks (Postulate ZS.11): matter keeps a
clock offset by the constant λ = H0loc /H0early , set when
matter first binds and falls behind the expansion, so the
CMB-anchored channels return H0early = 67.52 and loloc
cal probes H
√0 = 72.09 (Eq. (479)). Bound systems
read a0 = 2 α cH0loc = 1.1966 × 10−10 m/s2 , consistent
with SPARC within its systematic error, and the booking
C = 1.027 ± 0.029 of the Extended Derivations’ fivepresentation Λ theorem, SH0ES’s 2022 value re-expressed
and not a third arm, lies 0.9σ from C = 1. The postulate
is a ratio between frames, not a screen, so the screen
exclusion above is untouched. Its size λ is measured. The
absolute a0 scale (Euclid/DESI lensing RAR) tests it once
baryonic masses are known to about 2% and distances
are taken on each clock’s own H0 (App. ZS).
The G-H0 Link: Derived Exponent, Measured Coefficient
Constraint: (H0 tP )2 = C α57 ; exponent 57 derived
(Ngen = 3 a discrete input); on the early clock the
coefficient C = 0.87784 ± 0.00266 is measured once, on
bound-matter clocks C = 1 (Postulate ZS.11)
Status: H0early = 67.52 ± 0.09, concordant across five
CMB-anchored channels, +0.2σ from Planck;
H0loc = α28.5 /tP = 72.09 from the measured G with zero
parameters, 0.2σ from the Cepheid-free local mean, 1.5σ
below SH0ES’s full sample
Interpretation: One universe, two clocks
(Postulate ZS.11): the CMB-anchored channels read the
early clock and local probes the bound-matter clock; the
ψ-screen-bias reading is excluded (background ψ adds no
screen beyond the expansion, App. ZS)
Test: The postulate fails if bright sirens or
mass-sheet-robust time delays settle at 67.5, if early-ruler
and local ladders converge with distance, or if the
absolute a0 settles at the early-clock value on distances
computed with H0early (Euclid/DESI lensing RAR, with
baryonic masses known to about 2%); its price is a
laboratory-unit age of 12.92 Gyr against ages of the
universe of 13.5–13.8 Gyr that one programme infers
from globular clusters with a ΛCDM formation delay

D.

Epoch structure of the G–H0 invariant: coupling
versus state function

The invariant GℏH02 /c5 = C α57 (Eq. (470)) admits
two epoch readings, and only one of them survives. Read
as an all-epoch dynamical identity,
G(t) =

C α57 c5
ℏH(t)2

(480)

it forces a photon-frame drift Ġ/G = −2Ḣ/H =
+3H0 Ωm ≈ +6.4 × 10−11 yr−1 (matter-dominated, Ωm ≈

128
TABLE XLVIII. Hubble constant: DFD’s two clocks (Postulate ZS.11) vs. observations.
Source
DFD early clock, H0early (measured C)
DFD bound-matter clock, H0loc (C = 1)
SH0ES JWST combined
SH0ES JWST Cepheids
SH0ES JWST TRGB
SH0ES JWST JAGB
SH0ES HST+JWST, full sample
Local Distance Network
CCHP TRGB (HST+JWST)
CCHP JAGB (JWST)
Planck ΛCDM

H0
Uncert.
∆/σ (early)
67.52
±0.09
—
72.09
—
—
Local distance ladder (JWST)
72.6
±2.0
−2.5σ
73.4
±2.1
−2.8σ
72.1
±2.2
−2.1σ
72.2
±2.2
−2.1σ
73.49
±0.93
−6.4σ
73.50
±0.81
−7.4σ
70.4
±1.9
−1.5σ
67.8
±2.7
−0.1σ
CMB-inferred (model-dependent)
67.4
±0.5
+0.2σ

∆/σ (loc)
—
—

Ref.
This work
Eq. (479)

−0.3σ
−0.6σ
0.0σ
−0.1σ
−1.5σ
−1.7σ
+0.9σ
+1.6σ

[105]
[105]
[105]
[105]
[106]
[107]
[108]
[108]

—

[58]

Units: km/s/Mpc. ∆/σ ≡ (H0DFD − H0obs )/σobs , with H0DFD = H0early (early) or H0loc (loc), Eq. (ZS4).

0.31, H0 = 67.5 km/s/Mpc), exceeding the lunar-laserranging and binary-pulsar bound |Ġ/G| ≲ 10−13 yr−1 by
∼ 103 ; and the atomic-frame rescue (a cosmological screen
evolving as 2ψ̇cosmic = −3H0 Ωm ) is unavailable, because
the FRW flat-direction theorem (Thm. AE.1, App. AE)
makes the homogeneous cosmological ψ̇ non-dynamical.
That self-tension would be a conditional self-consistency
falsifier. It is resolved by the following distinction, which
the elliptic constraint reading (Axiom WS.1, App. WS)
makes structural rather than optional.
Theorem XIX.3 (Epoch consistency / anchor fork; conditional on Axiom WS.1). Under Axiom WS.1, the constitutive transition scale a⋆ in µ(|∇ψ|/a⋆ ) tracks the instantaneous cosmological state,
√
a0 (z) = 2 α c H(z),
(481)
and the running sign ∂z a0 > 0 is forced: the “anchored”
branch of the anchor fork—an a0 frozen at a formationepoch value—is structurally unavailable.
Proof core (the no-memory argument). The elliptic constraint sector is instantaneous and carries no intrinsic
dynamical memory: there is no local field degree of freedom in which a past epoch’s H could be stored. The scalar
has no mass term (unit-shift protection: ψ → ψ + const is
a units redefinition, so no potential can anchor a remembered scale), and under Axiom WS.1 it has no independent
temporal kinetics in the local
√ sector. The stationarity
functional that derives a0 = 2 α cH0 (Theorem N.14, Appendix N) is evaluated on the instantaneous background,
so its output references the current screen flow. A frozen
a0 would require a storage mechanism the theory does
not possess; the tracking branch is therefore forced. The
grade is theorem conditional on Axiom WS.1: the result
inherits that axiom’s status and no more.
Corollary XIX.4 (Ġ = 0: coupling versus state function). G is an action coupling (the source normalization
of the field equation): it is epoch-independent, Ġ = 0, and

the lunar-laser-ranging bound is passed trivially. a0 is a
state function (a constitutive normalization by the cur√
rent screen flow): it tracks H(z), with ȧ0 = 2 α c Ḣ ̸= 0.
The identity GℏH02 /c5 = C α57 is a boundary-condition
statement in the Axiom PO class (App. Q)—it selects
the present-epoch H0 in Planck units—not an all-epoch
dynamical law; Eq. (480) is accordingly not a dynamical
law; it is only the definition of the yardstick. The two
predictions, Ġ = 0 and ȧ0 ̸= 0, are independent falsifiers.
Note Ġ = 0 is GR-degenerate: it is a consistency check,
not a discriminator; the discriminating partner is ȧ0 ̸= 0.
Under Corollary XIX.4, G at every epoch equals today’s
value: it is not weaker by ∼ 109 at recombination. The
cavity–atom LPI comparison (Section XII) remains a
live clock-sector test of the photon-frame/atomic-frame
difference, but it is not required to cancel a Ġ drift—there
is none.
a. First data confrontation: MUSE-DARK III. The
next paragraphs confront the data on the Ωm = 0.274
background of the one-clock C=1 convention; every Ωm =
0.274, γE = 1.038, and 1.7σ statement there belongs to
that background, and the measured-clock values are those
of the re-evaluation below (Ωm ≃ 0.312, γE = 1.098, 2.1σ).
The tracking prediction is now confronted by the first
measurement able to see the sign. MUSE-DARK III [109]
reports a linear growth of the RAR acceleration scale,


−10
a0 (z) = 1.00 ± 0.04 + 1.59+0.11
m s−2 ,
−0.10 z × 10
−10
a0 (z ≈ 1) = 2.38+0.12
m s−2 .
−0.10 × 10

(482)
At face value this is the sign DFD predicts, ∂z a0 > 0,
and it is the one discriminator against static MOND,
which predicts no growth. It rests on the survey’s bins,
their published errors and its baryon census. Refitted
galaxy by galaxy from the public products, the rise reproduces; within the survey a frozen a0 lies about 1σ from
it once galaxy-level errors are counted, and anchored to
SPARC’s z = 0 value, the galaxy-level bins still exclude
a frozen a0 at about 3σ, and DFD’s zero-parameter law

129
fits them on the bound-matter clock. With molecular
gas tied to the survey’s disc masses the rise stays; tied
to SED-based masses, or with SED stellar masses in the
census, the trend is flat or falling (computed; App. A0Z,
§A0Z 6). The magnitude comparison is pending. DFD
predicts a0 (z)/a0 (0) = H(z)/H0 exactly: 1.78 at z=1 on
the measured clock (Ωm = 0.312; 1.71 on the one-clock
C=1 background’s Ωm = 0.274); a conditional bracket
[1.79, 2.83] does not apply—the matter-only 2.83 endpoint
is not a rival DFD prediction, and under Theorem XIX.3
the prediction collapses to the H(z)/H0 line. At face
value the published linear-fit growth is steeper than H(z)scaling (e.g. 2.05 predicted at z=1 from the SPARC local
value 1.20 × 10−10 on the C=1 background, 2.14 on the
measured clock, vs 2.38 ± 0.11 measured, a 95% interval),
but the paper’s per-bin uncertainties are untabulated
and the fit’s local anchor (1.00 × 10−10 ) sits below the
SPARC local value, a systematic not yet assessed—so the
magnitude verdict awaited the binned data.
The published bins follow the epoch prediction,
and counted galaxy by galaxy the rise is contested.
The binned points have since been digitised and confronted directly [110] (published, MNRAS 551(4), 2026
September 7). Fitting the continuous exponent family
a0 (z) = A(1 + z)γ , which contains the frozen (γ = 0) and
matter-density (γ = 3/2) hypotheses as special cases and
closely brackets Hubble tracking, gives
γ = 0.78 ± 0.15,

(483)

excluding a frozen a0 at 5.3σ on the published bin errors
by the properly nested test (∆χ2 = 27.7 for one degree
of freedom, rising to 6.6σ with the local SPARC z ≃ 0
anchor, which returns γ = 0.83 ± 0.13) and the matterdensity scaling at 4.8σ (on the exponent’s error; 4.7σ by
the same nested test); H(z)-tracking is the informationcriterion-preferred one-parameter description (χ2 = 5.2/3,
GOF p = 0.16, AICc 9.2 against 17.1–17.2 for the twoparameter forms and 27–33 for the alternatives), with
no bin deviating by more than 1.7σ. The “steeper” impression recorded above is anchor-dependent and does
not survive a floating amplitude: measured against the
binned relation itself the growth is, if anything, mildly
shallower than H(z), the fitted exponent sitting 2.1σ
below the effective slope of the Planck E(z) that the analysis assumed, published as γE ≃ 1.10 over 0.5 ≲ z ≲ 1.3
(our own evaluation of that same background over the
fitted range gives 1.087)—and only 1.7σ below γE = 1.038
evaluated on the one-clock C=1 background Ωm = 0.274.
Two caveats travel with this closure. First, the confidence convention of the source’s plotted bars is not
given there, and Ref. [110] therefore reports both readings.
Read as 95% intervals the bars shrink and the published
exclusions strengthen—γ = 0 to ≳ 10σ and γ = 3/2 to
9.5σ on the exponent’s error (9.2σ by likelihood ratio),
with the exponent family tightening to γ = 0.78 ± 0.08—
while exact E(z) tracking moves into goodness-of-fit tension (χ2 = 19.9/3, p = 1.8 × 10−4 ), the growth then being
shallower than E(z) at the published ∼4σ on the Planck

background; the same offset evaluated on the one-clock
C=1 background’s Ωm = 0.274 is 3.4σ. The exclusions of
γ = 0 and γ = 3/2 hold under either reading; the consistency of Hubble tracking does not. Second, the fitted tracking amplitude A = (1.40±0.03stat )×10−10 m s−2 sits 17%
above the canonical SPARC normalisation 1.20 × 10−10 —
0.8σ on SPARC’s systematic-dominated uncertainty, but
+6.7σ on the published statistics alone (+6.3σ using this
corpus’s own re-fit of the same bins, A = 1.4010 ± 0.0321;
+5.3σ once the SPARC statistical bar is included). The
reference itself quotes only the systematic-scale comparison; the statistics-only figures are evaluated here. It is
a mass-scale-conditional pass, and the survey’s own onesided molecular-gas systematic is large enough to cover the
required +0.094 to +0.103 dex of baryons (Stat. A0Z.6).
Falsifiers: a measured ∂z a0 ≤ 0 falsifies Theorem XIX.3;
a confirmed Ġ/G detection at lunar-laser-ranging sensitivity falsifies Corollary XIX.4; a high-redshift frozen a0
falsifies the epoch promotion. The two kinematic conditions apply to measurements whose baryon census counts
molecular gas, whose pressure-support correction is stated,
and whose errors are counted galaxy by galaxy. Massive
discs at 0.6 < z < 2.5 with molecular gas counted meet
the third at face value, a0 ≃ 1.2 × 10−10 m s−2 at z ≃ 2.2
against 3.7 for tracking, and lean toward the first (a
within-sample γ = −0.86+0.46
−0.48 on statistical errors, −0.20
without the discs with Vrot /σ0 < 2.1). They do so on
scaling-relation gas masses and the standard pressuresupport correction; baryonic masses about 0.2 dex lower
at z ≃ 2 than those fitted in Ref. [111] (0.3–0.5 dex below
the census of SED stars and scaling-relation gas), or a
pressure-support correction 2.4 times the standard one, remove the tension (App. A0Z, §A0Z 7). The domain-local
screen of App. ZT would need 10 h−1 Mpc domains about
6 to 11 standard deviations denser than average, which
their measured environments do not show (Stat. A0Z.12).
At high redshift the epoch promotion is under test, not
confirmed.
b. The same bins refitted on DFD’s own background.
The comparison just quoted evaluates E(z) on Planck
ΛCDM, because a theory-neutral analysis has no alternative. DFD does: the same α57 closure that supplies a0 also
supplied, at the one-clock C=1 convention,
H0 = 72.0897
√
and Ωm = 0.2749, so a0 (z) = 2 α cH(z) is a zeroparameter curve up to one overall amplitude. (At the measured coefficient the DFD background is H0early = 67.52,
Ωm ≃ 0.31 — essentially the Planck row already tested
above; the amplitude question is the absolute-a0 test
of the two-clock postulate, App. ZS.) On the current
measured-clock background, refitting the four bins with
Ωm ≃ 0.312 held fixed and only the amplitude free gives
A = 1.394 ± 0.032, χ2 = 5.36/3, AICc = 9.36, and
γE = 1.098 (2.1σ above the measured 0.782 ± 0.149).
The 4.00/3, AICc 8.00 result belongs to the C = 1,
Ωm = 0.2749 background. Hubble tracking remains viable
and the preferred one-parameter description (App. A0Z,
Table CX). Three qualifications apply to this comparison. (i) The predefined P = 0.426 false-positive con-

130
trol and ∆AICc = −1.18 apply to the Ωm = 0.2749 row of
the one-clock C = 1 background and cannot be claimed
as a current improvement. (ii) The current effectiveexponent offset is 2.1σ before the alternative 95% errorbar reading; the external-field variant remains a separate
regime test. (iii) Against the current survey-fitted amplitude, the early-clock a0 (0) = 1.1208 × 10−10 is lower by
8.5σstat (about 2σ counted galaxy by galaxy), a coherent
stellar-mass shift of about +0.095 dex, and the boundmatter a0 = 1.1966 × 10−10 , against the amplitude on
distances computed with H0loc , is lower by 0.094–0.103 dex
(8.5–9.2σstat ; Stat. A0Z.6)—where a frozen a0 requires
+0.220 to +0.354 dex per bin, 2.3 to 3.8 times more, plus
a redshift tilt that no common-mode rescaling supplies,
but that a fuller baryon census does: with molecular gas
tied to SED-based masses the refitted trend is about flat,
tied to the survey’s own disc masses the rise stays and
DFD’s law fits it, and with SED stellar masses it falls
(computed; Stat. A0Z.6, §A0Z 6). The registered z = 2
and z = 3 falsifiers are tabulated in Table CXI.
c. Relation to the companion note. The companion
√
note [112] stated the epoch extension a0 (z) = 2 α cH(z)
as a program-grade conditional result (forced modulo
the contemporaneous-normalizer identification). Theorem XIX.3 upgrades it: under Axiom WS.1 the identification itself is forced by the no-memory argument, so
DFD lands on a forced fork branch, not a chosen one.
The result remains conditional on Axiom WS.1 and is
stated at that grade throughout this corpus. The binned
kinematic confrontation has since been carried out in a
separate, theory-neutral analysis that does not invoke
DFD, published in MNRAS 551(4), 2026 September 7
(Letter; doi:10.1093/mnras/stag1485) [110]; the epoch
extension has therefore met data, and it is contested. Lowmass discs at 0.33 < z < 1.44 show the predicted rise on
the published bins; counted galaxy by galaxy, a frozen
a0 lies about 1σ from it within the survey and about 3σ
from it anchored to SPARC’s z = 0 value, with DFD’s
zero-parameter law fitting on the bound-matter clock,
and the sign of the trend depends on how the molecular
gas is added. Massive discs at 0.6 < z < 2.5, on a census
that counts molecular gas, do not show the rise beyond
z ≃ 1.5 at face value (App. A0Z, §A0Z 6 and §A0Z 7).
Direct molecular-gas masses, kinematic masses not limited by a halo-model prior and per-galaxy analyses with
galaxy-level errors and a0 (z) imposed decide it, the last
for the massive discs only with an independent bound
on the χ halo or rotation curves beyond 3Re ; JWST and
ALMA rotation curves at z ≳ 3, and ultimately SKA H i
kinematics, extend the test to where E(z)-tracking at the
fitted amplitude and the survey’s linear form diverge: on
this corpus’s evaluation at the survey’s fitted amplitude,
Hubble tracking gives a0 ≃ 4.20 at z = 2 and 6.32 at
z = 3, against 4.18 and 5.77 for the survey’s linear form
and 3.38 and 4.23 for the measured exponent family.

E.

The homogeneous optical-time Hamiltonian
(background expansion)

Theorem AE.1 establishes that the homogeneous mode
of the static field action is a flat direction: with the source
written as the density contrast ρ − ρ̄, a homogeneous configuration carries no net source and the Euler–Lagrange
equation places no constraint on ψ̄(t). This is the correct
statement for the perturbation (galactic/AQUAL) sector,
and it is the reason the cosmic expansion rate does not
follow from the spatial field equation. It does not imply
that DFD lacks an expansion law: the background rate
is set by the Einstein–Hilbert term of Proposition WS.5,
which fixes the homogeneous mode that the scalar action
leaves free.
a. The optical-time bridge. The relation 1 + z = eψ/2
holds for a static field. For a homogeneous field that
changes in time, a null geodesic of the matter-coupling
metric gives 1+z = e(ψ̄0 −ψ̄e )/2 instead. In the static arena
with absolute time the homogeneous field is the expansion
itself, ψ̄ = 2 ln A, where A is the scale factor of the
Friedmann–Robertson–Walker metric −c2 dτ 2 + A2 dx2 in
proper time τ , with 1+z = A0 /A; background observables
depend on ψ̄ only through A (Theorem ZS.3). Define the
optical expansion variable ψopt ≡ −2 ln A. Then, exactly,
H = − 12 ψ̇opt ,

eψopt = A−2 .

(484)

In that arena the homogeneous field is on the fast-light
branch (App. Q 6), with ψ̄ < 0 in the past and optical
speed c e−ψ̄ > c: that fast-light past is the ordinary
expansion read on the arena clock, where light moves at
c/A2 , and it opens no horizon; the variable ψopt , about
45 at nucleosynthesis, is not the field and does not set the
optical speed.
b. The energy first integral. Under premise G1 of
Proposition WS.5 the Einstein–Hilbert term acts on
the uniform background of the scalar field, replacing
the printed Newtonian term there. DFD has no lapse
to vary: its time is absolute, and the dictionary fixes
gtt = −c2 e−ψ̄ . The Einstein–Hilbert Lagrangian on the
homogeneous matter-coupling metric has no explicit time
dependence, so its energy is conserved, and with a perfect
fluid on the matter-coupling metric and the vacuum term
Λc2 /3 = ΩΛ H02 (R = ΩΛ C the measured coefficient of
App. AU) that first integral is, exactly (computed), the
homogeneous optical-time equation
8πG
Λc2
ρ + kc2 eψopt −
= 0
3
3
(485)
with k the integration constant of the first integral, paired
with the continuity equation, which in optical-time variables reads


p
p
⇐⇒ ρ̇ + 3H ρ + 2 = 0.
ρ̇ = 32 ψ̇opt ρ + 2
c
c
(486)
2
Hopt ≡ 14 ψ̇opt
−

131
2
2
Via (484) Eq. (485) is identically H 2 = 8πG
3 ρ − kc /A +
2
Λc
3 : the background rate is the conserved Einstein–
Hilbert energy written in the optical expansion variable.

Theorem XIX.5 (Self-consistency of the optical-time
sector). The first integral (485) and continuity (486),
with the bridge (484), reproduce the second (acceleration)
Friedmann equation with the correct relativistic source,


4πG
Λc2
Ä
3p
1 2
1
= − 2 ψ̈opt + 4 ψ̇opt = −
ρ+ 2 +
. (487)
A
3
c
3
Proof. Differentiate (485) in t, substitute (486) for ρ̇, eliminate kc2 eψopt using (485), and insert Ä/A = Ḣ + H 2 =
2
− 21 ψ̈opt + 14 ψ̇opt
from (484). The result is (487) identically
(symbolically verified, residual 0). The active gravitational
mass ρ + 3p/c2 follows from the Einstein–Hilbert energy
for any fluid: the sector is the full Friedmann system, not
its matter special case.
Accelerated expansion of A beyond the vacuum term
therefore needs ρ + 3p/c2 < 0. The optical engine of
App. Q 6, which in this arena is accelerated expansion
of A, needs such a source while it runs and a stage that
reheats the universe after it (Rem. Q.11).
c. Consequence for the MOND acceleration scale. Because the background rate is the rate of the optical expansion variable, the optical transition scale is set by that
rate,
√
√
a0 (z) = 2 α c H(z) = α c |ψ̇opt | .
(488)
The redshift dependence of a0 is therefore not an imported
clock but the evolution of ψopt (t) governed by (485): the
MOND knee tracks the rate of the cosmic expansion, and
the sign ∂z a0 > 0 follows from |ψ̇opt | = 2H rising with z.
d. Scope (two-channel). The DFD-native content is
the bridge (484) and the identity (488): the expansion
rate and the MOND scale are expressed through the
optical expansion variable with no additional clock. Equation (485) is the conserved energy of the Einstein–Hilbert
term on the uniform background, so its coefficient 8πG/3
is the same G as in the tensor and vector sectors on
one clock (Postulate ZS.11 departs from this premise);
the law is derived given premise G1 (Proposition WS.5;
the spectral action and its Einstein–Hilbert term are
the construction of Chamseddine and Connes [113, 114],
and induced gravity is Sakharov’s idea [115]), reproduces
the Friedmann background and makes no new expansion prediction. The AQUAL equation has no homogeneous mode (Theorem AE.1) and is sourced by the
contrast ρ − ρ̄: the Einstein–Hilbert scalar term is the rest
channel’s term, counted once. Primordial helium tests
the shared coefficient: Yp = 0.2453 ± 0.0034 alone gives
GBBN /G = 0.98 ± 0.04 (computed; the nucleosynthesis
theorem of Ext. Deriv., App. AH uses the same rate). The
curvature constant k is the integration constant of the first
integral, not the flat arena’s zero curvature, and it is zero
under the stated clause of Axiom WS.1: while no baryons
are bound anywhere, and at least until they decouple

from the photons, gravity is the Einstein–Hilbert term
alone, and there only k = 0 keeps the scalar sector elliptic
and its evolution well posed (App. WS, Remark WS.7).

F.

The Parameter Structure

If Eq. (470) holds, DFD has the following structure:
TABLE XLIX. DFD input/output structure.
Category
Topological
Observational

Quantity
kmax = 60
Ngen = 3
α−1 = 137
H0 or G; the coefficient C
G or H0
v = 246 GeV

Derived
ρc /ρPl
All masses

All mixings

Source
χ(CP 2 , E)
Input (App. F)
CS quant.
Measured
Eq. (470)
√
MP α8 2π
(prefactor fitted)
Eq. (474)
α-hierarchy; seven
prefactors
data-selected
CP 2 geom. and
stated postulates

The resulting one-anchor tower of scales is displayed as
the α-exponent ladder of Fig. 12.
a. Parameter counting. The scale sector introduces
one measured continuous coefficient (the vacuum
coefficient R, of which the clock normalization C is today’s early-clock value; Rem. AP.6) and no continuous
parameter fitted to a predicted shape. Three further continuous calibrations live outside this sector and are stated
with it for completeness: the primordial amplitude As ,
the coefficient of ωb h2 , and the reionization optical depth
τ shared with ΛCDM, four in all, against ΛCDM’s six
free parameters (App. FK, Rem. FK.8). The discrete
topological sector is uniquely determined by Standard
Model structure:
• The isotropy charge 6Y fixes q1 = 3; hypercharge integrality alone admits q1 ∈ {3, 6}, and q1 = 6 would
give the weak doublets an even charge (Lemma F.9,
Rem. F.10)
• Minimal integer-charge lift gives O(9) = L⊗3
Y
• Five hypercharged chiral multiplet types fix n = 5
• Within E = O(a)⊕O⊕n , minimal-padding uniquely
selects (a, n) = (9, 5) with kmax = 60
One scale measurement (H0 or equivalently G) plus
the measured coefficient C determines all dimensionful
quantities via the invariant GℏH02 /c5 = C α57 .

132
α0 : MP = 1.22 × 1019 GeV (anchor)
α3 : MR = α3 MP ≈ 4.7 × 1012 GeV (seesaw)

α8 : v =

√

2π α8 MP = 246 GeV (EW)

scale Q = αk MP (depth ∝ k: log scale in α)

α19/2 : ΛDFD = α19/2 MP = 61.2 MeV (QCD)
√
α21/2 : me = 23 π α21/2 MP = 0.511 MeV

α28.5 : H0loc = α28.5 /tP = 72.09 km s−1 Mpc−1
(bound-matter clocks; 67.52 on the early clock)

One Measured Coefficient — Spectral-Action-Derived
Exponent (Appendix O)
The scale sector introduces one measured continuous
coefficient and no continuous parameter fitted to a
predicted shape (the theory’s three further calibrations,
the amplitude As , the ωb coefficient and the shared τ , are
not scale inputs; App. FK). Once the discrete topological
sector is fixed by Standard Model structure (kmax = 60,
Ngen = 3), the exponent in the dimensionless invariant
GℏH02
= C α57
(489)
c5
is forced by primed-determinant scaling (Corollary O.3).
Gaussian mode integration over the finite-dimensional
microsector derives the identification with the physical
hierarchy (Theorem O.7); the O(1) coefficient
C = 0.87784 ± 0.00266 is measured, and the eight-route
no-go proves it underivable from the printed corpus
(Rem. AP.6). One scale measurement (H0 or G) plus C
then fixes all dimensionful quantities.

XX.
A.

CONCLUSIONS

Summary of Density Field Dynamics

Density Field Dynamics is a scalar refractive-index theory of gravity defined by a single field ψ that determines:
• Optical propagation: Light travels through an
effective medium with index n = eψ , phase velocity
ceff = c/n, and nondispersive propagation in optical
bands.
• Test-mass dynamics: Free-fall acceleration a =
(c2 /2)∇ψ derives from the effective potential Φ =
−c2 ψ/2.
3
C α57 ≈ 1.7 × 10−123
α57 : ρc /ρPlanck = 8π

FIG. 12. The α-exponent ladder: one dimensionful anchor
(MP , equivalently one measurement of H0 or G) and the derived tower of scales Q ∼ αk MP . Rung depth is proportional
to the exponent k, i.e. a log scale in α. Rungs: √
the seesaw
scale MR = α3 MP , the electroweak scale v = 2π α8 MP
(Table XLIX), the QCD chain scale ΛDFD = α19/2 MP , the
√
locked electron map me = 23 π α21/2 MP , the Hubble rate
on bound-matter clocks H0loc = α28.5 /tP (Eq. (479); the
early
clock carries the measured O(1) coefficient, H0early =
√
28.5
Cα
/tP = 67.52, Rem. AP.6), and the vacuum-density
3
ratio ρc /ρPlanck = 8π
C α57 (Eq. (474)), spanning the full 122
orders of magnitude of the invariant Eq. (470).

• Clock rates: Proper time rates depend on position through ψ, with any channel-resolved speciesdependent coupling belonging to the extended nonmetric channel (null under the minimal action,
Thm. XI.2), organized by electromagnetic, strongsector, and composition-sensitive contributions.
• Gravitational radiation: Transverse-traceless
perturbations propagate at speed c on the optical cone, which light shares (App. WS), with the
standard quadrupole formula.
The theory is governed by a nonlinear field equation:
 


|∇ψ|
8πG
∇· µ
∇ψ = − 2 (ρ − ρ̄),
(490)
a⋆
c
with the µ-function interpolating between Newtonian
(µ → 1) and deep-field (µ → x) regimes at the acceleration scale a0 = 1.1966 × 10−10 m/s2 read on boundmatter clocks (Postulate ZS.11; 1.1208 × 10−10 on the
early clock). Under the adopted Rest-Mass Channel postulate (App. GR.3a), this screened equation governs the

133
gradient channel, confined to bound systems, while conserved rest mass sources an unscreened Poisson channel, the Einstein–Hilbert term’s own static scalar energy.
Two further adopted inputs act inside bound systems:
the floor composition of App. AV, in the form of one
momentum-conserving energy under which matter moves,
and environmental confinement, which keeps a subsystem
Newtonian inside a strong host field (App. NL). These
and the other inputs of the graded predictions are listed
in Tables CXXVII and CXXVIII.

B.

What DFD Accomplishes

a. Solar System and precision tests. DFD reproduces
all Solar System tests with PPN parameters γ = β = 1
(§IV). Light deflection, Shapiro delay, perihelion advance,
and Nordtvedt effect match observations to current precision. Through second post-Newtonian order the one- and
two-body dynamics is general relativity’s, with ϵ = 1 and
the second-order light-deflection coefficient 15π/4 (§IV;
App. WS).
b. Gravitational waves. The TT sector propagates at
c on the optical cone that light follows, cT = cγ on every
background (App. WS)—a structural result proven from
O(3) irreducible decomposition, not fine-tuning (§V C).
Within the CP 2 × S 3 spectral completion, both ψ and
hTT
ij are derived as irreducible components of the same
zero-mode parent tensor on the internal manifold (§V A 4).
A Lichnerowicz rigidity analysis proves no unwanted massless modes arise; the truncated microsector carries no size
or shape modulus (Appendix O). The theory carries two
tensor polarizations and satisfies the standard quadrupole
formula (§V). Binary pulsar orbital decay agrees at 0.2%.
LIGO/Virgo observations are consistent.
c. Strong fields. With the parent strain’s σ kept
DFD’s strong-field gravity is general relativity’s
(App. WS): black holes have Schwarzschild’s and Kerr’s
horizons and shadows, consistent with current EHT observations (§VI); static stars obey the TOV equations
(App. AT, Theorem AT.13); ringdowns are Kerr’s. Beyond first order in the spin the Kerr results rest on an
argument (App. WS).
d. Galactic dynamics. The µ-crossover produces flat
rotation curves, the baryonic Tully-Fisher relation Mbar ∝
vf4 , and the radial acceleration relation (§VII). The di√
mensionless law a0 /(cH0 ) = 2 α is derived from
the S 3 microsector, and µ(x) = x/(1+x) is the weakresponse form of the derived exponential response
(Appendix N); its physical value on bound-matter clocks
is a0 = 1.1966 × 10−10 m/s2 , with no measured coefficient
(Postulate ZS.11). Quantitative validation: In head-tohead comparison using SPARC galaxy parameters, DFD
beats Newton in 100% of galaxies tested; a dedicated
model-independent interpolation-family scan on all 175
SPARC galaxies further finds nopt = 1.15 ± 0.12 (95%
CI [1.00, 1.50]), placing DFD’s n = 1 inside the preferred
region and strongly disfavoring Standard MOND’s n = 2.

Near the Sun DFD predicts no wide-binary boost: the
Galaxy’s baryonic field there exceeds (ln 2)2 a0 , so under
the adopted environmental confinement local pairs are
exactly Newtonian, γ = 1, which the Sun’s long-period
comets also require (App. NL, Status NL.12); pairs beyond R ≃ 13–16 kpc keep the boost. The Gaia DR3
amplitude analyses are contested [51, 52]; DFD sides with
the analyses that find no signal, and a confirmed boost in
local pairs falsifies the axiom (§VII). Ultra-faint dwarfs’
extreme inferred mass-to-light ratios may be inflated by
measurement systematics (binary contamination, tidal
heating); χ subhalos are the alternative.
e. Cluster scales. Under the µ-law alone the clusters fall short of their measured masses (Obs/DFD
= 1.57 ± 0.08 relaxed, 1.99 ± 0.16 merging, at a0 =
1.20 × 10−10 m/s2 and at the bound-matter a0 = 1.1966 ×
10−10 m/s2 ), and the shortfall is the derived cold component χ (§XVI G): with χ adding its Newtonian pull and
a hydrostatic bias b = 0.21 on the relaxed X-ray masses,
all sixteen clusters need χ = 4.5 ± 0.6 Mb , 0.84 ± 0.12 of
the cosmic mix; under the floor composition of App. AV
they carry the ΛCDM budget. The amplitude follows
from the derived abundance under the adopted energy:
1.21 ± 0.05 of the cosmic mix is predicted inside r500 ,
against 1.28 ± 0.04 measured with no hydrostatic bias
(estimate; App. I). A no-χ closure by five literature correction factors does not hold (the baryonic items sum to
1.26, and the projection correction has the wrong sign for
lensing masses). The Bullet Cluster needs χ at 0.6–0.9 of
the cosmic mix when χ adds its Newtonian pull (1.1–1.2
under the floor composition). Galaxy groups fall short in
the same way on published X-ray masses and need χ too.
See Appendix I for the complete per-cluster analysis.
f. CMB and cosmology. A ψ-based CMB framework is presented (§XVI C):
• Peak ratio from the baryon loading with the derived
χ-matter (2.20 on the measured clock; analytic estimate 2.34)
• Peak location ℓ1 ≃ 220.4 from the measured-clock
no-screen forward solution (100θ∗ = 1.04110); no
global ∆ψ = 0.30 attribution is used
• Cold clustering carrier: the cosmological cold
dark matter (CMB third-peak height, large-scale
structure, and clusters, which need it at the amplitude the derived abundance gives under ΛCDM-like
assembly (1.21 ± 0.05 of the cosmic mix inside r500
predicted; estimate); App. GR) is the derived χmatter particle (Ωχ h2 ≃ 0.12; App. AV), whose
composition with the µ-law inside galaxies is the
floor law, and whose halos assemble as in ΛCDM
under the adopted momentum-conserving energy
(estimate). (The projectable “dust branch” w → 0,
c2s → 0 cannot carry an a−3 clustering charge to
recombination; χ does; see App. J.) The linear perturbation operator and Geff growth law are written
explicitly (Sec. XVI L); full survey-pipeline P (k)
matching remains a numerical program item.
These mechanisms address what standard cosmology at-

134
tributes to “dark matter” (Ωc = 0.26) and “dark energy”
(ΩΛ = 0.69). The analytic framework is extensive, but
the full precision confrontation with cosmological perturbation pipelines remains an active program item rather
than a finished replacement for every standard analysis
tool.
g. Parameter-free predictions. The α-relations
(§VIII) provide parameter-free predictions:
√
a0 = 2 α cH0loc (within SPARC’s systematic error)
(491)
kα = α2 /(2π)

(pure-α bounded)

(492)

ka = 3/(8α)

(consistent with RAR)

(493)

h. Standard Model parameters from topology. Appendix Z demonstrates that Standard Model parameters
emerge from the topology of CP 2 × S 3 :
Derived structural results (grades as stated per
item):
• α−1 = 137.036 from Chern-Simons quantization,
convention-locked (Appendix K 1; the finite-level
bracket coefficient’s provenance is open, Rem. X.4;
the identification of the hypercharge-trace coefficient
with 1/e2 is not derived, §X I)
• sin2 θW = 3/13 from gauge partition + the stated
trace normalization (0.23% agreement)
√
• αs (MZ ) = 0.1187 from ΛQCD = MP α19/2 + 4π
matching (selected scale; 0.8σ)
• θ̄ = 0 (tree level; loop-induced ≲ 10−16 ) from the
det-orthogonal tree-level protection together with
the η-invariant argument, with θYM = 0 fixed by
the spectral action given that nothing is added to
it for the gauge
fields (Appendix L)
√
8
• v = MP α√
2π: exponent from microsector scaling,
prefactor 2π fitted (a no-go on deriving it; App. Z)
• Ngen = 3 (one stated input, used once; no index on
the internal space supplies it, Rem. F.21)
• εH = 3/60 = 0.05 from channel counting (Appendix H)
• Generation = left Z3 phase sectors (Proposition Y.12)
Comparisons:
• b/τ = 1.98 (obs 2.35, 16% off) — from bin scan
(0, 2)/(1, 2)
• b/t = 0.018 (obs 0.024, 24% off) — same mechanism
• c/t = 0.0073 (obs 0.0073, 0.8% off) — from bin
(2, 0)/(1, 0)
• CKM: (31, 108, 43
2 , 49)×α pattern, apex fixed by the
Euler-projection postulate (App. AO); γ = 66.3◦ ,
J = 2.97 × 10−5
Remaining (numerical refinements):
• Nine fermion masses at 1.42% leading-order mean
error via explicit Af (Theorem K.4): the top and
charm derived, the other seven on seven dataselected prefactors (App. FM)
• Neutrino mass splittings with χ2 = 0.025 vs NuFIT
6.0 on the data-selected branch (Appendix X)
The charged-fermion sector carries no continuous knob

beyond α and v,
√ whose prefactor is fitted: the top mass
mt = (1 − α)v/ 2 = 172.74 GeV is forced, and the charm
with it, along with forced inter-generation ratios such as
mt /mb = 42 (the comparison divides the top pole mass
by mb (mb ); at one scale, Mt , the ratio is 59.7, so the
comparison awaits a one-scale test). The double ratio
D, which reads 27.7 in PDG’s mixed convention, is 29–
33 at one renormalization scale, and the same weights
miss a second ratio by 15% at any scale (App. Y), so it
is not among the forced ratios. The absolute chargedfermion spectrum other than the top and the charm is
not derived: the other seven species rest on seven dataselected prefactors (App. FM).

C.

The Critical Tests

Each experimental channel has a distinct role:
a. 1. Cavity-atom LPI test (§XII). The cavity–atom
channel remains important, but geometric cancellation
removes any order-unity tree-level slope. It is best viewed
as a precision residual test whose cleanest role is to probe
the surviving non-metric cavity/atom mismatch once the
constitutive-chain cancellation is accounted for.
b. 2. Clock anomalies (§XI). The clock program is
interpreted in a channel-resolved way. Same-ion optical
clocks test the pure α sector; cross-species atomic ratios
test composition-sensitive structure; and nuclear clocks
test the strong sector. Improved multi-species measurements remain among the sharpest falsifiers in the whole
DFD framework.
c. 3. Matter-wave T 3 signature (§XIII): not a DFD
prediction. The formula below is not a DFD phase:
2
ℏkeff
g 3
T .
(494)
m c2
Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII. Its
T 3 scaling, rotation sign flip and even k-parity do not
separate DFD from GR, which coincide through 1PN.
d. 4. Antimatter gravity (§XV). Matter–antimatter
differential acceleration probes C-odd sector couplings:
∆aH H̄
≈ 2|σH̄ − σH |.
(495)
a
At the metric level, DFD predicts ∆aH H̄ /a = 0 (matching
GR). Non-metric couplings to baryon/lepton number are
already bounded by ordinary-matter equivalence-principle
tests, to |∆aH H̄ /a| ≲ 10−11 (Sec. XV G).
e. 5. EM–ψ coupling (Appendix R). In a parametricmode model that needs a dynamical ψ mode, which the
local ψ of Axiom WS.1 lacks, and whose absolute normalization is open, the parameter λ controls electromagnetic
back-reaction on ψ:

∆ϕDFD =

|λ − 1| ≲ 3 × 10−5

(accidental bound from cavity stability).

(496)
An intentional 2ω modulation search could reach |λ − 1| ∼
3 × 10−9 —about four orders of magnitude tighter—with
upgraded apparatus.

135
D.

If DFD Is Confirmed

If laboratory tests confirm DFD predictions, the implications would be as follows:
1. Gravity is fundamentally optical/refractive,
not geometric. The metric tensor would be emergent from the parent strain on the flat arena, its
trace ψ and its traceless part, rather than fundamental.
2. The dark-sector carriers are identified, with
their grades stated. Flat rotation curves and
the radial-acceleration relation arise from the universal µ-crossover acting on the baryons, with no
fitted halo; the derived χ is present in galaxies too,
composed with the µ-law by the floor law, in ΛCDMlike halos whose inner parts must be cored for the
rotation curves (estimate; App. AV). The cosmological cold carrier is the topologically derived χ
field
√ (App. AV); its 32.005 eV coefficient uses the
158 multiplicity, one stated count, its abundance
is forced-amplitude/stated-convention, conditional
on the flux-period identification and the asserted
premise P4′ , Qχ = 1 clustering is conditional on the
adopted Rest-Mass Channel, and it has no photon
coupling (E = 0, derived; App. AV). Dark energy is
the α57 geometric vacuum energy with the measured
vacuum coefficient R (App. O).
3. Part of the Standard Model structure follows from topology. The electroweak group is
the isotropy group of CP 2 under one stated identification, and colour, the family content and the three
generations enter as inputs (App. F); the fermionmass hierarchy is derived (texture fitted) and the
CKM matrix follows from the selected counts and
the postulated apex.
4. The bare-parameter form of the hierarchy
problem is removed. The 17 orders of √
magnitude between MP and v follow from α8 ; the 2π
prefactor is fitted and radiative stability stays open
(App. H).
5. Strong CP at tree level (Theorem L.3). With
θYM = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields, θ̄ =
0 at tree level, loop-bounded below 10−16 . Tree
level: arg det(Mu Md ) = 0 (App. AO). Anomaly:
the fermion Clifford module on the even-dimensional
(8) mapping torus has spectral pairing and no zero
modes, forcing η = h = 0. No axion required.

E.

If DFD Is Falsified

DFD is falsifiable. The theory would be ruled out if:

a.

Core falsification.

• Matter-wave: not a falsification channel (null prediction, §XIII; DFD and GR coincide through 1PN)
• Antimatter ∆aH H̄ /a ̸= 0 at > 3σ (universal coupling violated)
b.

Indirect falsification.

• RAR deviates from µ-crossover prediction at > 3σ
(the galactic sector is wrong)
• GW speed differs from c at > 10−15 (the TT sector
is wrong)
• α-relations fail by > 20% on the clock each reads
(the theoretical framework is wrong)
c. What remains. If DFD is falsified, General Relativity remains the established theory. The galactic dark
matter problem would still require explanation (CDM,
other modified gravity). The clock anomalies, if confirmed,
would need alternative interpretation.

F.

Comparison with Alternatives

TABLE L. Comparison of DFD with alternative approaches.
GR+CDM MOND TeVeS f(R) AeST DFD
Solar System
GW speed = c
Binary pulsars
Rotation curves
Tully-Fisher
RAR tightness
Clusters
CMB peaks
Lab predictions

✓
✓
✓
✓ (DM)
? (DM)
?
✓
✓
—

✓
—
✓
✓
✓
✓
×
×
—

✓
×
✓
✓
✓
✓
×
∼
—

✓
✓
✓
×
×
×
✓
✓
—

✓
✓
✓
✓
✓
✓
∼
✓
—

✓
✓
✓
✓
✓
✓
✓
✓
✓

Notes: The cluster entry for DFD is “✓” because the
derived χ supplies the mass the µ-law leaves (0.84±0.12 of
the cosmic mix when it adds its Newtonian pull; App. I),
at the amplitude the derived abundance gives under
ΛCDM-like assembly (1.21 ± 0.05 of the cosmic mix inside r500 predicted; estimate). The rotation-curve, Tully–
Fisher and RAR entries for DFD are the µ-law on the
baryons; they hold with χ present when its ΛCDM-like
halos are cored (App. AV). The CMB entry for DFD is
“✓” because the peak ratio follows from the baryon loading
with the derived χ-matter and the peak location is reproduced by the current measured-clock forward solution
without a global ψ screen.
DFD’s distinctive features are: (1) a broad ψ-CMB
framework (the peak ratio has an analytic estimate; the
peak location is the measured-clock no-screen forward
result), (2) cluster-scale phenomenology addressed

136
in the same framework, (3) falsifiable laboratory
predictions spanning channel-resolved clocks (extended
non-metric channel), antimatter, and cavity–atom residuals, (4) parameter-light predictions via the α-relations
and topological microsector, and (5) a unification layer
collecting the fermion-mass, CKM, PMNS, and Higgsscale derivations in one place.

G.

a.

Outlook

Near-term priorities.

1. Nuclear-clock (Th-229/Sr) campaigns and Ooi-style
annual-phase reanalyses
2. Cross-species clock comparison campaigns (Hg/Sr,
Yb+ /Sr, Yb/Sr, Cs/Sr)
3. Same-ion null checks to keep the pure-α sector
pinned down
4. Matter-wave interferometry upgrade for the T 3
search (null prediction) and, longer-term, cavity–
atom residual roadmaps
b.

Medium-term goals.

1. Nuclear clock (Th-229) tests of strong-sector coupling

c. Long-term vision. DFD’s theoretical framework
names its main open items rather than hiding them
(§XVIII A), among them the origin of the primordial perturbations and the horizon problem, which enter through
one stated starting condition (the amplitude As is a calibrated number), the interacting quantum dynamics it
shares with every quantum field theory, and in the dark
sector the inner cores and masses of the χ halos, the modeby-mode rule of motion of the momentum-conserving
energy, the adopted floor law and environmental confinement, and the weak lensing of isolated galaxies beyond
their bound radius, where the halo model of ΛCDM also
falls below the measured points (App. GR); the Qχ = 1
χ-clustering channel is closed by the adopted Rest-Mass
Channel axiom, App. GR.3a, which confines the µ-law to
bound systems, the residual being to derive that axiom
from a deeper principle. We do not claim the corpus to
be free of internal inconsistency: it is large, parts of it are
inherited from earlier releases, and its internal review is a
continuing obligation rather than a completed one. What
we do claim, and what the reader can check, is that the
review is conducted in the open: corrections are made
at the statements they amend (the principal ones are
tabulated before the table of contents), and the inherited
statements flagged by a proof-pattern scan are graded by
name in §XVIII K. The remaining task is experimental
verification, closing the named open items, and continued
internal hardening of the live phenomenology modules.
If confirmed, the theory would represent a shift in our
understanding: gravity as optics, the electroweak group
from the inner space under one stated identification, and
a cosmology whose dark matter is derived rather than
fitted (the χ field) and whose dark energy has a derived
exponent and one measured coefficient (the α57 vacuum).

2. Space-based precision tests (ACES successor)
H.

Structural Separation: Gravity vs. Microsector

3. Independent verification of microsector derivations
4. Further cluster-by-cluster verification

We separate the testable gravity program from the
unification claims:

DFD Gravity: Robust and Testable
What stands independently:
• Two core postulates: n = eψ , Φ = −c2 ψ/2; the adopted axioms, among them the Rest-Mass Channel behind the
cosmological Qχ =1 clustering channel (App. GR.3a), and the identifications, selections and calibrations are listed in
Tables CXXVII and CXXVIII
• PPN parameters: γ = β = 1
• GW sector: cT = cγ exactly (one metric for light and gravitational waves; App. WS), two polarizations
• Galactic dynamics: µ-crossover, RAR, BTFR, and the SPARC shape-selection result near n = 1
• Cluster phenomenology: the µ-law plus the derived χ, at the amplitude its abundance gives under ΛCDM-like
assembly (estimate)
• Laboratory predictions: channel-resolved clocks (extended non-metric channel), antimatter, and cavity–atom residual
tests
Falsifiers: collapse of the channel-resolved structure of the extended clock channel and RAR/shape deviations at high
significance
If the microsector is wrong, DFD gravity stands.

137
Gauge Emergence: Conditional
What depends on CP 2 × S 3 framework:
• α−1 = 137.036 from the convention-locked microsector closed form (§X; bracket coefficient open)
• Electroweak group as the isotropy group of CP 2 , under one stated identification; colour and the family content as
inputs
• Ngen = 3 (one stated input, Rem. F.21)
• Top and charm masses, CKM magnitudes and PMNS angles from geometry, with the stated inputs of Table CXXVIII
• GℏH02 /c5 = Cα57 invariant
√ (exponent derived; the vacuum coefficient R measured)
• Higgs scale: v = MP α8 2π (exponent derived, prefactor fitted)
Falsifiers: Wrong fermion mass ratios, proton decay observation, HF = Cd derived from first principles (would shift α by
43 ppm)
If this fails, DFD gravity can be retained with α as input.

a. The firewall. The gravity program is constructed
to survive even if the gauge emergence program fails
entirely. The α-relations can be taken as empirical input
rather than topological output. The laboratory tests
(§XI–§XIII) depend only on the two postulates, not on
the microsector.

I.

Final Statement

things: either (i) derived from the core DFD field/action
system, or (ii) derived from an explicitly stated auxiliary
closure framework whose assumptions are displayed in
the text. Empirical benchmark modules and numerical
consistency checks are labeled as such and should not
be confused with core-field theorems. This convention is
deliberate: it preserves the monograph’s one-paper unity
while preventing auxiliary closure principles, benchmark
hierarchies, and open numerical pipelines from being
mistaken for hidden first-principles proofs.

a. Interpretive convention for claim strength.
Throughout this review, “derived” means one of two
DFD: Unified Framework + Falsifiable Predictions
Derived results (items marked ⋆ are theorem-grade with formal proofs; others depend on dictionary axioms or structural
assumptions):
• µ(x) = x/(1 + x), the weak-response form of the derived exponential response (Theorem N.8; App. BS)
√
• a0 /(cH
0 ) = 2 α derived from topological stationarity (Theorem N.14); its physical value on bound-matter clocks is
p
2α29 c7 /Gℏ = 1.1966 × 10−10 m/s2 (Postulate ZS.11; consistent with SPARC within its systematic error), and
1.1208 × 10−10 m/s2 on the early clock
⋆ Dust branch: K ′ (∆) = µ(∆) gives w → 0, c2s → 0 (Theorem Q.7); the cold clustering carrier is the derived χ-matter
(App. AV), not this projectable branch
• Strong CP: θ̄ = 0 at tree level, loop-bounded, with θYM = 0 fixed by the spectral action given that nothing is added to it
for the gauge fields (Theorem L.3)
• G–H0 invariant: (H0 tP )2 = Cα57 ; exponent 57 spectral-action-derived and C measured on the early clock (Appendix O);
on bound-matter clocks C = 1 (Postulate ZS.11) and H0loc = α28.5 /tP = 72.09 km/s/Mpc from the measured G
(Eq. (479))
• Clock coupling: kα = α2 /(2π) (Appendix P)
Quantitative matches:
• α−1 = 137.036 (21 ppm from the integers; the bracket coefficient is open)
√
• Higgs: v = MP α8 2π (exponent derived, prefactor fitted)
• Fermion masses: 1.42% leading-order mean error (9 particles; seven prefactors data-selected)
• CKM: λ = 31α = 0.2262 (integer pattern; CP 2 overlap motivation)
• PMNS: trimaximal TM1 (App. X); sin2 θ13 = 3α a stated conjecture
• CMB: the analytic R = 2.34 (6% above Planck’s 2.21; the χ-matter spectrum, run on the measured clock, gives 2.20,
within 0.6σ), ℓ1 = 220 (no postulated CDM; third-peak height from the derived χ-matter field, App. AV)
res
• ESPRESSO: conditional on ξLPI
Key problems addressed: UV completion (topology, proposed), Λ problem (α57 ), hierarchy (α8 ), strong CP (tree level,
loop-bounded), neutrino hierarchy (< 0.2σ, App. X).

138
No continuous parameter is fitted to a predicted shape, except the baryon density. The theory’s cosmology
carries four continuous calibration constants, the vacuum coefficient R (absolute scale), the amplitude As of the primordial
perturbations, the coefficient of ωb h2 (the one shape) and ΛCDM’s shared astrophysical τ , against ΛCDM’s six free
parameters, and three against its five effective parameters on the Planck likelihood itself (App. FK, Rem. FK.8); the particle
sector adds the fitted Higgs prefactor, the imported factor ηq and the Higgs quartic, conjectured to be 1/8 at a matching
scale that is not derived. The discrete topological sector carries no continuous freedom: the isotropy charge fixes q1 = 3, the
determinant-line lift gives O(9) (with Ngen = 3 a discrete input), and the five chiral multiplet types fix the padding (status
remarks: App. F). Within E = O(a) ⊕ O⊕n , minimal-padding uniquely selects (a, n) = (9, 5) with kmax = 60; the discrete
inputs are priced in the trials audit of App. CE. One scale measurement (H0 or G), together with the measured R, then
determines all dimensionful quantities.
New structural results:
⋆ The Padé identity (Theorem AA.1, Appendix AA): LGR (u) = [P1,1 (u)]2 in isotropic coordinates, where L ≡ c2 /|gtt | is
the lapse-squared scalar and Pm,m (u) denotes the [m, m] Padé approximant of eu , so that GR’s isotropic lapse-squared is
the m = 1 slot of a Padé hierarchy whose m → ∞ limit is the exponential exp(2u), DFD’s lapse-squared at ψ = 2u. The
identity reproduces β = 1 in both theories; γ = 1 for DFD follows separately from the physical metric. With the parent
strain’s σ kept DFD’s exterior is Schwarzschild’s (App. WS), so the pole is the event horizon of DFD’s black holes too
(Theorems AA.2, AA.3 establish the firewall from Yilmaz-type exponential metrics).
⋆ Uniqueness of CP 2 × S 3 (Theorem AB.1, Appendix AB): The internal manifold is not an ansatz but the unique
compact manifold (up to diffeomorphism) satisfying seven satisfiable vacuum axioms motivated by DFD’s optical
structure, three generations, and the framed transgression sector; dimension 7, Spinc and π3 = Z are outputs of the
theorem, not inputs; kmax = 60 follows through the conditional determinant-line count of App. F, given the generation
count, the flux quantum and the five multiplet types; and the SM gauge group is carried by the internal mode bundle
rather than by T X. The integers {3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137} are not free parameters: they are fixed by this
manifold together with the stated inputs and identifications of App. FK.
Companion notes:
• Epoch extension of a0 [112]: theorem-grade
conditional on Axiom WS.1 (Theorem XIX.3): the no-memory argument
√
forces the tracking branch a0 (z) = 2 α cH(z) — the anchored alternative is structurally unavailable — while the G–H0
invariant stays a present-epoch boundary condition with Ġ = 0 (Corollary XIX.4); the two are independent falsifiers. First
data: over 0.33 < z < 1.44 MUSE-DARK III [109] shows a rising scale, the sign that separates DFD from static MOND; a
theory-neutral analysis of its bins, published in MNRAS 551(4), 2026 September 7 [110], gives a0 (z) = A(1 + z)γ with
√
γ = 0.78 ± 0.15 (0.83 ± 0.13 with the SPARC z ≃ 0 anchor), excluding a frozen a0 at 5.3σ (6.6σ anchored) and a0 ∝ ρm
at 4.8σ (on the exponent’s error; 4.7σ by likelihood ratio) on the published bin errors, with H(z)-tracking the
information-criterion-preferred one-parameter description. These exclusions are amplitude-marginalised nested tests and
carry no H0 , h, Ωm or distance scale; the clock coefficient cannot move them. On DFD’s measured-clock background
(Ωm ≃ 0.312) the refit gives A = (1.394 ± 0.032) × 10−10 m/s2 , χ2 = 5.36/3 and γE = 1.098, 2.1σ above the measured
exponent. Two caveats apply: under the 95% reading of the source’s unlabelled bars the two exclusions strengthen (to
≳ 10σ and 9.5σ on the exponent’s error, 9.2σ by likelihood ratio) but the tracking consistency moves into tension
(χ2 = 19.9/3, p = 1.8 × 10−4 ), and the fitted amplitude sits 17% above the canonical SPARC normalisation, within its
systematic-dominated but not its statistical uncertainty. Refitted galaxy by galaxy from the survey’s public products, the
rise reproduces, but the galaxy-level uncertainty is several times the published bin errors, so within the survey a frozen a0
lies about 1σ away, while anchored to SPARC’s z = 0 value, the galaxy-level bins still exclude a frozen a0 at about 3σ,
and DFD’s zero-parameter law fits them on the bound-matter clock; the trend depends on how the molecular gas is added,
keeping the rise when tied to the survey’s disc masses and flat when tied to SED-based masses, and falling with stellar
masses from spectral energy distributions (App. A0Z). Massive discs at 0.6 < z < 2.5, whose census counts molecular gas,
return a0 ≃ 1.2 × 10−10 m/s2 at z ≃ 2.2, close to the local value, where tracking requires 3.7; DFD’s domain-local screen
would give values this low only in 10 h−1 Mpc domains about 6 to 11 standard deviations denser than average, which their
measured environments do not show (Appendix ZT). The running sign is therefore contested: direct molecular-gas masses,
kinematic masses not limited by a halo-model prior, per-galaxy analyses with galaxy-level errors and, for the massive
discs, an independent bound on the χ halo or rotation curves beyond 3Re decide it, and JWST and ALMA high-z
rotation curves, and ultimately SKA H i kinematics, extend it (see Sec. XIX and Appendix N).
• Minimal-sector baseline for λ [116]: tree-level no-drive theorem giving λbare = 1 for ideal symmetric single-mode
standing-wave cavities in the minimal optical-metric EM sector; Appendix R reads the accidental bound as a joint
constraint on the channels of λeff − 1.
The theory stands or falls on experiment. The decisive near-term tests are channel-resolved cross-species and
nuclear-clock campaigns, which discriminate the extended non-metric channel rather than the minimal action (Thm. XI.2),
followed by longer-horizon cavity–atom residual experiments. Matter-wave interferometry is a null check (DFD and GR
coincide through 1PN; §XIII).
√
DFD operationally ties the local galactic transition scale to the cosmic expansion rate through a0 = 2 α cH0 (Appendix N,
Theorem N.14). This is the empirically sharpest interpretive handle on the theory and is the structural identity that

139
√
Theorem XIX.3 (conditional on Axiom WS.1) extends to a0 (z) = 2 α cH(z), with a rising scale in low-mass discs at
0.33 < z < 1.44 on the published bins (γ = 0.78 ± 0.15, a frozen a0 excluded at 5.3σ on the bin errors, about 1σ away galaxy
by galaxy within the survey and about 3σ away galaxy by galaxy anchored to SPARC’s z = 0 value, and 2.1σ from DFD’s
own E(z) slope at the measured clock, the bars read as 1σ; MUSE-DARK III [109], analysis published in MNRAS 551(4),
2026 September 7 [110]), a trend that depends on how the molecular gas is added, and no rise in massive discs at
0.6 < z < 2.5: the extension is contested.
A scientific theory must make predictions that can be proven wrong. DFD does so. The community is invited to test it.

140
Appendix A: Notation and Conventions

This appendix provides a complete reference for all
notation used in the review. Consistent conventions facilitate reproducibility and comparison with other work.

1.

Fundamental Fields and Parameters

2.

Coordinate and Metric Conventions

ds̃2 = −

2

c dt
+ dx2 ,
n2

• δ φ̂k : ppE phase deformation at k-PN order. DFD:
δ φ̂k = 0 for compact binary accelerations ≫ a0 .

(Minkowski). (A1)

This matches the convention of Misner, Thorne &
Wheeler [117] and is standard in gravitational physics.
b. Optical Metric. The optical line element takes the
form:
2

• cT : Tensor mode propagation speed. DFD: cT = cγ
exactly: light and gravitational waves share one
metric (App. WS).
• h+ , h× : Plus and cross polarizations. DFD: identical to GR (no scalar GW modes in far zone).

a. Metric Signature. We use the (−, +, +, +) (mostly
positive) signature throughout:
ds2 = −c2 dt2 + dx2 + dy 2 + dz 2

scalar field ψ affects source dynamics but not GW propagation (see Sec. V B for construction, Sec. V C for rigorous
proof):

n = eψ .

(A2)

Light rays satisfy ds̃2 = 0. The coordinate speed of light
is c/n = c e−ψ .
c. Spherical Coordinates. For spherically symmetric
problems:
dx2 = dr2 + r2 (dθ2 + sin2 θ dϕ2 ).

a.

5.

Clock and LPI Parameters

6.

Galactic Dynamics Notation

Key Relations.

Vc2
r
GMbar (< r)
gbar =
r2
4
Vflat = GMbar a0
gobs =

(A3)

7.

(centripetal acceleration) (A7)
(Newtonian gravity)

(A8)

(BTFR, deep-field limit)

(A9)

Unit Conventions

2

The radial acceleration magnitude is a = (c /2)|dψ/dr|.
d. Index Conventions.
• Greek indices µ, ν, . . . ∈ {0, 1, 2, 3} for spacetime
• Latin indices i, j, . . . ∈ {1, 2, 3} for spatial components
• Repeated indices imply summation (Einstein convention)

3.

a.

Physical Constants

Derived Quantities.
2GM
rs =
c2
GM⊙
c2 r

(Schwarzschild radius)

(A5)
≈ −9.87 × 10

4.

at 1 AU

(A10)

1 M⊙ = 1.477 km = 4.926 µs.

(A11)

[M ] = [L]−1 = [T ]−1 ,
6

(Solar potential)
−9

[M ] = [L] = [T ],

When geometric units are used, this is stated explicitly.
c. Natural Units. For quantum considerations, ℏ =
c = 1 gives:

(A4)
Φ⊙ /c2 = −

a. SI Units. All equations in this review are written in SI units unless otherwise noted. This ensures
dimensional transparency and direct comparison with
experimental values.
b. Geometric Units. For some derivations, particularly those involving spacetime structure, it is convenient
to set G = c = 1. In these “geometric units”:

(A6)

Post-Newtonian and Gravitational Wave
Parameters

a. Gravitational Wave Parameters. DFD’s GW sector is constructed as a minimal transverse-traceless sector
that reproduces GR exactly in the radiative zone. The

1 eV = 5.068 × 10 m

−1

(A12)
= 1.519 × 10

15 −1

s

.

(A13)

d. Gaussian vs. SI Electromagnetism. For electromagnetic quantities, we use SI (rationalized) units. The
fine-structure constant is:
1
e2
≈
.
(A14)
α=
4πϵ0 ℏc
137
8.

Abbreviations and Acronyms

9.

Sign Convention Summary

For quick reference, the key sign conventions are:

141
TABLE LI. Primary field variables and coupling parameters in DFD.
Symbol Name

Definition/Value

Units

Fundamental field
ψ
Scalar refractive field
n
Refractive index
Φ
Effective potential

Primary gravitational d.o.f.
n = eψ
Φ = −c2 ψ/2

dimensionless
dimensionless
m2 /s2

Acceleration scales
a⋆
Characteristic gradient scale 2a0 /c2 = 2.663 × 10−27 m−1
m−1
√
loc
−10
2
m/s m/s2
a0
MOND acceleration scale
2 α cH0 = 1.1966 × 10
a
Physical acceleration
a = (c2 /2)∇ψ
m/s2
a2
Acceleration invariant
a2 ≡ a · a
m2 /s4
Coupling constants
ka
Self-coupling parameter
kα
Clock coupling
KA
Effective clock coupling

ka = 3/(8α) ≈ 51.4
kα = α2 /(2π) ≈ 8.5 × 10−6
channel-resolved; Eq. (291)

dimensionless
dimensionless
dimensionless

Interpolating function
µ(x)
Crossover function
ν(y)
Inverse function
x
Dimensionless argument

µ → 1 (x ≫ 1), µ → x (x ≪ 1)
y = xµ(x), x = yν(y)
x = |∇ψ|/a⋆ = a/a0

dimensionless
dimensionless
dimensionless

TABLE LII. Physical constants used in calculations. Values from CODATA 2018 unless noted.
Symbol Name
c
G
ℏ
α
α−1
H0
M⊙
R⊙
AU

Value

Units
8

Speed of light
2.99792458 × 10
m/s
Gravitational constant
6.67430(15) × 10−11
m3 kg−1 s−2
Reduced Planck constant 1.054571817 × 10−34
Js
Fine-structure constant 7.2973525643(11) × 10−3 (CODATA 2022)
dimensionless
Inverse α
137.035999177(21) (CODATA 2022)
dimensionless
Hubble constant
67.52 ± 0.09 (H0early ); 72.09 (H0loc , C = 1, Postulate ZS.11) km s−1 Mpc−1
Solar mass
1.98841 × 1030
kg
Solar radius
6.9634 × 108
m
Astronomical unit
1.495978707 × 1011
m

TABLE LIII. Post-Newtonian parameters. DFD predictions
match GR exactly.
Parameter Meaning
γ
β
ξ
α1
α2
α3
ζ1 –ζ4

GR DFD

Space curvature per unit mass
1
Nonlinearity in superposition
1
Preferred-location effects (PPN)
0
Preferred-frame (PFE)
0
PFE parameter 2
0
PFE parameter 3
0
Violation of momentum conservation 0

1
1
0
0
0
0
0

Sign Conventions
• Metric signature: (−, +, +, +)
• Potential sign: Φ < 0 in gravitational wells
• Field sign: ψ > 0 in gravitational wells (so n > 1)
• Relation: Φ = −c2 ψ/2, hence ψ = −2Φ/c2 > 0
• Acceleration direction: a = −∇Φ = (c2 /2)∇ψ
points toward mass
• Curvature: Not applicable (DFD uses flat
background)

These conventions ensure consistency with both the
Newtonian limit and standard GR formulations.

Appendix B: Detailed Derivations

This appendix provides step-by-step derivations of key
results referenced in the main text. Each derivation includes dimensional checks and identifies approximations
used.

142
TABLE LIV. Clock comparison parameters and sensitivities.
Symbol Definition
res
ξLPI
α
SA

KA
∆KAB
y

Typical Value

Residual cavity–atom LPI parameter DFD: screened residual; GR: 0
α-sensitivity of clock A
See Table LV
Effective clock coupling
channel-resolved Eq. (291)
Differential coupling
KA − KB
Fractional frequency
y = ∆ν/ν

TABLE LV. α-sensitivities for selected clock transitions.
Clock

Transition

Sα

1.

Second Post-Newtonian Light Deflection

Reference

Cs hyperfine 6S1/2 F=3→4 +2.83 [70]
Rb hyperfine 5S1/2 F=1→2 +2.34 [70]
H maser
1S hyperfine +2.00 [70]
1
Sr optical
S 0 → 3 P0
+0.06 [118]
2
Yb+ E2
S1/2 → 2 D3/2 +0.88 [118]
2
Yb+ E3
S1/2 → 2 F7/2 −5.95 [118]
+
1
Al
S 0 → 3 P0
+0.008 [118]

a.

Consider light propagating past a spherically symmetric
mass M at impact parameter b ≫ rs = 2GM/c2 . In DFD,
the refractive index is:
2GM
n(r) = eψ(r) ,
ψ(r) = 2 + O(rs2 /r2 ).
(B1)
c r
b.

TABLE LVI. Notation for galactic dynamics and rotation
curves.
Symbol Definition
Vc
Vflat
Vbar
gobs
gbar
Mbar
Σ
Υ⋆

Units

Circular velocity
km/s
Asymptotic flat velocity
km/s
Baryonic (Newtonian) velocity
km/s
Observed centripetal acceleration m/s2
Baryonic gravitational acceleration m/s2
Total baryonic mass
M⊙
Surface mass density
M⊙ /pc2
Stellar mass-to-light ratio
M⊙ /L⊙

Acronym Meaning
Density Field Dynamics
General Relativity
Parametrized Post-Newtonian
Local Position Invariance
Modified Newtonian Dynamics
Baryonic Tully-Fisher Relation
Radial Acceleration Relation
Gravitational Wave
Parametrized Post-Einsteinian
Effective Field Theory
Ultraviolet (high-energy)
Cosmic Microwave Background
Baryon Acoustic Oscillations
Spitzer Photometry and Accurate Rotation Curves
Lunar Laser Ranging
Very Long Baseline Interferometry

Ray Equation

From Fermat’s principle, the ray equation is:


d
dx
n
= ∇n.
ds
ds

(B2)

For small deflections, parameterize the path as x(z) =
(x(z), y(z), z) where z is the coordinate along the unperturbed ray. The transverse deflection satisfies:
1 ∂n
∂ ln n
d2 x
=
.
≈
dz 2
∂x
n ∂x
c.

TABLE LVII. Frequently used abbreviations.
DFD
GR
PPN
LPI
MOND
BTFR
RAR
GW
ppE
EFT
UV
CMB
BAO
SPARC
LLR
VLBI

Setup

(B3)

First-Order (1PN) Deflection

At first order, n ≈ 1 + ψ and we integrate along the
unperturbed straight line at x = b, y = 0:
Z +∞
∂ψ
dz.
(B4)
α(1) =
−∞ ∂x x=b
√
For ψ = 2GM/(c2 b2 + z 2 ):
∂ψ
2GM b
=− 2 2
.
(B5)
∂x
c (b + z 2 )3/2
The integral is standard:
Z +∞
dz
2
= 2.
(B6)
2 + z 2 )3/2
b
(b
−∞
Therefore:
α(1) =

4GM
c2 b

(B7)

Dimensional check:
[GM/c2 b] = m/m =
dimensionless ✓
This reproduces the GR result exactly, as required for
γ = 1.

143
d.

Second-Order (2PN) Deflection

At 1PN order:
Φ(r) = −

At 2PN, we need:
1. Higher-order expansion of the gradient: ∇(ψ +
ψ 2 /2 + . . .)
2. Path corrections from 1PN deflection
The 2PN correction arises from expanding n = eψ ≈
1 + ψ + ψ 2 /2:
∂ ln n
∂ψ
∂ψ
≈
+ψ
+ O(ψ 3 ).
(B8)
∂x
∂x
∂x
The additional contribution is:
Z +∞
∂ψ
(2)
α =
ψ
dz.
(B9)
∂x x=b
−∞
√
Substituting ψ = 2GM/(c2 r) with r = b2 + z 2 ,
ψ ∂x ψ = −(2GM/c2 )2 b/r4 and
2 Z +∞

dz
2GM
2πG2 M 2
(2)
b
=
−
.
α =−
2
2 2
c2
c4 b2
−∞ (b + z )
(B10)
The gradient term above is not the full 2PN deflection:
the bending of the ray path contributes at the same order,
so the coefficient must be obtained from the exact null
geodesic rather than asserted. With the parent strain’s σ
kept, DFD’s static exterior is the Schwarzschild metric in
DFD’s coordinates (App. WS), so the exact deflection is
Schwarzschild’s and the second-order coefficient is general
relativity’s [119, 120]:


15π GM
4GM
1+
α= 2
,
cDFD
= cGR
= 15π
2
2
4
c b
16 c2 b
(B11)
where c2 is the coefficient of (GM/c2 b)2 in α (evaluated
from the exact null geodesic; computed). The isotropic
exponential metric g00 = −e−ψ , gij = e+ψ δij with ψ =
2GM/c2 r, which sets σ to zero, would give 4π instead.
At the solar limb the second-order term is ∼ 11 µas and
presently unmeasured (Gaia operates at ∼ 20–30 µas and
avoids pointing near the Sun); a LATOR-class astrometric
mission would measure it. Falsifier: a measured secondorder solar-deflection coefficient different from 15π/4, such
as the 4π of the isotropic exponential metric, falsifies
DFD’s exterior.

2.

Perihelion Precession
a.

Effective Potential

For a test mass in the DFD field of a central mass M ,
the effective one-dimensional potential is:
L2
,
(B12)
2r2
where Φ = −c2 ψ/2 and L is the angular momentum per
unit mass.
Veff (r) = Φ(r) +

GM
G2 M 2
− 2 2 + O(c−4 ).
r
c r

b.

(B13)

Orbit Equation

Using u = 1/r, the 1PN geodesic equation of the physical metric (γ = β = 1, §IV) gives the Binet equation:
GM
3GM 2
d2 u
+u= 2 +
u .
(B14)
2
dϕ
L
c2
The last term causes precession. For a nearly circular
orbit with semimajor axis a and eccentricity e:
1
u≈
(1 + e cos ϕ).
(B15)
a(1 − e2 )
c.

Precession Rate

The perihelion advances by:
6πGM
6πG2 M 2
= 2
c2 L2
c a(1 − e2 )
per orbit. In terms of orbital period T :
∆ω =

ω̇ =

6πGM
c2 a(1 − e2 )T

(B16)

(B17)

Dimensional check: [GM/(c2 aT )] = m · s−2 /s =
rad/s ✓
d.

Mercury

For Mercury: a = 5.79 × 1010 m, e = 0.2056, T =
7.60 × 106 s.
ω̇Mercury = 42.98 arcsec/century,

(B18)

matching GR and observations.

3.

Baryonic Tully-Fisher from µ-Crossover
a.

Deep-Field Limit

In the deep-field regime where |∇ψ| ≪ a⋆ , the interpolating function satisfies µ(x) → x for x ≪ 1. The field
equation becomes:


|∇ψ|
8πG
∇·
∇ψ = − 2 ρ.
(B19)
a⋆
c

144
b.

Spherical Symmetry

For a spherically symmetric mass distribution with total
mass M :


′
1 d
8πGρ
2 |ψ | ′
(B20)
r
ψ =− 2 .
2
r dr
a⋆
c
In the asymptotic region (r → ∞), integrating over a
sphere:
4πr2 ·

(ψ ′ )2
8πGM
.
=
a⋆
c2

Therefore:
r
ψ′ =

2GM a⋆
=
c2 r2

√

2GM a⋆
.
cr

(B21)

Numerical verification (round illustrative benchmark H0 = 70 km/s/Mpc; H0loc = 72.09 gives the boundmatter a0 = 1.1966 × 10−10 m/s2 (Postulate ZS.11) and
H0early = 67.52 the early-clock 1.1208 × 10−10 , App. O):
√
α = 0.08542
(B29)
α = 1/137.036,
c = 2.998 × 108 m/s

H0 = 70 km/s/Mpc = 2.27 × 10
s
(B31)
√
2 α cH0 = 2 × 0.08542 × 2.998 × 108 × 2.27 × 10−18
(B32)
2

= 1.16 × 10−10 m/s .
(B22)

(B33)

Observed (SPARC): a0 = (1.20±0.02±0.24)×10−10
m/s2 .

b.
c.

(B30)
−18 −1

Relation II: ka = 3/(8α)

Asymptotic Velocity

The circular velocity is:
c2
cp
cp
Vc2 = r a = r · ψ ′ =
2GM a⋆ r2 /r2 =
2GM a⋆ .
2
2
2
(B23)
Therefore:
GM a⋆ c2
c2
· 2GM a⋆ =
.
(B24)
Vc4 =
4
2
Substituting a⋆ = 2a0 /c2 :
GM · (2a0 /c2 ) · c2
Vc4 =
= GM a0 .
2

(B25)

(B26)

Dimensional check: [GM a0 ] = m3 s−2 · m s−2 =
m s ✓
This is the Baryonic Tully-Fisher Relation with slope
exactly 4 in log-log space.
4 −4

d.

Zero-Point

For V in km/s and M in M⊙ :

1/4
Mbar
Vflat = 112.3 km/s
.
1010 M⊙
4.

α-Relation Derivations

a.

√
Relation I: a0 = 2 α cH0

Relation III: kα = α2 /(2π)

The pure electromagnetic-sector clock coupling is:
α2
.
(B35)
2π
This is the leading same-ion term inside the full channelresolved coupling of Eq. (291); the complete clock phenomenology also includes strong-sector and compositiondependent contributions (Sec. XI).
Numerical value:
5.325 × 10−5
(1/137.036)2
=
= 8.47 × 10−6 .
kα =
2π
6.283
(B36)
α
KA = kα · SA
,

d.

Using G = 6.67430 × 10−11 m3 kg−1 s−2 and the boundmatter a0 = 1.1966 × 10−10 m s−2 (Postulate ZS.11):
G a0 = 7.987 × 10−21 m4 kg−1 s−4 .

c.

(α)

Therefore:
4
Vflat
= GMbar a0

The coefficient ka enters the Ξ functional of App. N.
Numerical value:
3
3 × 137.036
ka =
=
= 51.39.
(B34)
8α
8

(B27)

(B28)

This relation connects the MOND acceleration scale to
fundamental constants and the Hubble rate.

where

kα =

Consistency Check

The three relations are not independent. Combining
Relations I and II:
√
3
3cH0
ka · a0 =
· 2 α cH0 = √ .
(B37)
8α
4 α
This provides an additional consistency check on the
parameter values.

145
5.

b.

Matter-Wave Phase Shift
a.

Phase Evolution

For a circular binary with masses m1 , m2 , separation
a, and orbital frequency ω:

For a matter wave with momentum p and mass m, the
phase accumulated along a path is:
Z
1
(E dt − p · dx) .
(B38)
ϕ=
ℏ
A species-dependent gravitational coupling Katom enters the local energy as:
E = mc2 +

p2
+ mΦeff ,
2m
b.

Binary Inspiral

Φeff = Φ(1 + Katom ). (B39)

Three-Pulse Interferometer

32G4 (m1 m2 )2 (m1 + m2 )
.
5c5
a5
The orbital decay rate:
P =

(B44)

64G3 m1 m2 (m1 + m2 )
.
(B45)
5c5
a3
For PSR B1913+16, the quadrupole formula with its
eccentricity factor predicts Ṗb = −2.403×10−12 , matching
observations at 0.2%.
ȧ = −

Appendix C: Interpolating Function Catalog

In a Mach-Zehnder configuration with pulse separation
T:

This appendix catalogs the interpolating functions µ(x)
used in DFD, their properties, and calibration procedures.

1. First pulse (t = 0): Beam split
1.

2. Second pulse (t = T ): Mirror
3. Third pulse (t = 2T ): Recombine

Any viable interpolating function must satisfy:

The standard gravitational phase is:
∆ϕgrav = keff g T 2 ,

General Requirements

1. Newtonian limit: µ(x) → 1 as x → ∞
(B40)

where keff is the effective wave vector and g is the local
gravitational acceleration.

2. Deep-field limit: µ(x) → x as x → 0
3. Monotonicity: dµ/dx > 0 for all x > 0
4. Smoothness: µ ∈ C ∞ (0, ∞)

c.

Species-Coupling Expression (Not a DFD Prediction)

A species-dependent coupling would add the expression:
2
ℏkeff

g 3
T · Katom .
(B41)
m c2
Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII.
∆ϕDF D =

6.

Gravitational Wave Emission
a.

Quadrupole Formula

The leading radiation comes from the time-varying
quadrupole moment:

Z 
1
Qij = ρ xi xj − δij r2 d3 x.
(B42)
3
The radiated power is:
G D ... ...ij E
P = 5 Q ij Q
(B43)
5c
This matches the GR quadrupole formula exactly, as
required for consistency with binary pulsar observations
at the 0.2% level.

5. Positivity: µ(x) > 0 for all x > 0
6. Strong-field approach rate: 1 − µ(x) must vanish faster than 1/x as x → ∞; a 1/x tail leaves
a constant anomalous acceleration a0 at every
planet. The derived exponential
√ response (App. BS,
Thm. BS.9) has 1 − µ = e− gbar /a0 or e−x on its
two branches (Cor. BS.14)
The argument is the dimensionless ratio:
|∇ψ|
a
x=
= ,
(C1)
a⋆
a0
where a = (c2 /2)|∇ψ| is the gravitational acceleration
and a0 = 1.1966 × 10−10 m/s2 is the characteristic acceleration scale on bound-matter clocks (Postulate ZS.11).
The Lagrangian gradient scale a⋆ = 2a0 /c2 ensures x is
dimensionless.

2.

Catalog of Functional Forms

3.

Simple Interpolating Function

The simple form (in DFD the weak-response approximant of the derived exponential response, App. N,

146
TABLE LVIII. Interpolating functions used in MOND/DFD
literature.
Name
Simple

µ(x)
x
1+x
√x
1+x2
1 − e−x

Standard
Exponential

1 √
ν(y) =
1−e− y
x
(1+xn )1/n
x
;1
1+x/2

RAR (ν form)
n-family
Toy

Trans.
Gradual

Ref.
FM12

Sharp

M83

Gradual

B04

Empirical

M16

Tunable

—

Piecewise

—

b.

Advantages:

• Historical standard
• Sharper transition matches some rotation curves
better
c.

Disadvantages:

• Slightly worse fit to RAR than simple form
• More complex analytically

FM12: Famaey & McGaugh; M83: Milgrom; B04: Bekenstein;
M16: McGaugh et al.

5.

The empirical fit to the SPARC Radial Acceleration
Relation is:

App. BS) is:
µsimple (x) =
a.

x
1+x

(C2)

gbar
√

gobs =
1−e

Properties:

• Asymptotic: µ → 1 − 1/x + O(x

−2

) as x → ∞

• Deep-field: µ → x − x2 + O(x3 ) as x → 0
• Transition width: ∆ log x ≈ 2 (gradual)
p
• Inverse: ν(y) = (1 + 1 + 4/y)/2
b.

RAR Empirical Function

−

(C4)

gbar /a0

This corresponds to an effective ν-function:
gbar
1
√ ,
y=
νRAR (y) =
.
(C5)
a0
1 − e− y
a. Calibration: McGaugh et al. (2016) [10] fit this
form to 2693 data points from 153 SPARC galaxies, obtaining:
2

a0 = (1.20 ± 0.02 ± 0.24) × 10−10 m/s ,

(C6)

where the first uncertainty is statistical and the second
systematic (the 20% normalization uncertainty in Υ⋆ ).

Advantages:

• Analytically tractable
• Smooth transition

6.

The n-Family

• Good fit to RAR data
c.

Disadvantages:

• May overpredict Newtonian deviations in intermediate regime
• Transition slightly too gradual for some galaxies

A one-parameter family interpolating between different
transition sharpnesses:
x
µn (x) =
(C7)
(1 + xn )1/n
• n = 1: Simple function
• n = 2: Standard function

4.

Standard Interpolating Function

• n → ∞: Step function at x = 1
a. Best fit to SPARC: n ≈ 1.0–1.5, favoring gradual
transition.

The standard (original MOND) form is:
µstandard (x) = √
a.

x
1 + x2

(C3)
7.

Properties:

• Asymptotic: µ → 1 − 1/(2x2 ) + O(x−4 ) as x → ∞
• Deep-field: µ → x − x /2 + O(x ) as x → 0
3

5

• Transition width: ∆ log x ≈ 1 (sharper)

1/2
• Inverse: ν(y) = 12 + ( 14 + y −2 )1/2

8.

Comparison of Properties
Calibration Procedure

√
The form of the acceleration scale, a0 = 2 α cH0 , and
the interpolating function are action-derived, not phenomenological inputs (App. AP, Thm AP.25; App. N,
Thm N.14). The procedure below is the empirical validation of that derived form against rotation-curve data, not
a fit that selects it:

147
TABLE LIX. Comparison of interpolating function properties.
Property

Simple Standard RAR

n = 1.5

b. EFT Interpretation. The specific form of µ(x) may
receive quantum corrections at UV scales. The low-energy
effective form is what is calibrated observationally.

√
− x

Newtonian approach 1/x
1/x2
e
1/x1.5
Deep-field approach
x
x
x
x
Transition sharpness Gradual Sharp Gradual Medium
Analytic tractability High
Medium
Low Medium

a.

Step 1: Select Galaxy Sample. Use galaxies with:

• High-quality rotation curves (HI 21cm + Hα)
• Well-determined distances (Cepheids, TRGB)

Appendix D: Experimental Protocols

This appendix specifies technical requirements for the
key experiments that can test DFD predictions. The goal
is to enable independent replication and provide guidance
for experimentalists.

1.

Clock Comparison Procedure

• Resolved stellar and gas mass distributions
a.

• Range of surface brightnesses and masses
b. Step 2: Construct Baryonic Model. For each
galaxy:
2
2
2
2
Vbar
(r) = Vdisk
+ Vbulge
+ Vgas
,

(C8)

using mass-to-light ratio Υ⋆ from stellar population models.
c. Step 3: Fit to Rotation Curve. Minimize:
X [Vobs (ri ) − VDFD (ri ; a0 , Υ⋆ )]2
.
(C9)
χ2 =
σi2
i
d. Step 4: Construct RAR. Plot gobs vs. gbar for all
radii in all galaxies. Fit the ensemble to determine the
universal interpolating function.
e. Step 5: Cross-Validation. Test on held-out galaxies and independent datasets (e.g., dwarf spheroidals,
ellipticals) to verify universality.

9.

Physical Interpretation

The interpolating function µ(x) encodes how gravity
transitions from the Newtonian regime to the deep-field
(MOND) regime. In DFD:

Measurement Overview

The clock anomaly test searches for species-dependent
gravitational coupling by comparing frequency ratios of
different clock types as Earth’s distance to the Sun varies
through the year.
a. Observable:
νA (t) − νB (t)
− ⟨yAB ⟩,
(D1)
yAB (t) =
νA
where A and B are clock types with different αsensitivities.
b. Expected Signal:
∆Φ⊙ (t)
,
(D2)
yAB (t) = (KA − KB )
c2
where ∆Φ⊙ (t) varies annually by 3.3 × 10−10 peak to
peak.

b.

Technical Requirements

c.

Recommended Clock Pairs

1. Primary: Cs hyperfine – Sr optical
• ∆S α = 2.83 − 0.06 = 2.77

• µ(x) arises from the field equation structure, not
fitted by hand

2. Enhanced: Yb+ E3 – Al+

• The transition at a0 reflects fundamental physics (if
α-relations hold)

• Larger signal amplitude

• The gradual transition (favored by data) suggests
continuous crossover rather than phase transition
√
a. Connection to α-Relations. If a0 = 2 α cH0 ,
then:
√
x = 1 ⇔ a = a0 = 2 α cH0 .
(C10)
The crossover scale is set by the geometric mean of electromagnetic (α) and cosmological (H0 ) scales.

• ∆S α = −5.95 − 0.008 = −5.96
• Both optical (reduced systematics)
3. Null control: Sr – Yb (1 S0 –3 P0 )
• ∆S α = 0.06 − 0.31 = −0.25
• Small ∆S serves as null check
d.

Data Analysis

a. Step 1: Time Series Construction. Record frequency ratio νA /νB vs. modified Julian date (MJD).

148
TABLE LX. Clock comparison technical specifications.
Parameter

Requirement

Current State

−16

Fractional stability
σy < 10
@ 1 day
Achieved (Sr, Yb+ )
−17
Systematic uncertainty < 10
Achieved (best optical)
Measurement duration > 1 year (ideally 2–3)
Standard campaigns
Sampling rate
Daily or better
Standard
Clock pair ∆S α
> 2 (maximize signal)
Cs–Sr: ∆S = 2.77
Environmental control mK temperature stability Standard
Vibration isolation
< 10−9 g @ 1 Hz
Standard

b.

Step 2: Template Fitting. Fit to:
Φ⊙ (t)
y(t) = A0 + A1 t + AΦ ·
+ systematics,
c2
where Φ⊙ (t) = −GM⊙ /r⊕ (t).
c. Step 3: Extract ∆K.
KA − KB = AΦ .

c.
(D3)

(D4)

the extended channel’s unscreened scale is 2kα |∆S α |.
(D5)

Systematic Error Budget

TABLE LXI. Systematic error budget for clock comparison.
Effect

1. Perform both analyses and report both results.
2. Quantify the covariance between drift and potential
coefficients in each case.

d. Step 4: Compare to Prediction.
(KA − KB )DF D = 0 (minimal action, Thm. XI.2);

e.

Recommended protocol.

3. A robust signal should appear in both approaches;
discrepancy indicates systematic concerns.
4. Preserve and publish raw ratios to enable independent reanalysis.
The windowed approach is particularly valuable when
exploring marginal hints, as aggressive global detrending
can project out exactly the annual structure one seeks to
test.

2.

Magnitude Mitigation

Blackbody radiation
∼ 10−16
Temperature control
Zeeman shifts
∼ 10−17
Magnetic shielding
Gravitational redshift
∼ 10−16 h−1 Height measurement
Reference cavity drift
∼ 10−17 /day Co-located comparison
Annual temperature cycle Variable
Monitor and correct
Tidal effects
∼ 10−17
Model and subtract

f.

Cavity-Atom Setup Requirements
a.

Experiment Concept

Compare an optical cavity (photon sector) to an atomic
clock (matter sector) while varying gravitational potential.
DFD predicts different responses, with GR predicting
GR
ξLPI
= 0 and corrected DFD predicting only a screened
res
residual ξLPI
once the constitutive-chain cancellation is
imposed.

Windowed vs. Global Analysis Strategies
b.

Two complementary approaches exist for extracting
annual gravitational signals:
a. Global year-long fit. Fit the full multi-year dataset
with a flexible drift model (polynomials, splines) plus the
gravitational template Φ⊙ (t). Advantages: robust statistics, clear identification of sinusoidal annual signal. Risk:
flexible drift models can partially absorb the gravitational
template, especially if the signal is weak.
b. Perihelion-windowed analysis. Analyze a focused
window (30–60 days) around perihelion, where Φ⊙ is
extremal. Use only linear drift within the window. Advantages: sensitive to the shape of the potential variation;
less prone to drift absorption. Risk: shorter baseline
increases degeneracy with instrumental drift.

Φ(h)

Key Configuration

Atomic Clock
Matter reference

νatom /νcavity

Optical Cavity
Photon reference
FIG. 13. Schematic of cavity-atom comparison.

149
3.

TABLE LXII. Cavity-atom test specifications.
Component

Requirement

Notes

a.

ULE or Si spacer
Temperature
stabilized
Atom clock
Sr or Yb optical
< 10−18 systematic
∆Φ/c2 variation
> 10−12
Height change
or orbital
Measurement duration > 104 s per height Statistics
Height separation
> 10 m (terrestrial) Tower or elevator

> 10
< 10−16 @ 1 s

c.

a.

Technical Specifications

The formula under test is:
ℏk 2 g
∆ϕDF D = eff 2 T 3 · Katom .
(D7)
m c
Status: not a DFD prediction (the matter-wave sector is
null); see the status note after Eq. (337) in Sec. XIII.
The formula is not a phase (its units are s2 /m) and sets
no phase-resolution target; the targets of Table LXIII do
not apply to DFD.

Height Comparison Method

b.

Interferometer Requirements

Configuration A: Tower Experiment

• Cavity at ground level

TABLE LXIII. Matter-wave interferometer specifications for
the null test of DFD.

• Atomic ensemble transported to height h

Parameter

• Compare via fiber link

Minimum Target

Free-fall time T
0.5 s
keff
107 m−1
Phase resolution 10−8 rad
Atom number
105
Systematic control 10−9 rad
87
Species
Rb

• ∆Φ/c2 = gh/c2 ≈ 10−14 per 100 m
b.

Formula Under Test (Not a DFD Prediction)

5

Cavity finesse
Cavity stability

d.

Matter-Wave Interferometer Specifications

Configuration B: Space Mission

• Cavity and atoms on same platform

Notes

2s
Limits signal
2 × 107 m−1 Two-photon Raman
10−10 rad
Shot noise limit
107
Statistics
10−10 rad
Gravity gradients
87
Rb, 85 Rb Comparison

• Vary orbital altitude
• ∆Φ/c2 ∼ 10−10 (LEO to higher orbit)

c.

• Enhanced signal but complex mission
e.

Dual-Species Configuration

To extract the species-dependent Katom :
1. Run identical interferometer with 87 Rb and 85 Rb

Observable

2. Both have same mRb to < 2%
3. Different S α values
d
dΦ



νatom
νcavity


=

ξLPI
,
c2

(D6)

GR
where ξLPI
= 0 and the corrected DFD expectation is a
small screened residual rather than an order-unity slope.

f.

Discrimination Significance

With current technology:
• 100 m height: ∆Φ/c2 ≈ 10−14
• Clock comparison at 10−18 : useful only for a
residual-level cavity–atom search
• Discrimination requires pushing into the
screened-residual regime rather than separating ξLPI = 0 from an order-unity value

4. Differential measurement cancels common-mode systematics

d.

T 3 Signature (Not a DFD Prediction)

The formula scales as T 3 , while:
• Standard gravitational phase ∝ T 2
• Gravity gradient phase ∝ T 4
• Rotation phase ∝ T 2
Through 1PN, DFD and GR carry the same phases
(§XIII), so this scaling does not discriminate between
them.

150
d.

TABLE LXIV. Matter-wave systematic errors.
Effect

Scaling

Mitigation

Quality Metrics

• χ2 /dof < 2 (good fit)

4

Gravity gradient
T
Gradient compensation
Coriolis force
T2
Rotation compensation
Laser wavefront
T2
High-quality optics
AC Stark shift
Independent Laser intensity control
Magnetic fields
T2
Magnetic shielding
Two-photon light shift T 2
Symmetric pulse

e.
4.

Systematic Control

• Υ⋆ consistent with stellar population models
• a0 consistent across galaxy sample
5.

Galaxy Rotation Curve Analysis
a.

• Residuals randomly distributed (no systematic
trends)

Data Requirements

• Rotation curve: HI 21cm and/or Hα emission
• Resolution: Beam size < 1 kpc at galaxy distance

Decision Matrix: Which Experiment to Prioritize

a. Recommendation: The near-term emphasis is on
nuclear clocks and cross-species clock analyses. Cavity–
atom work remains valuable as a long-horizon residual
test rather than the first binary discriminator. Matterwave interferometry is a null check (DFD and GR coincide
through 1PN).

• Velocity precision: < 5 km/s per point

Appendix E: Data Tables

• Radial extent: Out to ≳ 3 disk scale lengths
• Inclination: 30◦ < i < 80◦ (avoid edge-on/faceon)

This appendix collects numerical data used in the review
for reference and reproducibility.

1.
b.

Post-Newtonian Parameter Bounds

Baryonic Mass Model
2.

Binary Pulsar Timing Data

1. Stellar mass: From 3.6 µm photometry
Σ⋆ (r) = Υ⋆ · I3.6 (r)

(D8)

with Υ⋆ ≈ 0.5 M⊙ /L⊙ (disk)
2. Gas mass: From HI 21cm + correction for He
Σgas = 1.33 · ΣHI

(D9)

2
2
Vbar
(r) = V⋆2 (r) + Vgas
(r)

(D10)

3. Total:

c.

a.

Notes:

• Ṗbobs corrected for Shklovskii effect and Galactic
acceleration
• GR prediction uses measured masses from other
post-Keplerian parameters
• DFD predicts identical Ṗb to GR (same quadrupole
formula)

DFD Fitting Procedure
3.

2
Compute gbar (r) = Vbar
(r)/r

a. Step 1:
b. Step 2: Apply interpolating function:


gbar (r)
gobs (r) = gbar (r) · ν
a0
c. Step 3: Convert to velocity:
p
VDF D (r) = r · gobs (r)
d.

Minimize χ2 :
X [Vobs (ri ) − VDF D (ri )]2
χ2 =
σi2
i

a.
(D11)
b.
(D12)

Sensitivity Definition:
∂ ln νA
α ∂νA
α
SA
≡
=
.
∂ ln α
νA ∂α
Optimal Pairs for DFD Test:

(E1)

1. Cs – Al+ : ∆S = 2.82 (large baseline)
2. Yb+ E3 – Al+ : ∆S = −5.96 (largest, opposite
signs)

Step 4:

with free parameters: a0 (or fixed), Υ⋆ , distance.

Clock Sensitivity Coefficients

(D13)
3. Cs – Sr: ∆S = 2.77 (readily available)

151
TABLE LXV. Experimental decision matrix for DFD tests.
Experiment

Signal

Timescale

Cost

Discriminating Priority

−18

Clock anomaly (extended channel; null under the minimal action)
≲ 1 × 10
1–2 yr
Low
Cavity-atom residual
screened residual Long-term Medium
Matter-wave T 3 (null prediction)
null (= GR)
3–5 yr
Medium
Galaxy RAR
< 0.15 dex
Done
Low
GW ppE
δ φ̂ = 0
Done
N/A

Yes
Yes
No (null)
No
No (confirms)

High
Medium
Medium
Complete
Complete

TABLE LXVI. Experimental bounds on PPN parameters. DFD predicts GR values.
Parameter GR/DFD Bound
γ−1
β−1
|α1 |
|α2 |
|α3 |
|ξ|
|ζ1 |
|ζ2 |
|ζ3 |
|ζ4 |

0
0
0
0
0
0
0
0
0
0

SPARC Galaxy Sample Statistics

5.

Gravitational Wave Constraints

The GWTC-3 ppE bounds are those of Table IX; DFD
predicts δ φ̂k = 0.
a. Speed of Gravity: GW170817/GRB 170817A constraint [126]:
cT − c
−3 × 10−15 <
< +7 × 10−16 .
(E2)
c
DFD prediction: cT = cγ exactly: light and gravitational
waves share one metric (App. WS).

Physical Constants Summary

7.
8.

a.

DFD Parameter Summary
Experimental Timeline

Falsification Threshold:

• Cavity–atom residual: a dedicated null at the
screened-residual target would constrain or remove
that channel
• Matter-wave: not a falsification channel (null prediction, §XIII)
• RAR: Scatter > 0.3 dex would falsify

Reference

(2.1 ± 2.3) × 10
Cassini
[35]
(−4.5 ± 5.6) × 10−5 LLR
[121]
< 4 × 10−5
Pulsar timing
[37]
< 2 × 10−9
Pulsar spins
[5]
< 4 × 10−20
Pulsar accel.
[122]
< 10−3
Binary pulsars
[123]
< 2 × 10−2
Lunar orbit
[36]
< 4 × 10−5
Binary pulsars
[5]
< 10−8
Newton’s 3rd law [124]
—
Not independent —

4.

6.

Method
−5

Appendix F: Rigorous Foundations for Gauge
Emergence

This appendix presents mathematically rigorous derivations supporting the gauge emergence mechanism described in §XVII. Sections F 1–F 6 contain complete proofs;
Section F 8 presents a physically motivated conjecture.

1.

Minimality of the (3, 2, 1) Partition

Proposition F.1 (Minimality). Among all block partitions (n1 , . . . , nk ) of CN whose U (N )-stabilizer contains
exactly two simple non-Abelian factors SU (3) and SU (2)
and a singlet sector, the unique minimal partition is
(3, 2, 1) with N = 6. Its stabilizer U (3) × U (2) × U (1) has
two abelian factors modulo the overall phase; the single
hypercharge is fixed by the isotropy reading of Remark F.2.
Proof.
For aQ partition (n1 , . . . , nk ), the stabilizer is
Q
U
(n
i) =
i
i [SU (ni ) × U (1)] modulo diagonal U (1).
Necessity of three blocks: A two-block partition (na , nb )
gives stabilizer SU (na ) × SU (nb ) × U (1). This has no
singlet sector: every vector transforms non-trivially under
at least one SU factor. Hence k ≥ 3.
Necessity of block sizes 3, 2, and 1: Two blocks must
have dimensions 3 and 2 to yield SU (3) × SU (2). The
third block provides the singlet sector; minimality requires
n1 = 1.
Minimality of N = 6: Any partition with k ≥ 3 blocks
including sizes 3 and 2 has N ≥ 3 + 2 + 1 = 6. The
partition (3, 2, 1) achieves this bound.
Uniqueness: The only partition of 6 with blocks of sizes
3, 2, and 1 is (3, 2, 1) itself.

152
TABLE LXVII. Binary pulsar systems used for gravitational tests.
Pb [hr] Ṗbobs

System

PSR B1913+16 7.752
PSR J0737-3039 2.454
PSR J1738+0333 8.515
PSR J0348+0432 2.460
PSR J1141-6545 4.744

−12

Type S α

kα S α (ext., unscreened) Ref.

Microwave (hyperfine)
133
Cs 6S1/2 F=3→4 HFS
87
Rb
5S1/2 F=1→2 HFS
1
H
1S1/2 F=0→1 HFS

+2.83 2.4 × 10−5
+2.34 2.0 × 10−5
+2.00 1.7 × 10−5

[70]
[70]
[70]

Optical
87
1
Sr
S0 → 3 P0
E1
171
Yb 1 S0 → 3 P0
E1
27
+
1
3
Al
S0 → P0
E1
171
Yb+ 2 S1/2 → 2 D3/2 E2
171
Yb+ 2 S1/2 → 2 F7/2 E3
199
Hg+ 2 S1/2 → 2 D5/2 E2

+0.06 5.1 × 10−7
+0.31 2.6 × 10−6
+0.008 6.8 × 10−8
+0.88 7.5 × 10−6
−5.95 −5.1 × 10−5
−3.19 −2.7 × 10−5

[118]
[118]
[118]
[118]
[118]
[118]

Nuclear (proposed)
229
Th Nuclear isomer M1/E2 ∼ 104

∼ 0.1

[125]

TABLE LXIX. SPARC sample properties (Lelli et al. 2016).
Property

Agreement
−12

−2.398 × 10
−2.403 × 10
0.2%
−1.2478 × 10−12 −1.2478 × 10−12 0.004%
−2.59 × 10−14
−2.77 × 10−14
6.5%
−13
−2.73 × 10
−2.58 × 10−13
6%
−4.03 × 10−13
−3.86 × 10−13
4%

TABLE LXVIII. Sensitivity coefficients for atomic transitions.
(α)
α
The pure-α leading term is KA = kα ·SA
with kα = 8.5×10−6 ;
the full channel-resolved coupling includes additional strongsector and composition terms (Eq. (291)).
Atom Transition

ṖbGR

Value

Number of galaxies
175
Number of RAR data points 2693
Distance range
2 – 150 Mpc
Luminosity range
107 – 1011 L⊙
Vflat range
20 – 300 km/s
Morphological types
Sa – Irr
RAR fit results
a0 (best fit)
Observed scatter

(1.20 ± 0.02 ± 0.24) × 10−10 m/s2
0.13 ± 0.02 dex

BTFR results
Slope
Intrinsic scatter

3.98 ± 0.08
0.11 ± 0.02 dex

Why N > 6 is excluded: Any partition with N > 6
either has larger block sizes (giving wrong gauge groups)
or additional blocks (giving more than two non-Abelian
factors). Since we seek the minimal N , enumeration
beyond N = 6 is unnecessary.
For completeness, we verify that no partition with
N ≤ 6 other than (3, 2, 1) satisfies all requirements:

N Partition SU factors
5 (3, 2)
5 (2, 2, 1)
6 (4, 2)
6 (3, 3)
6 (3, 2, 1)
6 (2, 2, 2)

Singlet? Status

SU (3) × SU (2) No
SU (2) × SU (2) Yes
SU (4) × SU (2) No
SU (3) × SU (3) No
SU(3) × SU(2) Yes
SU (2)3
No

✗
✗
✗
✗
✓
✗

Remark F.2 (The electroweak group as the isotropy group
of CP 2 ). The minimality proposition assumes the factors it selects. The electroweak group follows from the
internal space instead, under one stated identification.
The charge-trace step of the closed form for α (§X) needs
the hypercharge to act on CP 2 as a traceless element of
su(3), and it fixes only that: every traceless charge gives
the same (k + 3)/k, and a generic one is stabilised by
a torus, not by a U (2). The stated identification takes
the hypercharge to be the direction diag(1, 1, −2), up to
conjugation and scale the only one that commutes with
an SU (2) inside SU (3), normalised so that the isotropy
charge is 6Y , and it takes the electroweak group to be
the structure group of the isotropy bundles at the vacuum point p of CP 2 , acting on the fibres of the twisting
bundles that carry the matter fields, not on the internal
spinor and not by isometries of CP 2 (an isometric action
would rotate two of the generation profiles of App. H into
each other). The charge trace acts instead on the Toeplitz
spaces H 0 (CP 2 , O(k)), through the isometric lift of the
same generator, which equals the fibre charge only over p;
that the two are one hypercharge is part of the identification. Every point of CP 2 is equivalent under SU (3), so
the choice of p adds no content, and diag(1, 1, −2) is the
centre of the isotropy group U (2) of CP 2 = SU (3)/U (2)
at p. Under this charge a homogeneous line bundle O(m)
carries 2m and an irreducible rank-two homogeneous bundle carries an odd charge. With the charge equal to 6Y ,
the multiplets with odd 6Y , which are exactly the weak
doublets, must be rank-two isotropy bundles, so the weak
SU (2) is the isotropy SU (2) and the electroweak group
is U (2) = (SU (2) × U (1))/Z2 , the global form that every
Standard Model field obeys. That is the Z2 part of the
Standard Model’s Z6 , a check the identification passes; the
Z3 part comes with colour. In this reading the (2, 1) blocks
of the partition are the isotropy split C3 = ℓ⊥
p ⊕ ℓp (ℓp is
the line p), whose stabiliser S(U (2) × U (1)) = U (2) has
one abelian factor; the tangent space Tp = Hom(ℓp , ℓ⊥
p)
carries isotropy charge 3. The reading is not an elec-

153
TABLE LXX. Physical constants used in calculations (CODATA 2018 unless noted).
Constant

Symbol Value

Speed of light
c
Gravitational constant G
Planck constant
h
Reduced Planck
ℏ
Fine-structure
α
Electron mass
me
Proton mass
mp
Solar mass
M⊙
Astronomical unit
AU

Uncertainty

299792458 m/s
exact
6.67430 × 10−11 m3 kg−1 s−2
1.5 × 10−5
−34
6.62607015 × 10
Js
exact
1.054571817 × 10−34 J s
exact
7.2973525643 × 10−3 (CODATA 2022) 1.6 × 10−10
9.1093837015 × 10−31 kg
3.0 × 10−10
−27
1.67262192369 × 10
kg
3.1 × 10−10
1.98841 × 1030 kg
4 × 10−5
1.495978707 × 1011 m
exact

TABLE LXXI. Summary of DFD parameters and their values.
Parameter

Symbol Value

Source

Independent observational comparator
SPARC acceleration scale aSPARC
(1.20 ± 0.02stat ± 0.24sys ) × 10−10 m/s2 SPARC RAR
0
From α-relations (parameter-free)
Self-coupling
ka
Clock coupling
kα
Hubble relation
—

51.4
3/(8α)
8.5 × 10−6
α2 /(2π)
√
a0 = 2 α cH0loc = 1.1966 × 10−10 m/s2 Derived law; bound-matter clock

From theory structure
GW speed
PPN γ
PPN β
LPI residual

c, the optical cone (App. WS)
1 exactly
1 exactly
screened residual

cT
γ
β
res
ξLPI

troweak unification in the isometry group SU (3), which
would give sin2 θW = 3/4: the couplings come from the
frame stiffnesses (Theorem F.25).
Remark F.3 (Colour and the family content are inputs).
Colour cannot be a symmetry of CP 2 × S 3 when the
hypercharge acts on CP 2 as the charge trace needs: the
symmetries commuting with any nonzero element of the
su(3) of CP 2 lie, up to conjugation, in U (2) × SO(4),
whose simple factors are copies of SU (2), so they contain
no SU (3); those commuting with the electroweak U (2)
form U (1) × SO(4). Taking colour as the isometry group
of CP 2 instead would make the hypercharge central there,
which removes the +3 shift of the charge trace and lowers
α−1 to 136.9 (130.5 if only the numerator k + 3 is dropped
and d = k + 4 kept). From the integers alone, 136.9
lies 0.08% (800 ppm) below the measured value, against
21 ppm for the adopted reading. The exclusion rests on
the charge trace being the origin of the shift, which is how
§X D computes it; read as the degree of the determinant
line, which Rem. X.3 keeps as a consistency check, the
shift would survive for a central charge and give 137.033.
Colour, SU (3) acting on a separate three-dimensional
block, is therefore an input, as it is in the Standard Model.
The representations of one generation and their chirality
are inputs as well. Symmetries of a compact internal
space do not by themselves give chiral fermions [127], and
in this reading the flux twist on CP 2 does not supply the
chirality of a generation either: the spinc indices of the
five left-handed multiplets Q, uc , dc , L and ec are 3, 0, 3,

Optical metric
Conformal structure
Field equation
Constitutive-chain cancellation + channel dependence

−1 and 10 (computed).

2.

The SU (N ) Selection Lemma

Lemma
F.4
(Dimension-Casimir Coincidence).
Among compact simple Lie groups, the condition
dim(fundamental rep) = h∨ (dual Coxeter number) holds
if and only if G ∼
= SU (N ) for some N ≥ 2.
Proof. Direct verification from the classification of simple
Lie algebras [128, 129]:
Cartan

Group

h∨

An−1
SU (n)
n
Bn SO(2n + 1) 2n − 1
Cn
Sp(2n)
n+1
Dn
SO(2n) 2n − 2
G2
G2
4
F4
F4
9
E6
E6
12
E7
E7
18
E8
E8
30

dim(fund) Match?
n
2n + 1
2n
2n
7
26
27
56
248

✓
✗
✗
✗
✗
✗
✗
✗
✗

The exceptional isomorphisms Sp(2) ∼
= SU (2) and
SO(6) ∼
= SU (4) reduce to the An case.

154
TABLE LXXII. Projected timeline for DFD experimental tests.
Test
Near-term (1–3 yr)
Clock (Cs/Sr)
Multi-clock
Medium-term (3–7 yr)
Cavity–atom
Matter-wave T 3 (null prediction)
Nuclear clock
Long-term (>7 yr)
Space optical
Space atom int. (null prediction)

Time

Sens.

Status

2025–26
2025–26

null (minimal action)
K ∼ 10−6

Underway
In progress

2030+
2027–30
2028–32

Residual level
10−10 rad
K ∼ 10−3

Long-horizon
Devel.
R&D

2030+
2032+

K ∼ 10−7
10−11 rad

Concept
Concept

Remark F.5. This lemma concerns only the fundamental
representation. SM fermions transform in fundamentals
of SU (3) and SU (2), so higher representations need not
be considered.

3.

The Spinc Flux Quantization

a. Setup. CP 2 is a compact complex surface with
H 2 (CP 2 ; Z)
R = Z · H where H is the hyperplane class
satisfying CP 2 H 2 = 1. Since w2 (T CP 2 ) = c1 mod 2 =
3H mod 2 = H ̸= 0, CP 2 does not admit a spin structure
but does admit a spinc structure with determinant line
bundle Ldet = K −1 = O(3) and c1 (Ldet ) = 3H [130, 131].
Definition F.6 (Hypercharge Twist). Let L be a line
bundle on CP 2 with c1 (L) = H, and let q1 ∈ Z>0 be the
U(1) flux quantum. For a representation of hypercharge Y ,
1/2
the spinc -twisted hypercharge line is Lq1 Y ⊗ Ldet , whose
total degree q1 Y + 3/2 is the object required to be integral
or half-integral (Lemma F.8); the bare factor Lq1 Y is a
formal half-integral symbol and is not individually a line
bundle for Y ∈ {1/6, −1/2} at q1 = 3 (there is no O(1/2)
on CP 2 ).
Remark F.7 (Open collision between the hypercharge dictionary and the kmax bundle — stated, not resolved).
The microsector bundle E = O(9) ⊕ O⊕5 of Lemma F.11
assigns the five matter types degree-0 summands, whereas
the hypercharge dictionary above assigns them degrees
q1 Y ∈ {1/2, 2, −1, −3/2, −3} at q1 = 3 — two of which
are non-integral, so those lines cannot individually exist as
bundles. The two are read off different bundles: the kmax
count uses E as a rank-6 twist bundle whose summands
carry SM labels, not as the direct sum of the individual
hypercharge lines. Reconciling the two readings is an
open item; kmax = 60 is quoted from the E-reading
throughout. (Under the alternative reading in which the
five O(0) summands are genuine trivial bundles, degree
= q1 Y = 0 would force Y = 0 for all five — contradicting
their definition as exactly the hypercharged types.) The
two non-integral degrees belong to the weak doublets Q
and L, which are rank-two bundles of odd first Chern class

rather than lines (Remark F.10), so they exist as bundles; reconciling the degree-zero summands of E with the
hypercharge degrees stays open. In the isotropy reading
(Remark F.2) the five types, written as the left-handed
multiplets Q, uc , dc , L and ec of Lemma F.11, are homogeneous bundles of slopes 12 , −2, 1, − 23 and 3 with spinc
indices 3, 0, 3, −1 and 10, not five summands of index 1.
Lemma F.8 (Integrality Condition). For the spinc Dirac
index to be well-defined for all SM hypercharges Y ∈
{1/6, 2/3, −1/3, −1/2, −1, 0}, the combination q1 Y + 3/2
must lie in 21 Z for all Y .
Lemma F.9 (q1 = 3 is Uniquely Minimal). The unique
minimal positive integer q1 satisfying Lemma F.8 is q1 =
3.
Proof. Direct computation:
TABLE LXXIII. Charge combinations for various hypercharge
assignments.
q1 Y = 1/6 Y = 2/3 Y = −1/3 Y = −1/2 Y = −1 All ∈ 12 Z?
1
2
3
4
5
6

5/3
11/6
2
13/6
7/3
5/2

13/6
17/6
7/2
25/6
29/6
11/2

7/6
5/6
1/2
1/6
−1/6
−1/2

1
1/2
0
−1/2
−1
−3/2

1/2
−1/2
−3/2
−5/2
−7/2
−9/2

✗
✗
✓
✗
✗
✓

Only q1 = 3 and q1 = 6 satisfy the condition; q1 = 3 is
minimal.
Remark F.10 (The flux quantum q1 = 3 from the isotropy
reading). The condition of Lemma F.8 is necessary, not
sufficient: it admits both half-odd and integer values of
q1 Y +3/2, and a spinc twist of a line bundle on CP 2 needs
the half-odd coset. The flux quantum follows from the
isotropy reading of Remark F.2, given its identification.
With the hypercharge the isotropy charge 6Y , a multiplet
of hypercharge Y lives in a homogeneous bundle of slope
3Y , so q1 = 3. The allowed values are q1 = 3λ with λ odd;
q1 = 6 is excluded, since it would give the weak doublets
an even charge. Only λ = ±1 lets the isotropy group act
faithfully on the Standard Model fields (for odd λ ≥ 3 a Zλ

155
of its centre acts trivially), so the hypercharge circle is the
isotropy circle itself and q1 = 3. At q1 = 3 every Standard
Model multiplet is a well-defined field on the non-spin
CP 2 : the singlets as spinc twists and the doublets as
rank-two bundles of odd first Chern class. A uniform
spinc structure on line bundles alone would need q1 = 6;
that is the value when the weak doublets are line bundles
and the weak SU (2) does not act on CP 2 , while in the
isotropy reading the doublets are rank-two bundles. The
spin-Z4 B−L structure of Sec. F 7 is a second consistent
completion. Within the isotropy reading the derivation
does not use α. The same q1 = 3 is the choice for which the
determinant-line chain of Lemma F.11 lands on kmax = 60
and hence on α−1 = 137.036 to 21 ppm from the integers
(the last 21 ppm rests on the open bracket coefficient;
Rem. F.12). Between readings α does the selecting: with
the degree a = Ngen q1 that Lemma F.11 constructs, the
reading with line-bundle doublets, q1 = 6, would give
kmax = 195 and α−1 ≃ 450, and q1 = 1 would give
kmax = 15; with the degree read instead as Ngen c1 (CP 2 ),
which equals it only at q1 = 3, where LY = O(3) = K −1
(§X), α would not test q1 .
Lemma F.11 (Conditional Finite-Microsector Count (Determinant-Line Lift)). Let X = CP 2 with canonical spinc
structure Ldet = K −1 = O(3), and let L = O(1) with
c1 (L) = H. Take the flux quantum q1 = 3 (Rem. F.10), so
the minimal hypercharge line bundle is LY := Lq1 = O(3).
Then the minimal globally well-defined integer-charge lift
is the triple tensor power
L⊗3
Y = O(9).
Here the cube arises ( α-free) as the degree of the determinant line over the three-generation kernel G, where
Ngen = 3 enters as a discrete input (the observed family number, numerically coincident with χtop (CP 2 ) = 3;
see Rem. F.21): det(G ⊗ LY ) = L⊗3
= O(9), so a =
Y
Ngen q1 = 9 (see proof ). The microsector twist bundle
is E(a, n) := O(a) ⊕ O⊕n with cutoff the closed
spinc


a+2
index kmax := χ(X, E) = χ(O(a)) + n = 2 + n. With
a = 9 forced as above and n the SM matter-type count, the
unique solution is (a, n) = (9, 5) (the rival (8, 15) is excluded structurally: 8 ̸= Ngen q1 is not a determinant-line
degree, and n = 15 = 5 × 3 re-counts the three generations
already consumed inside O(9)):
E = O(9) ⊕ O⊕5 ,

χ(E) = χ(O(9)) + 5 = 55 + 5 = 60.

Interpreting n = 5 as the five hypercharged chiral matter
multiplet types per generation {Q, uc , dc , L, ec } fixes the
decomposition.
Proof. With a = 9 fixed by the determinant-line
lift


(Lemma statement), χ(O(9)) = 11
=
55,
and
the
five
2
SM matter-type lines give n = 5, so χ(E) = 55 + 5 = 60.
The decomposition (a, n) = (9, 5) is forced (not merely
minimal-padding): a = Ngen q1 = 9 is the determinantline degree, and the determinant functor sends O(3)⊕3
(χ = 30) to the tensor-cube line O(9) (χ = 55), so the
direct sum is not a rival. The rival (8, 15) is excluded
because 8 ̸= Ngen q1 (no integer rank k gives k · 3 = 8)

and 15 = 5 × 3 double-counts the protected generations.
The physical interpretation of the two integers:
• a = 9: Conditional on the fixed-kernel reading of
the generation space (G = ker DF of the single
rigid vacuum operator, a fixed coefficient space;
Remark F.12), the charged sector is det(G ⊗ LY ) =
dim G
det(G) ⊗ L⊗
= O(3)⊗3 = O(9). Note this
Y
determinant is the algebraic (Knudsen–Mumford)
functor — an integer-degree line carrying no measure, metric, or Grassmann structure — so no “measure identification” premise enters at this step (cf.
the two-layer resolution in Remark F.12). Here
det(G) ≃ O holds automatically, with no curvature premise: G is the kernel of a single fixed elliptic operator (the vacuum is rigid — the moduli
space is a point), hence a fixed 3-dimensional coefficient space C3 rather
than a bundle over CP 2 ,


3
and det C ⊗ O(3) = (Λ3 C3 ) ⊗ O(3)⊗3 = O(9)
by one line of exterior algebra (Λ3 C3 is a constant
line). Thus a = Ngen q1 = 3 · 3 = 9, α-free given
the two discrete inputs. Spinc integrality already
makes LY = O(3) globally well-defined; the genuine fractional structure of 3Y is Z2 (which would
give a square O(6)), and the triple comes from
rank(G) = Ngen = 3 via the determinant functor.
• n = 5: The number of distinct hypercharged chiral
multiplet types per generation in the minimal Standard Model: {Q, uc , dc , L, ec }. (The right-handed
neutrino νR has Y = 0 and does not contribute to
the hypercharge-twist sector.)

Remark F.12 (Independence of the Derivation Chain).
The logical structure of the derivation is:
SM → q1 = 3 → a = 9 → kmax = 60 → α−1 = 137.036
(F1)
Within the isotropy reading, α appears only at the end of
this chain as an output, not as an input. The chain begins
with Standard Model hypercharge assignments (which
are fixed by experiment independently of α), proceeds
through the electroweak identification and the conditional
determinant-line count (Rem. F.10, Lemma F.11), and
only produces the closed-form value of α via Chern-Simons
quantization at kmax = 60.
Inside the reading, then, the chain runs SM → topology
→ α without using α, given the construction of the conditional count, including its rule that n counts the five hypercharged types (Rem. F.7). The choice between readings
is where α enters: with the degree a = Ngen q1 it excludes
the reading with line-bundle doublets (Rem. F.10).
b. Status: derivation plus match. The above is an
α-free chain whose logical form is: kmax = 60 follows
from one integer derived under the electroweak
identification and two inputs, and the closed form
matches the measured fine-structure constant to
21 ppm from the integers alone; the last 21 ppm

156
rests on the open bracket coefficient. The three
entries are: q1 = 3 (derived from the isotropy charge
6Y under that identification, Rem. F.10), Ngen = 3 (input, the observed family number, Rem. F.21), and the
SM type count n = 5 (input, the hypercharged multiplet types). Given these, the determinant-line arithmetic
a = Ngen q1 = 9, χ(O(9)) + 5 = 60 is exact and rivalfree within the conditional count of Lemma F.11 ((8, 15)
is construction-incoherent; the alternatives q1 ∈ {1, 6},
which give kmax ∈ {15, 195}, are excluded inside the
isotropy reading by the isotropy charge and between readings by the measured α; the relation of the degree-zero
summands to the hypercharge degrees is open, Rem. F.7).
This is not a bare-CP 2 topology theorem, and it does not
by itself predict the value of α (which additionally requires
the coupling-origin theorem). The question — is the relevant microsector measure the fermionic Quillen determi′
nant line, or the bosonic Gaussian determinant det{} (K)
of App. O? — conflates two same-named objects. The
degree a = 9 is the algebraic Knudsen–Mumford determinant functor, det(G ⊗ LY ) = Λ3 C3 ⊗O(3)⊗3 = O(9) — an
integer degree carrying no measure, metric, or Grassmann
structure. The suppression α57 is the commuting complex′
Gaussian ratio Z(α−1 )/Z(1) = α57 with det{} (K) in
the denominator, eigenvalue-cancelling (App. O). The
fermionic (Berezin/Quillen norm-square) reading — determinant in the numerator, giving α−57 — is excluded, not
merely disfavored: the fundamental fields of the master
action are bosonic, Berezin–Toeplitz quantization of the
even Kähler microsector yields CCR (not CAR) statistics, the 57 mode amplitudes are x-independent finite
variables whereas every anticommuting variable of ZDFD
carries an L2 (M, SM ) leg (type mismatch), and the sign
of the observed hierarchy falsifies α−57 . The two layers are compatible and distinct: the algebraic line fixes
the count, the bosonic Gaussian fixes the suppression —
with the five-way factorization of ZDFD a derived conditional lemma (see the QG partition-function theorem in
the Extended Derivations, App. AH), unconditional in
its Grassmann-assignment and ϕi -absence legs. Of the
three entries, q1 is derived given the electroweak identification; Ngen and the type count are inputs shared with
the Standard Model, and no index on the internal space
supplies Ngen (Rem. F.21). The status is: kmax = 60 is
derived from these entries within the conditional count of
Lemma F.11. Within the isotropy reading the chain does
not use α, and the empirical match, 21 ppm from the
integers, is what excludes the reading with line-bundle
doublets (Rem. F.10). Closure of the reading question:
the rival evaluation-map reading is closed exhaustively
— for every embedding of ker DF = span{|0⟩, |20⟩, |40⟩}
(App. AH, microsector basis) into H 0 (CP 2 , O(9) ⊕ O⊕5 ),
the evaluation map either has constant rank 3, forcing a
trivial O⊕3 subbundle and the determinant answer a = 9
identically, or drops rank on a nonempty degree-9 curve
so that no rank-3 subbundle exists (dichotomy lemma,
Extended Derivations App. AH); hence a = 9 in every
well-defined reading, and the fixed-kernel conditionality

reduces to its definitional content. The saturation variant
of the rank-2 case reproduces the α-excluded kmax = 195
alternative.

4.

The Spinc Dirac Index on CP 2

a. Index formula. For a spinc 4-manifold M with
determinant line bundle Ldet , twisted by a vector bundle
V [130]:
Z
index(DV ) =
ch(V ) · ec1 (Ldet )/2 · Â(M ).
(F2)
M

b.

Characteristic data for CP 2 .

• c1 (T CP 2 ) = 3H, c2 (T CP 2 ) = 3H 2
• Pontryagin class: p1 = c21 − 2c2 = 3H 2
• Â-genus: Â(CP 2 ) = 1 − p1 /24 = 1 − H 2 /8
• Spinc exponential: e3H/2 = 1 + 3H/2 + 9H 2 /8
c. Index for the SU (3) instanton bundle. Let E3 be
an SU (3) instanton bundle with rank 3, c1 (E3 ) = 0, and
c2 (E3 ) = k3 H 2 . Then:
ch(E3 ) = 3 − k3 H 2 .

(F3)

Computing the index:
Z
9H 2
H2
index(DE3 ) =
(3 − k3 H 2 )(1 + 3H
2 + 8 )(1 − 8 )
CP 2

 27−8k
3
=
− 38 = 3 − k3 .
8
(F4)
For k3 = 1: index = 2 (integer, as required).
5.

Generation Count and Flux-Product Rule

Theorem F.13 (Künneth Factorization [102]). For a
product manifold M1 × M2 with product bundle E = E1 ⊠
E2 :
M1 ×M2
index(DE
) = χ(M1 ; E1 ) · χ(M2 ; E2 ).

(F5)

Theorem F.14 (Spectral Flow on S 3 from Winding Number [103]). For the Dirac operator on S 3 coupled along a
gauge path of SU (2) winding number k2 ∈ π3 (SU (2)) = Z,
the spectral flow equals the winding:


IS 3 (k2 ) = sf DS 3 ; gk2 = k2 .
(F6)
Remark F.15 (Correct index-theoretic home of IS 3 (k2 )).
Since S 3 is odd-dimensional, the static elliptic index vanishes identically and the round S 3 Dirac operator has no
harmonic spinors (Lichnerowicz gap |λ| ≥ 3/2R); IS 3 (k2 )
is therefore not a kernel dimension. Its rigorous content is the Atiyah–Patodi–Singer spectral flow along the
degree-k2 gaugeRpath (equal to the index on the mapping
torus S 3 × S 1 , ch2 = k2 ), equivalently the contact/CR
Toeplitz index of the winding-k2 symbol on the Hardy
space of S 3 (Venugopalkrishna 1972; Boutet de Monvel

157
1979). Both formulations give exactly k2 ; this theorem is
used below only in that spectral-flow sense.
Remark F.16 (Quantum Level Shift). The factor (k + 2)
appearing in the SU(2) Chern–Simons weight function
2
π
w(k) = k+2
sin2 k+2
arises from the quantum (one-loop)
∨
level shift k → k + h where h∨ = 2 is the dual Coxeter
number for SU(2). This is a standard result in WZW/CS
theory [132].
Definition F.17 (Generation Count). Let RSM =
{QL , uR , dR , LL , eR } be the chiral SM representations.
The generation count is:
Ngen := gcd{|index(DR )| : R ∈ RSM }.

(F7)

Theorem F.18 (Flux-product rule, a bookkeeping identity). For M = CP 2 × S 3 with flux configuration
(k3 , k2 , q1 ):
Ngen = |k3 · k2 · q1 |.

(F8)

Proof. By Künneth factorization, the index factors over
the product. The S 3 factor contributes k2 (Dirac index
from winding number). On CP 2 , the index for a representation with SU (3) dimension d3 and hypercharge Y
has the polynomial form:
ICP 2 (d3 , k3 , Y ) = d3 · [A(k3 ) + B(k3 ) · q1 Y + C · (q1 Y )2 ].
(F9)
The weighted hypercharge sum over one SM family
vanishes (gravitational-U (1)Y anomaly cancellation):
X
d3 (R) · d2 (R) · Y (R) = 1 + 2 − 1 − 1 − 1 = 0. (F10)
R

This ensures consistent topological structure.
The rule keeps only the mixed term of the index polynomial, so it is bookkeeping within the stated construction:
it reproduces the input Ngen = 3 and does not derive it
(Rem. F.21).

Remark F.21 (Status: Ngen = 3 is an input, not a derivation). The flux-product rule above is a bookkeeping identity within the stated construction, not an Atiyah–Singer
index theorem: the rule retains only the mixed d3 d2 Y
term of the index polynomial, the S 3 factor enters as spectral flow (Rem. F.15), not a kernel, and the per-multiplet
spinc indices on CP 2 at flux (1, 1, 3) do not share a common factor of 3. An exhaustive search over all currently
known index-theoretic routes — the plain elliptic index
on K = CP 2 × S 3 (identically zero in odd dimension),
the static kernel (zero, by Lichnerowicz on S 3 ), Berezin–
Toeplitz/spectral-flow assemblies (which yield 10 or 60,
never 3), the Z6 center-flux completion of GSM (nonexistent on CP 2 : every SM multiplet is Z6 -neutral), and the
spin-Z4 B−L structure of Sec. F 7 (per-multiplet indices
{0, 0, 0, 1, 1, 1}, provably non-uniform for every flux) —
fails to produce Ngen = 3 from the geometry. Accordingly,
Ngen = 3 enters DFD as a discrete input fixed by observation, in the same epistemic class as the SM representation content itself — as it does in every extant theory. The
numerical coincidence Ngen = χtop (CP 2 ) = 3 is an identification, not an index theorem (the untwisted spinc index
on CP 2 is Td = 1), and the selections k3 = k2 = 1 rest on
the BPS minimality assumption of Theorem F.19 (kr ≥ 1
posited). The input enters once, in the requirement
χtop (Mc ) = 3 that selects CP 2 (App. AB, axiom V5);
the generation kernel, the exponent 57 = 60 − 3 and the
determinant-line degree use the same input. No index on
the internal space supplies it: every elliptic differential
operator on the odd-dimensional CP 2 × S 3 has index
zero; on CP 2 the Euler number 3 is the index of the
de Rham operator, which counts harmonic forms rather
than spinor zero modes, and the twisted Dirac indices
equal to 3 belong to the isometry triplet, which carries
electroweak charges that generations do not. A fourth
sequential generation would falsify the identification.

7.
6.

Uniqueness of Minimal Flux

Global Fermionic Consistency: the Spin-Z4 B−L
Structure and the Necessity of νR

Theorem F.19 (Energy Minimization). Subject to the
spinc constraint q1 = 3 and non-trivial gauge structure
(k3 , k2 ≥ 1), the unique global minimum of the Yang-Mills
energy is (k3 , k2 , q1 ) = (1, 1, 3).

Because CP 2 is not spin (w2 = H ̸= 0), placing the full
chiral fermion content on the internal geometry requires
a gauge-sourced spin structure. This subsection records
which structure works — and its physical consequence.

Proof. The BPS energy bound is:

Theorem F.22 (Spin-Z4 B−L consistency). In the allleft-handed convention, the SM + νR multiplets carry

2

EBPS = 8π (κ3 |k3 | + κ2 |k2 | + κ1 |q1 |),

(F11)

where κr > 0. With q1 = 3 fixed, EBPS (k3 , k2 ) =
8π 2 (κ3 k3 +κ2 k2 +3κ1 ) is strictly increasing in both k3 and
k2 . The minimum over {k3 , k2 ≥ 1} is achieved uniquely
at (k3 , k2 ) = (1, 1).
Corollary F.20 (Three Generations). For minimal flux
(k3 , k2 , q1 ) = (1, 1, 3):
Ngen = |1 · 1 · 3| = 3.

(F12)

3(B−L) = { Q : +1, uc : −1, dc : −1, L : −3, ec : +3, ν c : +3 } ,

(F13)
uniformly odd across all six multiplet types. Consequently
the structure Spin ×Z2 U (1)B−L (spin-Z4 ), with half-odd
B−L flux on CP 2 , renders every SM+νR multiplet simultaneously globally well-defined on the non-spin internal
geometry. By contrast: (i) hypercharge cannot serve this
role with one uniform structure if every multiplet is a
line bundle, since the parities of 6Y = {1, −4, 2, −3, 6, 0}
are mixed; when the weak doublets are rank-two isotropy

158
bundles (Rem. F.10), every multiplet is consistent with
the canonical spinc structure, and the spin-Z4 structure
is one of two consistent completions; (ii) the Z6 center of
GSM = (SU (3) × SU (2) × U (1))/Z6 cannot, since every
SM multiplet is Z6 -neutral (which is precisely why the
quotient is admissible), so center flux shifts twists only by
integers — the wrong parity (cf. Tong, JHEP 07 (2017)
104).
Corollary F.23 (νR on the spin-Z4 completion). On the
spin-Z4 B−L background, cancellation of the Z16 global
anomaly requires sixteen Weyl fermions per generation,
so the right-handed neutrino must exist in each family
(Wang–Wen, Phys. Rev. Research 2, 023356 (2020)). The
requirement holds on this completion; on the canonical
spinc completion of the isotropy reading (Rem. F.10) it
does not arise. DFD contains νR in either case, because
its neutrino sector uses it: MR = α3 MP (App. P), with
the effective-mass prediction mβ = 9.15 meV (App. X).
Remark F.24 (B−L cannot replace hypercharge in the
determinant-line chain). The B−L line cannot substitute
for LY in Lemma F.11: the B−L-only determinant-line
chain gives χ(E) = 61 (the νR , carrying B−L ̸= 0 but
Y = 0, enters the count), and integral hypercharge on the
spin-Z4 background forces q1 ∈ 6Z, giving kmax = 195 —
never 60. Likewise no B−L background flux renders the
q1 = 3 line-bundle twists uniformly admissible (the mixed
6Y parities obstruct this for every flux value). The B−L
structure is therefore one consistent completion, on which
it mandates νR , while the hypercharge determinant line
at the derived q1 = 3 fixes the microsector count; the two
play distinct, compatible roles.

8.

The Self-Coupling Coefficient ka (Model)

Methodological Note
The following is the physically motivated heuristic for the
coefficient ka = 3/(8α). The theorem form is stated in
App. G (Thm G.1) and App. AP (Thm AP.19, which
takes the frame ratio n3 /n2 as an axiom:
1
3
ka = (n3 /n2 ) αM = 32 · 4α
= 8α
), so this section is
retained as the heuristic, not as the primary justification.

a. Physical basis. The DFD scalar ψ couples to gauge
fields through the physical metric of Eq. (12). The EM
sector in the magnetic-dominated regime and the nonAbelian frame stiffnesses contribute to the ψ self-coupling.
b. Model for the coefficient. The ψ self-coupling receives contributions weighted by gauge group structure:
CA (SU (n3 )) 1
n3 1
ka =
·
=
·
.
(F14)
CA (SU (n2 )) 4α
n2 4α
Under electromagnetic duality (Dirac quantization), α →
αM = 1/(4α).

c.

Result.

With (n3 , n2 ) = (3, 2):
ka =

d.

3 1
3
·
=
≈ 51.4
2 4α
8α

(F15)

Physical interpretation.

• Factor n3 /n2 = 3/2: ratio of SU (3) to SU (2)
Casimirs
• Factor 1/(4α): magnetic coupling from duality
9.

Frame Stiffness from Ricci Curvature

The relation κr = nr κ0 is graded a motivated assumption (§XVIII K); the argument below motivates it.
Theorem F.25 (Frame Stiffness from Geometry). Let
gauge fields arise as Berry connections associated with
the internal mode bundle Hint = C3 ⊕ C2 ⊕ C over spacetime (Sec. XVII B). The gauge sectors correspond to the
unitary frame groups of the complex blocks Vr ⊂ Hint of
complex rank nr ; the geometry that enters is that of the
projectivised fibre P (Vr ), not a splitting of T (CP 2 × S 3 )
(which admits no (3, 2, 1, 1) partition, Lemma AB.13).
Then the frame stiffness satisfies:
κr = nr · κ0 .

(F16)

Proof. Step 1: The Berry connection Ar for sector r is
valued in su(nr ).
Step 2: The energy functional for Berry connection
fluctuations:
Z
1
⟨δψ|δψ⟩,
(F17)
E[Ar ] =
2
where the inner product uses the Fubini-Study metric on
P (Vr ).
Step 3: For Vr of complex dimension nr , the Ricci
curvature of CP nr −1 is:
Rij̄ = nr · giFS
j̄ .

(F18)

Step 4: The energy cost of a unit rotation scales with
Ricci curvature: Erotation ∝ nr .
Step 5: Defining κr as this energy cost: κr = nr κ0 .
a.

Explicit values.
Sector Fibre P (Vr ) Ric factor κr
SU(3)
SU(2)
U(1)

10.

CP 2
CP 1
CP 0

3
2
1

3κ0
2κ0
κ0

Proton Stability

Theorem F.26 (Proton Stability). In gauge emergence
with internal space CP 2 × S 3 , the internal winding is

159
conserved for configurations that stay continuous (Remark F.27); no local operator or perturbative quantum
gravity correction changes it, and if baryon number is
violated only by the electroweak anomaly, which changes
it in steps of ±3, single-nucleon decay is forbidden.
Proof. Definition (soliton reading): Baryon number as
winding. The S 3 internal space is fixed (not a Higgs
vacuum manifold). Field configurations at fixed time
define maps:
3
3
ϕ : Sspatial
→ Sinternal
,

n = deg(ϕ) ∈ Z.
(F19)
The topologically protected, quantized invariant is the
winding degree n = deg(ϕ) ∈ π3 (S 3 ) = Z. The physical
proton is the minimal nonzero winding (n = 1); the factor
3 in “B = 3n” is the quark/colour bookkeeping tying
one unit of internal winding to a three-quark baryon, and
does not quantize the phenomenological baryon number in
steps of three (single baryons are not forbidden). Recent
RHIC measurements of baryon-number versus electriccharge stopping favour a single nonperturbative gluonic
“baryon junction” as the carrier of B over additive valencequark counting [133–135]; this is consistent with treating
B as an attribute of the field configuration rather than
an additive per-quark charge, though the QCD junction
is a distinct object from the internal S 3 winding and the
measurement does not itself test exact conservation.
Step 1 (Local operators): Any local operator O(x) modifies ϕ in a bounded region. The winding number integral:
Z
1
n=
ϵijk Tr(ϕ−1 ∂i ϕ · ϕ−1 ∂j ϕ · ϕ−1 ∂k ϕ) (F20)
24π 2
is continuous and integer-valued. Local perturbations
cannot change n.
Step 2 (No internal-winding sphalerons): In gauge emergence the internal S 3 is the internal flavor space itself —
fixed geometry, not a dynamical vacuum manifold—so
there is no internal-winding sphaleron. This does not forbid the ordinary electroweak SU (2)L sphaleron, which lives
on the spacetime gauge bundle—a distinct S 3 that merely
shares π3 = Z with the internal one. The electroweak
sphaleron changes B + L in units of ∆(B + L) = ±6
while preserving B − L, i.e. ∆B = ±3; it is the standard
process invoked for the (B − L) → B conversion (factor
28/79) in the baryogenesis/leptogenesis chain elsewhere
in this work (the “shared-π3 double-duty” theorem). The
internal and electroweak S 3 are different spaces; only
the former is frozen. If baryon number is violated only
by the electroweak anomaly, it therefore changes only in
steps of ∆B = ±3, so single-nucleon decay, ∆B = ±1, is
forbidden.
Step 3 (Quantum gravity): The “folk theorem” (Misner,
Banks, Seiberg) states quantum gravity violates global
symmetries. On the soliton reading B is not a global
symmetry but a topological winding number, and a gravitational violation would require topology change of the
B = 3n,

internal S 3 , suppressed by:
M̄ 2 c2 rp2
ΓB-violation ∼ exp − P 2
ℏ

!


∼ exp −1038 . (F21)

Counted by quarks, as the selection rule of Step 2 counts
it, baryon number is a global symmetry of the printed
action. DFD’s black holes radiate at Hawking’s rate
(Cor. AP.36), so a gravitational violation is not excluded;
its rate is not computed, and a dimensional estimate gives
τp ≳ 1042 yr, beyond Hyper-Kamiokande’s reach.
Remark F.27 (The soliton behind the winding). Step 1
holds for configurations that stay continuous. For the
frame stiffness of §XVII D alone, a unit-winding texture
of size L has energy proportional to L, so it can shrink
at no cost (Derrick 1964), and under the corresponding
evolution such textures can collapse and change their
winding at a point in finite time (Shatah 1988; Turok
and Spergel 1990). The hidden-local-symmetry form does
not help: with A = Ω the curvature vanishes and every
winding costs no energy. A quartic term with positive
coefficient, for which a texture of size L costs energy
proportional to 1/L, removes the collapse; the energy
then satisfies E ≥ 12π 2 |B| in its natural units, a unitwinding minimum exists (Esteban 1986; Kapitanski and
Ladyzhenskaya 1983), and the spherically symmetric one
has E = 1.2314 × 12π 2 with its two-derivative and quartic
parts equal, as Derrick’s identity requires (computed).
The printed action does not contain such a term, so
the soliton reading of the proton rests on a posit, not a
derivation. The single-nucleon selection rule below does
not use the soliton.
a. Falsifiability (a clean DFD-vs-GUT adjudicator).
DFD gives Γ = 0 in every single-nucleon channel (p →
e+ π 0 , p → ν̄K + ): the electroweak anomaly changes
baryon number only in steps of ∆B = ±Ngen = ±3,
so the ∆B = ±1 of single-nucleon decay is forbidden. It
needs Ngen = 3 (input; Rem. F.21), not the value of kmax ,
and holds if baryon number is violated only by the electroweak anomaly, not by a GUT ∆B = 1 mediator. Every
GUT, by contrast, predicts a positive single-nucleon rate
(SU(5) ∼ 1030–31 yr, SO(10) ∼ 1034–36 yr). Hence the
observation of any single-nucleon proton decay at HyperKamiokande, DUNE, or JUNO falsifies gauge emergence,
while a continued null result is what DFD shares with the
Standard Model without a unified group. (The only Bviolating channel the premise leaves is the multi-nucleon
∆B = 3 electroweak sphaleron, unobservably suppressed.)
11.

UV Robustness of Topological Results

Theorem F.28 (UV Stability). The integer results, the
input Ngen = 3 and the winding n of continuous configurations, are stable against:
1. Higher-loop corrections
2. Non-perturbative effects

160
3. Quantum gravity corrections (below Planck-scale
topology change)
Proof sketch. Anomalies: The Adler-Bardeen theorem
guarantees anomaly coefficients are one-loop exact. They
depend on the representation content, an input shared
with the Standard Model (Rem. F.3).
θ parameter: the tree-level θ = 0 (θYM = 0 fixed by
the spectral action given that nothing is added to it for
the gauge fields; Theorem L.3) is not an integer invariant:
it receives loop corrections at the Standard-Model level
only (Appendix L).
Generation number: Ngen = 3 is a discrete input identified with χtop (CP 2 ) = 3 (Rem. F.21); an integer input
does not run.
Baryon number: Winding in π3 (S 3 ) = Z is topologically protected for continuous configurations; the
electroweak anomaly changes baryon number only by
∆B = ±3 (Thm. F.26, Step 2).
a. Summary. Topological invariants don’t receive radiative corrections because they are integers. The gauge
emergence predictions are as robust as any result in quantum field theory.
12.

Summary: Rigorous vs. Conjectural

a. The logical chain. The manifold is established
by Thm. AB.1 (App. AB), which takes Ngen = 3 as
an input (axiom V5); the mode-bundle partition (3, 2, 1)
(Prop. F.1) enters as a dictionary entry, not a selector
(App. AB, axiom V6). The chain below is bookkeeping:
it reproduces the input Ngen = 3 and does not derive it
(Rem. F.21).

√
with determinant −ĝ = c e2ψ and inverse components
ĝ 00 = −e2ψ /c2 , ĝ ij = e−2ψ δ ij .
Justification: This auxiliary metric ĝ is a computational
device for deriving gauge coupling relations in covariant
form. The fundamental DFD arena remains flat (R3 , t)
with the Gordon optical interval; gauge fields ultimately
propagate on the same causal structure as light.
b. Yang-Mills action. For gauge sector r ∈ {3, 2, 1}:
√
Z
−ĝ
(r)
4
(r) (r)
SYM = − d x 2 ĝ µα ĝ νβ Fµν
Fαβ .
(G2)
4gr
c. Electric-magnetic decomposition. Defining Ei =
F0i and Bi = 12 ϵijk Fjk :
e2ψ 2 c e−2ψ 2
Br .
E −
2gr2 c r
2gr2
Variation with respect to ψ.
(r)

LYM =

d.

(r)

∂LYM
e2ψ
c e−2ψ 2
Br .
= 2 Er2 +
∂ψ
gr c
gr2
2.

3 Rem. F.10

(G4)

The Magnetically Dominated Regime

a. Physical setting. In astrophysical environments
where DFD effects are observable (galactic outskirts),
electromagnetic fields are magnetically dominated: E 2 ≪
c2 B 2 .
b. Dominant contribution. In this regime, Eq. (G4)
simplifies to:
(r)

∂LYM
cB 2
≈ 2r (1 − 2ψ).
∂ψ
gr
3.

2

(G3)

(G5)

Frame Stiffness Structure

Thm. F.19

CP × S −−−−−−→ q1 = 3 −−−−−−−→ (1, 1, 3)
Thm. F.18

(F22)

−−−−−−−→ Ngen = 3

a. Frame stiffness from gauge emergence. From Appendix F, the gauge couplings arise from frame stiffnesses:
M2
,
κr = κ0 · nr ,
(G6)
κr
where M is the frame mass scale, κ0 is a universal stiffness,
and nr is the block dimension.
For the (3, 2, 1) partition: n3 = 3, n2 = 2, n1 = 1.
gr2 =

Appendix G: Derivation of α-Relations from Gauge
Emergence

This appendix provides the derivation of the DFD αrelation ka = 3/(8α) from the gauge emergence framework
established in Appendix F.
1.

The Gauge-ψ Lagrangian

a. Auxiliary covariant metric for gauge calculations.
For the gauge emergence derivations in this appendix, we
employ an auxiliary 4D covariant metric that differs from
the Gordon-style optical interval ds̃2 = −c2 dt2 /n2 + dx2
used in the main text [§II A]. The main-text interval has
flat Euclidean spatial sections; here we use an exponentdoubled auxiliary ansatz:
ĝµν = diag(−c2 e−2ψ , e2ψ , e2ψ , e2ψ ),

(G1)

4.

Derivation of ka = 3/(8α)

Theorem G.1 (Self-Coupling Coefficient). In the gauge
emergence framework with (3, 2, 1) partition and magnetically dominated regime, the DFD self-coupling coefficient
is:
n3 1
3
ka =
·
=
≈ 51.4.
(G7)
n2 4α
8α
Proof. The proof proceeds in four steps.
Step 1 (Backbone-doorway structure): The gauge backreaction on ψ is mediated by the SU(2) sector (the “doorway”), while the self-coupling strength is determined by

161
TABLE LXXIV. Status of gauge emergence results.
Result

Status

Method

(3, 2, 1) minimal partition
Proposition
Explicit classification (dictionary entry)
Electroweak group U (2)
Identified
Isotropy group of CP 2 (Rem. F.2)
Colour SU (3), family content Stated inputs
As in the Standard Model (Rem. F.3)
SU (N ) selection
Lemma
Lie algebra table
q1 = 3
Derived, if identified Faithful isotropy charge 6Y (Rem. F.10)
Ngen = |k3 k2 q1 |
Bookkeeping rule
Thm. F.18, Rem. F.21
(1, 1, 3) unique minimum
Theorem
Energy minimization
Ngen = 3
Stated input
Observed family number (Rem. F.21)
κr = nr κ0
Motivated assumption Ricci curvature (Thm. F.25)
τp = ∞
Conditional
Anomaly selection rule (Thm. F.26)
UV stability
Theorem
Adler-Bardeen + topology (Thm. F.28)
ka = 3/(8α)

Derived

Gauge emergence (App. G, App. AP)

the SU(3) sector (the “backbone”). The ratio of contributions is n3 /n2 = 3/2.
Step 2 (Electromagnetic duality): In the magnetically
dominated regime, the relevant coupling is the magnetic
fine-structure constant:
1
αM =
,
(G8)
4α
arising from Dirac quantization: α · αM = 1/4.
Step 3 (Combination): The self-coupling combines
these factors:
n3
3
3 1
ka =
=
.
(G9)
· αM = ·
n2
2 4α
8α
Step 4 (Numerical verification): With α ≈ 1/137.036:
3 × 137.036
ka =
= 51.39.
(G10)
8
a.

Physical interpretation.

• The factor 3/2 = h∨ (SU(3))/h∨ (SU(2)) is the ratio
of dual Coxeter numbers.
• The factor 1/(4α) reflects magnetic dominance in
the ψ-gauge coupling.
• ka measures how strongly ψ self-interacts through
gauge field backreaction.

5.

the branch of Postulate Y.10, give arg det(Mu Md ) = 0
(App. AO).
Anomaly: The CP mapping torus has dimension
dim TCP = dim M + 1 = 8 (even). Its fermion bundle is
a graded Clifford module, so the Dirac operator is odd
under chirality (ΓDΓ−1 = −D), forcing exact ±λ spectral pairing, and the product-background gap removes
zero modes. Hence η(DTCP ) = 0 and ACP = 1 (Theorem L.3, Appendix L); loop corrections are bounded at
the Standard-Model level.
a. Falsifiability. Detection of QCD axions with coupling gaγγ in the KSVZ/DFSZ range would falsify this
prediction.
6.

Derivation of kα = α2 /(2π)

Proposition G.3 (Clock coupling coefficient (conditional)). In DFD with gauge emergence, the speciesdependent clock coupling coefficient is:
α2
≈ 8.5 × 10−6 .
(G12)
2π
Note: A derivation conditional on the one-gauge-vertex
axiom, using the Schwinger mechanism, is given in Appendix P.
kα =

Proof. The proof proceeds in four steps.
Step 1 (Photon-ψ vertex): The photon propagator on
the optical metric acquires ψ-dependence. At one loop,
the photon-ψ vertex has strength:

Strong CP Prediction

Theorem G.2 (Strong CP Suppression). In gauge emergence with internal space CP 2 × S 3 and minimal flux
(k3 , k2 , q1 ) = (1, 1, 3), with θYM = 0 fixed by the spectral
action given that nothing is added to it for the gauge fields,
and given Postulate Y.10:

g2
4πα
α
=
=
.
(G13)
8π 2
8π 2
2π
Step 2 (Atomic energy structure): Atomic energy levels
depend on the Coulomb interaction:

(tree level; loop-induced ≲ 10−16 ).

En ∝ α2 · (me c2 ) · f (n, l, j).

θ̄ = 0

(G11)

Proof sketch. At tree level: The SU(3) gauge field is a
Berry connection on CP 2 with quantized instanton number k3 = 1. Det-orthogonal mass matrices, positive on

λγψ =

(G14)

Step 3 (ψ-modification): The ψ-modification of atomic
levels:
α
δEn = En · SA
·

δα
α

(G15)

162
where δα/α = λγψ · α · ψ = (α2 /2π)ψ.
Step 4 (Result):
kα =

TABLE LXXV. Complete α-relations with derivation status.

α2
.
2π

(G16)

a. Extension to other gauge sectors.
generalizes to all gauge couplings:

The formula

αi2
g2
, αi = i .
(G17)
2π
4π
For the strong sector with αs ≈ 0.118, under the same
one-gauge-vertex axiom as kα (App. P), so that ks is a
matching coefficient of the extended channel:

Relation Formula Value
Derivation
√
√
loc
−10
2
a0
2 α cH0 1.1966 × 10
m/s n2 · α · cH0loc ; early clock 1.1208
2
−6
kα
α /(2π)
8.5 × 10
Prop. G.3 (cond.)
ka
3/(8α)
51.4
Theorem G.1
θQCD

—

0 (tree level)

τp

—

∞

ki =

αs2
≈ 2.2 × 10−3 .
(G18)
2π
This gives the nuclear clock enhancement factor:
αs
ks STh
KTh
|R| =
≈
(G19)
α .
Kopt
kα Sopt
ks =

7.

Proton Stability Prediction

(stable at zero temperature).

(G20)

Proof sketch.
1. In gauge emergence, there is no unified gauge group to break; gauge symmetries emerge
from Berry connections.
2. No X, Y bosons from GUT symmetry breaking
exist.
3. The electroweak anomaly changes B only by ±3
(Thm. F.26); a single proton has no decay channel.
a.

Contrast with GUTs.
Model

And α appears in characteristic powers:
√
• a0 : α (geometric mean)
• kα : α2 (one-loop)
• ka : 1/α (magnetic duality)
Appendix H: Higgs and Yukawa Sector from Gauge
Emergence

Theorem G.4 (Proton Stability). In gauge emergence
with (3, 2, 1) partition and internal space CP 2 × S 3 , if
baryon number is violated only by the electroweak anomaly:
τp = ∞

Theorem G.2
(θYM = 0 fixed
by the spectral
action given
that nothing is
added to it for
the gauge fields)
Theorem G.4 (conditional)

τp prediction

SU(5) GUT
1030−31 years
SO(10) GUT
1034−36 years
Gauge emergence ∞ (conditional)
b. Falsifiability. Observation of proton decay at any
rate τp < 1040 years would falsify gauge emergence.

This appendix treats the Higgs mechanism, Yukawa hierarchy, CKM mixing, and neutrino masses in the gauge
emergence framework. The topological results of Appendix F determined representation content; here we
address the mass spectrum.

1.

Higgs Emergence from the (3, 2, 1) Structure

Theorem H.1 (Higgs Doublet). The Standard Model
Higgs doublet emerges as the off-diagonal connector between the C2 and C1 sectors of the (3, 2, 1) partition.
Proof. The internal Hilbert space Hint = C6 with (3, 2, 1)
partition has density matrix:


ρ3 X32 X31
†
ρ = X32 ρ2 H  .
(H1)
†
X31
H † ρ1
The off-diagonal block H connecting C2 and C1 is:
• A 2 × 1 complex matrix (2-component vector)
• Transforms as 2 under SU (2) (from C2 index)
• Singlet under SU (3) (no C3 involvement)

8.

Summary of Results

a. The unified structure. All relations involve the
(3, 2, 1) block dimensions:
• a0 : factor n2 = 2
• ka : ratio n3 /n2 = 3/2

• Carries U (1)Y charge from relative phase
These are precisely the Higgs quantum numbers:
(1, 2, +1/2).

163
a. Higgs potential. The frame stiffness energy L =
−κ0 ψ · S[ρ] expanded around the vacuum ρ0 = 13 13 ⊕
1
2 12 ⊕ 1 gives:
2

2

4

V (H) = −µ |H| + λ|H| ,

where µ , λ > 0 would be set by the frame stiffnesses.
No printed form of S[ρ] realizes this: neither the von
Neumann entropy nor the purity gives a potential of
this shape, so the potential, and with it the prefactor of
v and the √
quartic, is not computed. The minimum at
⟨H⟩ = (0, v/ 2)T breaks SU (2) × U (1)Y → U (1)EM .

a. Setup. The internal space M = CP 2 × S 3 carries three generations, a stated input identified with
χtop (CP 2 ) = 3 (App. F, Rem. F.21); with SU (3) flux
k3 = 1 the three zero modes below are one per generation.
Proposition H.2 (Generation Localization). In homogeneous coordinates [z0 : z1 : z2 ] on CP 2 , the three generation wavefunctions are:
ψ (2) ∝ z1 ,

ψ (3) ∝ z2 .

(H3)

These are localized at the three “vertices” [1 : 0 : 0],
[0 : 1 : 0], [0 : 0 : 1].
The wavefunctions are holomorphic sections of O(1)
(the hyperplane bundle). The three vertices are distinct
points of CP 2 , so this picture needs the electroweak group
to act on the fibres of the twisting bundles and not by
isometries of CP 2 , as the identification of Remark F.2
states; an isometric action of the isotropy group of a
vertex would rotate the other two generations into each
other.

3.

Yukawa Hierarchy from Overlap Integrals

Theorem H.3 (Yukawa Couplings). The Yukawa coupling for generation n is:
Z
Y (n) = gY
ψ̄ (n) (z) · ϕH (z) · ψ (n) (z) dµF S , (H4)
CP 2

where ϕH (z) is the Higgs profile on CP 2 and dµF S is the
Fubini-Study measure.
a. The hierarchy mechanism. Assume the Higgs is
localized near vertex 3 (the third generation):
2

|ϕH (z)|2 ∝ e−|w| /σ

2

kX
max
1
|k⟩.
kmax k=1

(H6)

Let a generation vertex i couple equally to a subset Γi of
Ngen channels, with normalized state
X
1
|k⟩.
(H7)
|i⟩ := p
Ngen k∈Γ
i

Define the Higgs localization width by the squared overlap
εH := |⟨i|H⟩|2 .

(H8)

Ngen
3
=
= 0.05
kmax
60

(H9)

Then

Zero-Mode Localization on CP 2

ψ (1) ∝ z0 ,

|H⟩ := √

(H2)

2

2.

connector state as the uniform superposition

(H5)

in affine coordinates w = (z0 /z2 , z1 /z2 ).
Theorem H.4 (Channel-Counting Derivation of εH ).
Let Hch ∼
= Ckmax be the channel Hilbert space with ormax
thonormal basis {|k⟩}kk=1
. Define the (normalized) Higgs

εH =

Proof. Using orthonormality of the channel basis,
X
1
1
√
⟨i|H⟩ = p
⟨k|k⟩
Ngen kmax k∈Γ
i
(H10)
r
Ngen
Ngen
=p
=
.
kmax
Ngen · kmax
Squaring yields εH = Ngen /kmax = 3/60 = 0.05.
Lemma H.5 (Regular-Representation Forcing of the εH
Premises). Suppose the channel Hilbert space carries the
regular representation of the icosahedral group, Hch ∼
=
C[A5 ], with basis {|g⟩ : g ∈ A5 } and kmax = |A5 | = 60
— the identification already used structurally by the finite
Yukawa construction of Appendix Y, whose Higgs kernel
ℓ(g)
b ch = P
H
g∈A5 εH |g⟩⟨g| is written in a basis indexed by
group elements. Then:
(i) (Premise (a) forced.) The trivial representation occurs in the regular representation with multiplicity exactly one (Schur: each irreducible ρ occurs
with multiplicity
dim ρ), and its invariant line is
P
spanned by g |g⟩. Hence the unique A5 -invariant
unit vector, up to
Pphase, is the uniform superposition |H⟩ = √160 g∈A5 |g⟩: an A5 -invariant Higgs
connector must be the uniform state assumed in
Theorem H.4.
(ii) (Premise (b) forced.) Let Z3 = ⟨c⟩ ⊂ A5 be
the generation subgroup (c an order-3 element, cf.
the Z3 generation projectors of Appendix Y). A
generation vertex supported on a Z3 -coset orbit
Γi = {g, cg, c2 g} is invariant under the Z3 lefttranslation action iff it is the

 equal-weight state
|i⟩ = √13 |g⟩ + |cg⟩ + |c2 g⟩ : equal coupling to
Ngen = 3 channels is forced, not assumed.
(iii) (Generation universality derived.) The overlap
|⟨i|H⟩|2 = 3/60 is independent of the coset representative g; all three generations share a single εH , a
consequence here rather than an implicit assumption
of the mass-hierarchy pattern.

164
Hence εH = Ngen /kmax = 3/60 is forced given Hch ∼
=
C[A5 ].
L
Proof. (i) Decompose C[A5 ] = ρ (dim ρ) ρ over the five
P
irreducibles of A5 (dim ρ = 1, 3, 3, 4, 5; ρ (dim ρ)2 = 60).
The trivial irreducible appears with multiplicity
P dim(1) =
1, and its isotypic component is the line C · g |g⟩. Any
A5 -invariant vectorPlies in that line; normalizing gives
2
|H⟩. (ii) A state k=0 ak |ck g⟩ is invariant under left
translation
√ by c iff a0 = a1 = a2 ; normalization fixes
ak = 1/ 3 up to a common phase.√ (iii) Each of the
three basis kets of |i⟩ p
contributes 1/ 60 to the overlap:
⟨i|H⟩ = √13 · √360 = 3/60 for every representative g.
Squaring gives εH = 3/60 = 0.05, coset-independent.
Remark H.6 (Residual of the εH Closure). Lemma H.5 reduces both premises of Theorem H.4 to a single structural
input: Hch ∼
= C[A5 ]. That identification is the one Appendix Y already builds its entire finite Yukawa operator
on (dim Hch = 60 = |A5 |, basis indexed by g ∈ A5 , Cayley
connector, conjugacy-class species projectors); within the
corpus it is used, not derived. Deriving C[A5 ] from the
CP 2 × S 3 harmonic spectrum is the remaining open item.
The global Yukawa normalization gY (single anchor from
mτ /mµ ) remains a stated input, unchanged.
b.

Significance.

This derivation:

• Uses only fixed integers: kmax = 60 (derived),
Ngen = 3 (a stated input)
• Requires no mass data
• Is falsifiable: different microsector connectivity ⇒
different εH
c. Up/down distinction. Up-type quarks couple to
H̃ = iσ2 H ∗ , down-type to H. A complex phase in ϕH (z)
gives different effective couplings:
Yu ̸= Yd
4.

(within each generation).

(H11)

b. CP violation. The CP-violating phase is supplied
by Postulate Y.10 (App. AO).
The Jarlskog invariant:
∗ ∗
J = Im(Vus Vcb Vub
Vcs ) ∼ 3 × 10−5 .

5.

(H14)

Neutrino Masses from See-Saw

Theorem H.8 (Lepton Number Status). In gauge emergence:
• The internal winding n is conserved for continuous configurations (topological, π3 (S 3 ) = Z); singlenucleon decay is forbidden if baryon number is violated only by the electroweak anomaly (Thm. F.26)
• Lepton number L is not topologically protected
• Majorana masses are allowed
a. The see-saw mechanism. Right-handed neutrinos
νR (gauge singlets) have Majorana mass. Appendix P
identifies the scale by determinant scaling on the Ngen = 3
generation space:
MR = MP α3 = 4.74 × 1012 GeV

(H15)

(Theorem P.4). This is lower than the naive estimate
Mint ∼ 1014 –1016 GeV but still in the see-saw regime.
The light neutrino mass:
2
MD
(20 GeV)2
∼ 0.1 eV.
(H16)
∼
MR
5 × 1012 GeV
b. Large PMNS mixing. Unlike CKM (small mixing),
PMNS has large angles because:

mν ≈

• Charged leptons: localized like down quarks
• Neutrinos: right-handed νR have different localization pattern
The misalignment gives large θ12 , θ23 and small θ13 —
qualitatively matching observation.

CKM Mixing from Geometry
6.

Theorem H.7 (CKM Structure). The CKM matrix
arises from misalignment between up-type and down-type
mass eigenbases:
VCKM = ULu† ULd ,

(H12)

where ULu,d diagonalize the respective Yukawa matrices.
a. Small mixing from localization. Off-diagonal
Yukawa elements require overlap of different generation
wavefunctions:
−dij /σ

Mij ∼ e

,

(H13)

where dij is the geodesic distance between vertices i and
j on CP 2 .

a.

Summary of Mass Sector

Free parameters remaining.

derived:
1. v = 246 GeV
√ (EW scale) — Exponent
√
v = MP α8 2π = 246.09 GeV (the 2π prefactor
is fitted)
2. εH = 0.05 (Yukawa base) — Derived: εH =
Ngen /kmax = 3/60 (Theorem H.4)
3. λ ∼ 0.22 (Cabibbo) — set by vertex distance d/σ
(pattern, not derived)
4. MR = MP α3 = 4.74 × 1012 GeV — Identification
(Appendix P)

165
TABLE LXXVI. Standard Model mass sector from gauge
emergence.
Feature

Mechanism

Status

Higgs doublet (2, 1) off-diagonal
Theorem H.1
EWSB
Frame stiffness potential Not computed
Mass hierarchy Zero-mode localization Theorem H.3
CKM structure Overlap geometry
Theorem H.7
CP violation
Geometric CP offset
Postulated
(Postulate Y.10,
App. AO)
Neutrino mass See-saw mechanism
Theorem H.8
PMNS mixing Different localization
Explained

b.

Data sources.

• X-ray gas masses:
Vikhlinin et al. (2006),
Simionescu et al. (2011)
• Stellar masses: Gonzalez et al. (2013)
• Lensing masses: Clowe et al. (2006), Bradac et
al. (2006, 2008), Merten et al. (2011), Jee et
al. (2014), Kim et al. (2021)
• SZ masses: Planck Collaboration (2016)
2.

Complete Results Table

Table LXXVII presents the complete analysis for the
16 clusters.

Predictions.

1. Neutrinos: Majorana (neutrinoless double beta decay)
2. Light neutrino masses: Σmν = 61.5 meV, mβ =
9.15 meV on the adopted branch (App. X)
Assessment (Complete Analysis)
The gauge emergence framework derives the Standard
Model mass structure. The bare-parameter √
form of the
hierarchy problem is eliminated: v = MP α8 2π
(exponent derived, prefactor fitted; radiative stability
against the external matter loop remains the universal
open problem — App. AH scope). The topological results
(anomalies, the mass-hierarchy exponents) are derived,
and α has a closed form; Ngen = 3 is a discrete input.

Appendix I: Full Cluster Sample Analysis

This appendix provides the complete dataset and analysis for the galaxy cluster study presented in Section VII K.

1.

a.

Dataset Description

We analyze 16 galaxy clusters from published X-ray,
optical, and lensing surveys; the four galaxy groups are
compared on published group masses in the last subsection:
• Relaxed clusters (10): A1795, A2029, A478,
A1413, A2204, Coma, Perseus, A383, A611, MS2137
• Merging clusters (6): Bullet (1E 0657-56), A520,
El Gordo, MACS0025, A2744, RXJ1347
• Galaxy groups (4): Virgo, Fornax, NGC5044,
NGC1550

TABLE LXXVII. Complete cluster sample analysis with
µ(x) = x/(1 + x). Column y ≡ gN /a0 is the Newtonian
baryp
onic acceleration in a0 units; Ψ = ν(y) = (1 + 1 + 4/y)/2
is the inverse-MOND boost satisfying a µ(a/a0 ) = gN . Computed at a0 = 1.20 × 10−10 m/s2 ; the bound-matter a0 =
1.1966 × 10−10 m/s2 gives the same class means.
Cluster

Mg M∗ Mb Mtot r500
(1014 M⊙ )
(Mpc)

y

Ψ O/D

Relaxed
0.67 0.12 0.79 5.50
1.05 0.18 1.23 8.50
0.85 0.14 0.99 6.80
0.62 0.11 0.73 5.20
0.95 0.16 1.11 7.80
0.85 0.15 1.00 7.00
0.55 0.10 0.65 5.80
0.32 0.06 0.38 2.80
0.45 0.08 0.53 4.20
0.38 0.07 0.45 3.50

1.24
1.45
1.35
1.20
1.40
1.40
1.25
0.95
1.05
1.00

0.060 4.62 1.51
0.070 4.37 1.58
0.063 4.51 1.52
0.059 4.65 1.53
0.066 4.43 1.59
0.059 4.64 1.51
0.048 5.08 1.76
0.048 5.08 1.47
0.056 4.76 1.66
0.052 4.93 1.60

Merging
Bullet
1.15 0.20 1.35 11.5
A520
0.65 0.11 0.76 6.20
El Gordo 2.10 0.35 2.45 21.0
MACS0025 0.48 0.08 0.56 4.80
A2744
1.30 0.22 1.52 14.0
RXJ1347 1.40 0.24 1.64 15.0

1.50
1.20
1.85
1.10
1.60
1.65

0.070 4.32 1.97
0.061 4.57 1.79
0.083 4.00 2.14
0.054 4.84 1.77
0.069 4.34 2.12
0.070 4.31 2.12

A1795
A2029
A478
A1413
A2204
Coma
Perseus
A383
A611
MS2137

3.

Statistical Summary (the µ-Law Alone)

TABLE LXXVIII. Statistical summary by cluster type under
the µ-law alone (no χ).
Category

N Mean(Obs/DFD)

σ

Relaxed clusters 10
Merging clusters 6

1.57
1.99

0.08
0.16

All clusters

1.73

0.23

16

166
Note: Obs/DFD above one is the mass the µ-law does not
supply from the baryons; it is the derived χ component
(Conclusions of this appendix). With χ adding its Newtonian pull and the relaxed class carrying a hydrostatic bias
b = 0.21, all sixteen clusters need χ = 4.5 ± 0.6 Mb ,
0.84 ± 0.12 of the cosmic mix, at the bound-matter
a0 = 1.1966 × 10−10 m/s2 , as at the a0 of the table.
4.

Alternative µ1/2 Function

TABLE LXXIX. Cluster analysis with µ1/2 (x) = x/(1 +
Cluster

Ψobs ΨDFD (n = 0.5) Obs/DFD Status

A1795
A2029
A478
A1413
A2204
Coma
Perseus
A383
A611
MS2137

7.0
6.9
6.9
7.1
7.0
7.0
8.9
7.5
7.9
7.9

Relaxed Clusters
6.68
6.36
6.54
6.71
6.44
6.70
7.24
7.24
6.85
7.05

1.04
1.09
1.05
1.06
1.09
1.05
1.23
1.03
1.16
1.11

✓
✓
✓
✓
✓
✓
✓
✓
✓
✓

Bullet
A520
El Gordo
MACS0025
A2744
RXJ1347

8.5
8.2
8.6
8.6
9.2
9.1

Merging Clusters
6.30
6.61
5.90
6.95
6.32
6.29

1.35
1.23
1.45
1.23
1.46
1.45

✓
✓
✓
✓
✓
✓

√

x)2 .

Summary

5.

• r500 determination: 5–10% from overdensity definition
Combined systematic uncertainty on Obs/DFD ratio:
∼20–30%.

6.

Note: This section is included for completeness. DFD
does not adopt the n = 0.5 interpretation: the adopted
µ(x) = x/(1 + x) holds from galaxies to clusters, and the
cluster shortfall under it is the χ component (Conclusions
of this appendix).
√
Table LXXIX shows results using µ(x) = x/(1 + x)2 .
Its larger boost at cluster accelerations absorbs into µ the
mass that the derived χ supplies.

Well-fit (0.7–1.5)
Relaxed mean

• Total mass (lensing): 5–10% from calibration
and projection

16/16
1.09 ± 0.06

Systematic Uncertainties

The analysis incorporates the following systematic uncertainties:
• X-ray gas mass: 10–15% calibration uncertainty
• Stellar mass: Factor 1.5–2 from IMF uncertainty
(subdominant)
• Total mass (hydrostatic): 10–30% bias from
non-thermal pressure

Conclusions

a. Clusters need χ. Under the µ-law alone the sixteen
clusters fall short of their measured masses: Obs/DFD
= 1.57 ± 0.08 (relaxed) and 1.99 ± 0.16 (merging) in
Table LXXVII. The relaxed masses are hydrostatic Xray masses and the merger masses are lensing masses; a
hydrostatic bias b = 0.21 of the relaxed masses brings
the two classes to the same ratio (the CCCP calibration
measures b = 0.22 ± 0.07, Hoekstra et al. 2015). The
shortfall is the derived cold component χ (App. AV). If
χ adds its Newtonian pull to the baryons’ µ-law field,
as App. GR prints the two channels, all sixteen clusters, with that bias and at the bound-matter a0 , need
χ = 4.5 ± 0.6 Mb inside r500 , 0.84 ± 0.12 of the cosmic mix
Ωχ /Ωb ≃ 5.4 (computed). Under the floor composition
of App. AV gravity in clusters is Newtonian plus χ and
the budget is the ΛCDM one: 1.28 ± 0.16 of the cosmic
mix inside r500 at b = 0 and 1.51 ± 0.14 at b = 0.21,
the usual baryon depletion inside r500 (computed). The
amplitude follows from the derived abundance. Under
the adopted rule of motion, the momentum-conserving
energy of App. AV, cluster halos assemble as in ΛCDM
(estimate), and ΛCDM hydrodynamics gives massive clusters a baryon depletion inside r500 of Yb = 0.85 ± 0.03,
nearly independent of the physics included (Planelles et al.
2013); with fc = ωb /(ωb + ωχ ) = 0.157 this puts χ inside
r500 at (1 − Yb fc )/[Yb (1 − fc )] = 1.21 ± 0.05 of the cosmic
mix (estimate). The sixteen clusters give 1.28 ± 0.04 with
no hydrostatic bias, which cool-core clusters show (Mahdavi et al. 2013), and 1.51 ± 0.04 with b = 0.21 (errors of
the mean); with the calibration uncertainties of the gas,
stellar and lensing masses their measured masses are 1.10
(0.97–1.23) and 1.30 (1.13–1.47) times the predicted ones
(computed). The relaxed and merging classes differ by a
factor 1.18 in baryon fraction, which a hydrostatic bias of
0.15 or merger masses 15% high would remove. χ boosted
like the baryons would over-predict these masses 1.43fold (computed). The required χ peaks at 0.15–0.3 r500 ,
more concentrated than the gas, as cold dark matter is
(computed).
b. The five-factor correction stack does not close the
gap. A no-χ closure by five correction factors, Ci =
Bi JPDE,i Ti Mi Pi (Eq. (200)), is not supported (computed; next subsection): the baryonic-completeness factor
B sums to 1.26 on its own item list, not 1.30–1.45, and
to 0.96–1.10 for the items that lie inside r500 ; the substructure factor is real, but for the baryonic substructure

167
that the µ-law responds to it is JPDE ≃ 1.04; the temperature factor T absorbs the relaxed/merging split, which
is the hydrostatic bias above; the merger factor M is
applied to lensing masses, which carry no hydrostatic
nonequilibrium; and the projection factor P > 1 has the
wrong sign for lensing masses, which simulations find
biased low. A first-principles account awaits a structureformation run under the adopted momentum-conserving
energy (App. AV) and a cluster-by-cluster nonlinear DFD
hydrodynamic and lensing solver that includes χ.

7.

The Five Correction Factors, Examined

a. Baryonic completeness, B. The literature items
offered for missing cluster baryons are the warm-hot intergalactic medium, ∼10% of the gas mass [136, 137]; clumping and missed diffuse gas, net ∼5% of the gas [138]; intracluster light, ∼25% of the stellar mass [139, 140]; hot gas
beyond r500 , ∼10% of the gas [141]; and a bottom-heavy
IMF in cluster ellipticals, ∼10% of the stars [142]. Together they give B = 1.26, not 1.30–1.45. The warm-hot
medium and the gas beyond r500 lie outside the aperture
in which the masses are compared; the items that apply
inside r500 give B = 0.96–1.10 (computed).
b. Nonlinear substructure averaging, JPDE = 1 +
0.39 fsub . A direct 3D nonlinear AQUAL/DFD solver
(Brada-Milgrom 1995 method, face-centered µ discretization, validated against algebraic-MOND for spherical
NFW within 1%) gives:
JPDE (fsub ) ≃ 1.00 + 0.39 fsub ,
JPDE ≃ 1.07–1.12 for fsub = 0.15–0.30.

(I1)

The mechanism is real: the gas-weighted enhancement
in the diffuse intracluster medium exceeds the smoothcluster mean-field value because mass redistribution into
clumps reduces the smooth-component gradient at gas
positions, and the slope is robust across cluster mass
(M200 ∈ [3, 12] × 1014 M⊙ ), subhalo number (Nsub = 50–
200), and subhalo size (rsub = 30–100 kpc). The µlaw responds to the baryons, however, so fsub is the
member galaxies’ share of the baryons, ≃ 0.10, which
gives JPDE ≃ 1.04 (computed).
c. Temperature factor, T = 1 + 0.04(TX /keV − 7). T
spans 0.90–1.30, but a 10–30% temperature-calibration
error moves an X-ray gas mass by only 2–7% (Mgas ∝ T 1/4
in the soft band) and a lensing mass not at all. Its
correlation with the raw Obs/DFD (r = 0.77 over all
sixteen) comes from the merger flag (r = 0.87), not from
temperature within a class (r = 0.01 among the relaxed
clusters). T therefore absorbs the relaxed/merging split,
which is the hydrostatic mass bias: the relaxed masses
are hydrostatic X-ray masses, the six merger masses are
lensing masses (Clowe et al. 2006; Bradač et al. 2008;
Merten et al. 2011; Jee et al. 2014), and b = 0.21 equalizes
the two classes (WtG 1−b = 0.688±0.072, von der Linden
et al. 2014; CCCP 0.78 ± 0.07, Hoekstra et al. 2015;
computed). SZ-derived masses, which do not depend on

the X-ray temperature, exceed X-ray inferences for hot
clusters by ∼20% [143, 144], consistent with a hydrostatic
bias of that size.
d. Merger factor, M . M is sized by the non-thermalpressure envelope of merger simulations (Nelson et al.
2014), a bias of hydrostatic masses; the six merger masses
are lensing masses, to which it does not apply. Whether a
ψ field that has not equilibrated on the merger timescale
changes the DFD prediction for mergers is not computed.
e. Projection factor, P . For the sky-plane lensing
mergers the simulation literature finds NFW-fit weaklensing masses biased low by 5–10% on average (Becker
& Kravtsov 2011; Rasia et al. 2012), so a projection
correction raises those masses: P > 1 has the wrong sign.
The stack therefore does not close the shortfall of Table LXXVII; the clusters need χ
√ (Conclusions above).
Evaluating the derived a0 (z) = 2 α c H(z) at each cluster’s redshift, rather than a0 at z = 0, raises the boost
of the higher-redshift systems and lowers their ratios (El
Gordo, at z = 0.87, by about 20%), so it bears on the
relaxed/merging split and on the bias inferred from it; it
is an open consistency item of this table under the additive composition. Under the floor composition clusters
are Newtonian plus χ, and a0 drops out of their budget.
8.

Galaxy Groups

On published X-ray masses the µ-law falls short of
galaxy groups as it does of clusters. At the bound-matter
a0 = 1.1966 × 10−10 m/s2 , with distances computed with
H0loc and stellar masses estimated from K-band light or
from published stellar-fraction relations (computed on
published inputs):
• Sun et al. (2009), 23 Chandra groups: median
Obs/DFD = 1.22–1.36 at r500 (the range spans two
stellar-mass calibrations), and 1.76–2.30 at r2500 ,
none below 1.
• Gastaldello et al. (2007), 16 groups: median
Obs/DFD = 1.79 at the outermost measured overdensity; 15 of the 16 lie above 1, the exception being
the disturbed group RGH80.
• Virgo, Fornax, NGC 5044, NGC 1550: Virgo
1.19–1.48 at r500 = 663 kpc (Simionescu et al. 2017);
Fornax 2.7–4.0 at 19 kpc on its hydrostatic mass
(Su et al. 2017) and 1.02–1.95 at 393 kpc; NGC 5044
1.54–1.76 at r1250 and NGC 1550 1.77–2.01 at r2500
(Gastaldello et al. 2007), and 1.12–1.32 and 0.89–
1.08 at r500 on extrapolated profiles.
The result is set by the groups’ measured baryon fractions,
fgas = 0.05–0.11 at r500 (median 0.08) plus 0.02–0.11 in
stars (Sun et al. 2009), which the µ-law boost ν ≃ 6–7
does not lift to the measured masses. A MOND analysis
of X-ray groups with the same µ(x) = x/(1 + x) likewise
finds 60–80% of the dynamical mass at the last measured
radius unaccounted for by the baryons [54]. The shortfall

168
is the derived χ, as in clusters: under the floor composition
of App. AV, 20 to 22 of the 23 groups of Sun et al. are
Newtonian plus χ inside r500 , at 1.2–1.7 of the cosmic mix
(the clusters above: 1.28–1.51); if χ adds its Newtonian
pull the budget is 0.2–0.5 of the cosmic mix (computed;
the stellar masses are estimates). None of the four named
groups lies inside a larger bound region, so the RestMass Channel’s bound-system confinement gives them
almost no external field from their neighbours (App. GR,
Remark GR.12), and none is needed.

2.

a. The asymmetry factor. The ratio of odd to even
peak heights is determined by the asymmetry factor A:

2
Hodd
1+A
.
(J3)
=
Heven
1−A
b. Factor decomposition. We decompose A into four
physically distinct contributions:
A = fbaryon × fISW × fvis × fDop .

Appendix J: Derivation of the ψ-CMB Solution

This appendix gives the analytic baryon-loading calculation of the peak ratio R ≈ 2.34. The global ψ-lensing
derivation of the peak location in §J 5 is not a DFD prediction: exact conformal invariance and the homogeneousmode theorem (App. ZS) rule out that mechanism. The
observable holds on the measured-clock forward solution:
100θ∗ = 1.04110 and ℓ1 ≃ 220.4, with no global screen.
1.

Peak Height Asymmetry

a.

(J4)

Baryon Loading Factor fbaryon

The baryon-photon oscillator with baryon loading Rb
produces asymmetry:
Rb
fbaryon = √
.
(J5)
1 + Rb
a. Numerical value. With Rb = 0.6 (from BBN):
0.6
0.6
= 0.474.
(J6)
fbaryon = √
=
1.265
1.6

The ψ-Acoustic Oscillator

a. Setup. Consider a baryon-photon fluid in ψgravity. The temperature perturbation Θ ≡ δT /T obeys:
Θ̈ + c2s (ψ)k 2 Θ = −

k2
Φψ ,
1 + Rb

(J1)

where:
√
• cs (ψ) = c(ψ)/ 3 is the sound speed with c(ψ) =
c0 e−ψ
• Rb = 3ρb /(4ργ ) ≈ 0.6 is the baryon-to-photon density ratio
• Φψ = Φ/µ(x) is the ψ-enhanced gravitational potential
b. Solution structure. The general solution has the
form:
Θ(k, τ ) = A(k) cos(krs ) + B(k) sin(krs ) + (driving term),
(J2)
R
where rs = cs (ψ) dτ is the sound horizon.
c. Peak/trough pattern.
• Odd peaks (n = 1, 3, 5, . . .): compressions (maxima
of |Θ|)
• Even peaks (n = 2, 4, 6, . . .): rarefactions (maxima
of |Θ|, with Θ of opposite sign)
In standard cosmology, baryon loading causes compressions to be enhanced relative to rarefactions, producing
the odd/even asymmetry.

b.

Integrated Sachs-Wolfe Factor fISW

The observed temperature perturbation includes the
Sachs-Wolfe and integrated Sachs-Wolfe terms:
Z
∆T
(J7)
= Θ + Φ + 2 Φ̇ dτ.
T
Ra. Cancellation. The SW term (Φ) and ISW term
(2 Φ̇ dτ ) partially cancel. This cancellation is taken to
be approximately 50%:
fISW ≈ 0.50.

(J8)

This value is constrained to be O(0.5) by physical
considerations.

c.

Visibility Function Factor fvis

Recombination is not instantaneous. The visibility
function g(τ ) = τ̇c e−τc has finite width ∆τ .
a. Effect on asymmetry. Finite-width recombination
smears out the sharp features in the angular power spectrum. The effect on the asymmetry is:

2
π 2 ∆τ
fvis = sinc(π∆τ /τ∗ ) ≈ 1 −
.
(J9)
6
τ∗
b.

Numerical value.

With ∆τ /τ∗ ∼ 0.1:

fvis ≈ 1 − 0.02 = 0.98.

(J10)

169
d.

Doppler Factor fDop

The Doppler contribution from baryon velocity perturbations is:
ΘDop = n̂ · vb ,

(J11)

where n̂ is the line-of-sight direction.
a. Effect on asymmetry. The Doppler term is 90◦
out of phase with the acoustic term. When projected onto
the line of sight and averaged, this reduces the effective
asymmetry:
fDop ≈ 0.90.
e.

(J12)

Total Asymmetry

The oscillation amplitude scales as:
|Φ|/µ
1
|F |
∝ .
|Θ| ∝ 2 2 ∝
(J19)
cs k
c2s
µ
All peaks (odd and even) are enhanced by 1/µ. In the
ratio:
H1
(1/µ)2
|Θodd |2
R=
∝
= 1 × (baryon physics).
=
H2
|Θeven |2
(1/µ)2
(J20)
The µ-enhancement drops out of the ratio. What survives in this analytic estimate is the baryon loading factor,
which depends only on Rb , a quantity fixed by BBN.
The peak ratio needs no postulated CDM particle, but
it does need a cold clustering component: CAMB with
baryons alone gives H2 /H1 = 0.39 against Planck’s 0.45
(Thm. AV.31). That component, which also sets the thirdpeak height, is the derived χ-matter field (App. AV), not
a postulated WIMP.

Combining all factors:
A = 0.474 × 0.50 × 0.98 × 0.90 = 0.209.

3.

(J13)

Definition. The peak ratio is:
H1
(first peak height)
R≡
=
.
(J14)
H2
(second peak height)
b. Relation to asymmetry. For the angular power
spectrum Cℓ , the peak heights scale as:

2
Hn ∝ (1 + (−1)n+1 A) .
(J15)
Hence:

c.

(1 + A)2
=
(1 − A)2



1+A
1−A

2
.

With A = 0.209:

2
1.209
R=
= (1.528)2 = 2.34
0.791

(J16)

Result.

a.

Why the 1/µ Enhancement Cancels

a. Mechanism. In ψ-gravity, the driving term is enhanced: Φψ = Φ/µ. But this enhancement affects both
odd and even peaks equally.
b. Mathematical demonstration. The acoustic equation (J1) has driving term:
F (k) = −

k2
k2 Φ
Φψ = −
.
1 + Rb
1 + Rb µ

Status. Every equation in this subsection that maps
297 → 220 with ∆ψ = 0.30 belongs to the excluded
global-screen construction and is not a DFD prediction.
It must not be used to shift θ∗ , rs , DA , or H0 . Local and
anisotropic lensing remain separately testable and are not
covered by this notice.
a. The historical problem. Standard GR calculations
without CDM give ℓ1 ≈ 297, not the observed ℓ1 ≈
220. This has been cited as “proof” that dark matter is
required.
b. The construction. This argument assumes GR
propagation with fixed c and straight-line photon paths.
In ψ-physics, light travels through a medium with varying
refractive index n = eψ , producing gradient-index (GRIN)
optics effects.

(J17)

d. Comparison with observation. Planck’s best fit
gives R ≈ 2.21 ([H1 , H2 , H3 ] ∝ [1, 0.45, 0.44], §J 9); the
analytic value lies 6% above it, while the full spectrum
of the derived χ-matter field, run with CAMB on the
measured clock, gives H1 /H2 = 2.20 (H2 /H1 = 0.4545,
+0.5σ; Thm. AV.31).

4.

The ψ-lensing peak-location construction (not a
DFD prediction)

Peak Ratio Derivation

a.

R=

5.

(J18)

Gradient-Index Optics

a. Basic physics. In a medium with spatially varying
n(x), light rays follow curved paths according to Fermat’s
principle. For a gradient ∇n, rays bend toward regions
of higher n.
b. Angular magnification. For a GRIN lens with n
varying along the line of sight:
θobs
nemit
=
.
(J21)
θemit
nobs
If nemit > nobs (higher n at source):
• θobs > θemit : angular scales are magnified
• Observed ℓ is smaller than “true” ℓ (since ℓ ∝ 1/θ)

170
b.

6.

Application to CMB

a. ψ-gradient. With n = eψ , the angular scaling
becomes:
θobs
= eψCMB −ψhere = e∆ψ .
θemit

b.

Peak location relation.
θtrue
= ℓtrue × e−∆ψ .
ℓobs = ℓtrue ×
θobs

a. Self-consistency of the ∆ψ = 0.30 construction
(§J 5; not a DFD prediction).

(J22)

(J23)

c.

Required gradient. To obtain ℓobs = 220 from
ℓtrue = 297:
220 = 297 × e−∆ψ ,

(J24)

e−∆ψ = 220/297 = 0.74,

(J25)

∆ψ = − ln(0.74) = 0.30.

Consistency Checks

(J26)

d. Physical interpretation. ∆ψ = ψCMB − ψhere =
0.30 means:
• ψ was 0.30 higher at CMB than today
• nCMB /nhere = e0.30 = 1.35 (35% higher refractive
index)
• cCMB /chere = e−0.30 = 0.74 (26% slower light
speed)

1. α-variation bounds. With α(ψ) = α0 (1 + kα ψ)
and kα = α2 /(2π) ≈ 8.5 × 10−6 (Sec. VIII C):
∆α
= kα ∆ψ ≈ 8.5 × 10−6 × 0.30 ≈ 2.5 × 10−6 . (J27)
α
This is ∼ 2.5 ppm—well within observational
bounds. The quasar α-variation literature constrains |∆α/α| ≲ 10−5 at z ∼ 2–3, and CMB
constraints are |∆α/α| ≲ 10−3 . The construction
satisfies both.
Note: The coupling kα = α2 /(2π) governs electromagnetic variation; this is distinct from the acceleration coupling ka = 3/(8α) ≈ 51 that appears in
galactic dynamics.
2. BBN compatibility. BBN occurs at T ∼ 1 MeV,
much earlier than CMB (T ∼ 0.3 eV). If ψ-evolution
is monotonic, ∆ψBBN could be larger, but BBN
physics depends primarily on nuclear rates, not
optical effects. The constraint is on αBBN , which
can accommodate O(10%) variations.
3. Late-time ψ. Today, ψhere ≡ 0 by convention.
Local physics is unaffected by the absolute value of
ψ—only gradients matter.
7.

Comparison with ΛCDM

This is a modest gradient—not fine-tuned.
a.

Feature comparison between ΛCDM and ψ-Cosmology.
Feature

ΛCDM

ψ-Cosmology

Peak ratio R
CDM-driven (Ωc )
Baryon loading (Rb ) and χ-matter
Peak location ℓ1 GR distances (with CDM) measured-clock no-screen forward solution

b. Key difference. ΛCDM introduces dark matter
particles to explain the CMB. DFD explains the same
observations using ψ-physics:
• Peak ratio: baryon loading (same Rb from BBN)
with the derived χ-matter
• Peak location: measured-clock no-screen forward
solution
The peak-ratio result introduces no fitted particle. The
separately derived χ carrier supplies the primordial coldclustering channel required by the higher-peak heights.
8.

Falsifiable Predictions

The ψ-CMB solution makes specific predictions beyond
the peak structure:

1. Distance duality consistency. Etherington’s
reciprocity holds exactly in DFD’s optical metric:
DL
= 1.
(J28)
(1 + z)2 DA
Any common screening of DL and DA cancels in the
ratio. Any detected violation would falsify DFD’s
single-metric framework.
2. Polarization consistency. Local/inhomogeneous
ψ lensing should affect E-mode and B-mode polarization consistently. It is not a global peak-location
map.
3. Higher peaks. The third peak and beyond follow
the same frozen-input acoustic forward solution;
their heights test the derived χ carrier and the

171
adopted rest-mass clustering channel, not a global
ψ-lensing shift.
a. Ultimate test. The current global test is the frozeninput acoustic forward calculation: 100θ∗ and the peak
heights must remain consistent without a mean ∆ψ screen.
The distinct local/inhomogeneous test is a positive, meansubtracted cross-correlation with foreground structure; it
does not permit a global peak relocation.
b. Status (resolved by the derived χ field): the thirdpeak height. The derivations above give the analytic estimate H1 /H2 ≈ 2.34 of the odd/even peak ratio (from the
baryon loading Rb , which is BBN-fixed) and the first-peak
location ℓ1 ≈ 220. They do not fix the absolute thirdpeak height. The gravity-sector driving indeed “cancels
in ratios” (§XVI C), but that is precisely why this evades
the height problem rather than solving it: in ΛCDM
the observed H3 /H1 ≈ 0.44 (H3 /H2 ≈ 1.0, Planck) is
set by the sustained gravitational potential wells of a
pressureless component that clusters at z ∼ 1100. Radiation driving plus baryon loading alone collapse the
third peak — a linearized photon-baryon acoustic solver
(dynamical Newtonian-gauge potential + Silk damping)
gives H3 /H1 ∼ 0.1, far below the observed ≈ 0.44, and a
direct no-CDM ψ-Boltzmann integration misses H2 /H1
and H3 /H1 by tens of σ. The optical device that would restore the third peak is the “dust branch” (w → 0, c2s → 0)
of the cosmology section — which is, by construction,
the same pressureless clustering matter that ΛCDM calls
cold dark matter, i.e. an effective-CDM component, not
a baryon-loading effect. Furthermore, DFD’s own dustbranch no-go analysis shows the bounded optical saturation response (µ ∈ [0, 1)) cannot carry an a−3 clustering
charge to recombination without going stiff/radiation-like
(it fails by scaling, parameter-free, not by amplitude).
The carrier of the cold a−3 charge is therefore not the
optical ψ-response, but a separate derived matter field:
the χ-matter field of App. AV, the harmonic b3 threeform on S 3 = SU (2) that the rigidity classification (the
parent-tensor theorem, App. AH, Extended Derivations)
does not list. With its mass and decay constant (and, in
the photon-baryon kernel, the abundance Ωχ h2 ≃ 0.12
that a cold component of that density would supply), χ
is cold, pressureless, and non-thermal at z ∼ 1100, so
it enters the photon-baryon kernel identically to a cold
component of the same Ωh2 and restores the Planck height
pattern [H1 , H2 , H3 ] ∝ [1, 0.45, 0.44] (CAMB-validated;
App. AV, Theorem AV.31). This closes the obstruction at
Derived grade — a derived particle, not an effective-CDM
fit — distinct from both the (ruled-out) optical response
and the (absolute-time-breaking) projectable-lapse route
below.
Status. Baryon loading gives the analytic peak-ratio
estimate, the measured-clock no-screen solution the peak
location, and the derived χ-matter field (App. AV) the
third-peak height and, with the baryons, the full peak
ratio (with the TT damping tail and TE/EE polarization
following from the same cold component). The clustering
charge that ΛCDM attributes to a postulated cold-dark-

matter particle is here the derived χ field of fixed abundance Ωχ h2 ≃ 0.12; the ψ-screen is retained only for the
galactic deep-MOND regime and is not part of the CMB
kernel. The resolution is graded Derived for the carrier
(mass and decay-constant exponents, and cold clustering
behavior in the kernel); it is not an effective-CDM fit of the
peak shape. The relic abundance is derived, conditional
on the flux-period identification (Lemma AV.6), the stated
count 158 and the asserted premise P4′ (Lemma AV.27):
post-inflation (App. AV, Step 5b) the misalignment angle
is removed, and the relic amplitude is the finite SU (2)60
CS/WZW vacuum Casimir expectation, Ωχ h2 = 0.1193
(−0.6σ from Planck); a ∼9–44× overshoot is a units (4π 2 )
artefact on a continuum-measure choice. The amplitude
is forced given two stated conventions (Casimir operator;
measure conditional on P4′ ; k(k + 2) from χ’s derived
Z2 + Sugawara), the relic bridge is exact (κrad ), and the
grade is conditional-derived on the named flux-period
identification, the stated count 158 and P4′ (App. AV).
c. The integration-constant-dust route (DFD-adjacent,
not minimal-DFD). The natural candidate for the missing pressureless component is a constraint integration
constant rather than a particle. In projectable Hořava–
Lifshitz gravity
(Mukohyama) the Hamiltonian constraint
R
is global, d3 x H = 0, and a pressureless a−3 “dark
matter as an integration constant” survives locally;
the mimetic construction of Chamseddine–Mukhanov
produces the same dust from a fixed-norm constraint
g̃ µν ∂µ ϕ ∂ν ϕ = −1. Such a residual would obey the colddark-matter Boltzmann hierarchy at linear order (w = 0,
c2s = 0, σ = 0) and so restore H3 . We record, however,
that this mechanism is not native to minimal DFD as
formulated here, for three structural reasons. (i) DFD’s
fundamental arena carries an absolute external time, not
a dynamical ADM lapse: no projectable N (t) is varied, so
no global Hamiltonian constraint is generated and no local
integration-constant residual ρI (x, t) is left over (an absolute clock is not a projectable dynamical lapse). (ii) The
ψ field equation is the local, pointwise screened-Poisson
law ∇· [µ ∇ψ] = −(8πG/c2 )(ρ − ρ̄), which pins ψ(x) to
the matter source and admits no free clustering residual
decoupled from ρ. (iii) The physical metric rescales its
temporal and spatial parts oppositely, g̃00 = −c2 e−ψ and
g̃ij = e+ψ δij (Eq. (12)), so it is not a conformal (mimetic)
rescaling of a base metric; and the preferred reference ψ̇0
in the temporal deviation invariant ∆ = (c/a0 )|ψ̇ − ψ̇0 |
explicitly breaks the time-reparametrization invariance
t → f (t) that the projectable mechanism requires.
The label is therefore: the integration-constant-dust route
is a viable DFD-adjacent completion—“DFD + a projectable optical-clock constraint sector”—and not a theorem of minimal DFD. Realizing it would require explicitly
adding R(a) N (t) as a variational lapse with the global constraint d3 x H = 0; (b) a proof that the resulting residual
propagates as cold dust without triggering the Hořava–
Lifshitz scalar-graviton strong-coupling pathology; and
(c) a forced (not Planck-number-matched) abundance. We
do not adopt the projectable-clock OIC route: an a−3

172
cold-dust integration constant is effective cold dark matter, and the dynamical projectable lapse it requires contradicts DFD’s absolute-time postulate — precisely the
non-DFD content (a different clock) that minimal DFD
exists to avoid. The coincidence ωI = (16/3) ωb ≃ 0.12 is
“a clue, not a derivation”: the cold abundance is derived
in App. AV by a different mechanism, not through the
19 = 3+6+10 split. Minimal DFD’s conclusion is therefore: the cold clustering component is neither a postulated
particle nor a projectable-lapse integration constant, but
the derived χ field — the harmonic b3 three-form, a matter field on the fixed background that leaves absolute time
untouched. The third-peak height is closed by its abundance Ωχ h2 ≃ 0.12, derived conditional on the flux-period
identification, the stated count 158, P4′ and two stated
conventions. The projectable-clock OIC route remains
only a rejected adjacent alternative.
9.

The Cold-Clustering Wall: a no-go theorem for
the third-peak height

The prose above asserts that the third-peak height
requires a pressureless clustering component, while the
peak location does not. That assertion is the main reason
DFD introduces the derived χ field rather than a further
optical or screening device. We now state and prove it as
a theorem. The theorem is the rigorous statement of the
“wall” onto which dozens of no-CDM mechanisms have
converged; it is what the derived χ-matter field (App. AV)
is engineered to satisfy, not evade.
a. Setup and definitions. Work in Newtonian gauge
on the spatially flat FRW background, in Fourier space
at comoving
R η wavenumber k, with conformal time η and
rs (η) = 0 cs dη ′ the sound horizon. The tightly coupled photon–baryon monopole obeys the forced acoustic
oscillator (corpus Eq. (J1), equivalently Hu–Sugiyama)
i
dh
(1 + Rb ) Θ̇ + c2s k 2 (1 + Rb ) Θ
dη
i
k2
dh
(1 + Rb ) Φ̇(k, η) ,
= − (1 + Rb ) Φ(k, η) +
3
dη
(J29)
with Rb = 3ρb /4ργ , c2s = 13 (1 + Rb )−1 , and Φ the metric
potential sourced by the total clustering stress through
the relativistic Poisson (Einstein 00) equation
X
k 2 Φ = −4πGa2
ρ̄s δs ≡ −4πGa2 ρ̄ ∆,
(J30)
s

the sum running over every species s present at recombination. The observed effective temperature at last scattering
is the Sachs–Wolfe combination (Θ + Φ)(k, η∗ ), and the
2
n-th acoustic peak height is Hn ∝ (Θ + Φ) evaluated at
kn rs (η∗ ) = nπ. We write zeq for matter–radiation equality, zrec ≃ 1090 for recombination, and define a species to
be cold-clustering if it has, at recombination, equation of
state ws ≃ 0 and sound speed c2s,s ≃ 0 (pressureless, so it
carries no Jeans scale above the CMB damping scale).

Theorem J.1 (Cold-Clustering Wall). Let the thirdpeak height ratio be lifted from the baryon-only value
(H3 /H1 )b ≈ 0.20 to the observed (H3 /H1 )obs ≈ 0.44
(H3 /H2 ≈ 1.0, Planck) while the peak locations kn rs =
nπ and the baryon-loading ratio structure are held fixed.
Then the driving potential Φ(k, η) in (J29) must contain
a contribution that is (i) non-decaying (a standing well,
Φ → const) on sub-horizon scales through last scattering, and (ii) sourced by a species that is cold-clustering
at recombination with matter–radiation equality preceding
recombination, zeq ≳ zrec . The minimal abundance of
that species is
Ωc h2 ≳ Ωr h2 (1 + zrec ) − Ωb h2 ≈ 0.023

(standing-well floor),

(J31)
and matching the observed height H3 /H1 ≈ 0.43–0.44
sharpens this to Ωc h2 ≈ 0.10–0.12. No radiation-pressuresupported component (c2s,s > 0) and no time-only modification of the homogeneous background can supply the
lift.
Proof. The argument has four steps. Steps 1–2 establish
that only a non-decaying (Φ →const) sub-horizon well lifts
H3 /H1 ; Steps 3–4 close the two evasion classes (radiationpressure-supported, and time-only-background).
Step 1 (the height is set by the surviving well, not
by the oscillator). The general solution of (J29) at
fixed k is the homogeneous acoustic oscillation about the
displaced zero point plus the particular (forced) response.
The forced piece is governed entirely by Φ(k, η) near
and after the mode’s horizon crossing. Decompose the
potential into the part that has decayed by recombination,
Φdec , and the part that survives as a standing well, Φ∞ ≡
Φ(k, η∗ ). The two parts act through distinct channels.
For a mode well inside the horizon at recombination
(krs ≫ 1, which includes the third peak, k3 rs = 3π ≈ 9.4),
the Hu–Sugiyama radiation-driving boost of the acoustic
amplitude over the no-driving baseline is sourced by the
decaying part: the near-resonant decay of Φdec at horizon
crossing pumps the oscillation amplitude. The surviving
well enters through the zero point: a sustained Φ∞ ̸= 0
displaces the oscillation zero point and adds coherently
to Θ + Φ at last scattering. Quantitatively, the thirdpeak power scales monotonically with the fraction of the
clustering source that is in a standing well at η∗ ; with
no standing well it collapses to the baryon-only baseline
H3 /H1 ≈ 0.20.
Step 2 (a standing sub-horizon well requires a
pressureless clusterer with equality before recombination). A standing well, Φ →const on sub-horizon
scales, is possible iff the dominant clustering source
in (J30) is pressureless. For a cold (c2s,s = 0) species
there is no Jeans term, the Meszaros/matter-era solution
gives Φ =const exactly (super- and sub-horizon alike),
and the well persists to recombination. This in turn
requires that the cold species already dominate the perturbation source at η∗ , i.e. that equality precede recombination, zeq ≳ zrec . Since 1 + zeq = Ωm h2 /Ωr h2 with
Ωr h2 ≃ 4.15 × 10−5 (photons+3ν), the floor zeq ≥ zrec

173
gives Ωm h2 ≥ Ωr h2 (1 + zrec ) ≈ 0.045, hence (J31).
(b)
Baryons alone give 1 + zeq = Ωb h2 /Ωr h2 ≈ 539 < 1 + zrec :
baryon-only equality occurs after recombination, so the
baryon well is still decaying at last scattering and cannot stand. Mapping the surviving-well depth (Step 1)
to the peak height through the Hu–Sugiyama radiationdriving boost, the observed H3 /H1 ≈ 0.43–0.44 is reached
only for Ωc h2 ≈ 0.10–0.12 (numerically verified against
the corpus CAMB anchors Ωc h2 =0→[1, 0.39, 0.205] and
Ωc h2 =0.12→[1, 0.45, 0.44]). This proves the existence and
the abundance claims.
Step 3 (radiation-pressure-supported mechanisms
cannot do it — they drive their own well to zero).
Suppose the proposed lift is carried by a component with
c2s,s > 0 (a relativistic or sound-supported fluid: photon
driving, a hot relic, the optical cs (ψ) envelope, an AQUALenhanced radiation term). In radiation/pressure domination the potential sourced by that component obeys,
exactly,
kη
sin x − x cos x
,
x≡ √ ,
(J32)
Φ(x) = 3 Φ0
3
x
3
whose super-horizon limit is Φ(0) = Φ0 (well intact) but
whose sub-horizon envelope is Φ ≃ −3Φ0 cos x/x2 →
0. The pressure that supports the component against
collapse simultaneously drives its own gravitational well
away once the mode enters the horizon (verified: |Φ/Φ0 |
falls as 0.024, 0.0027, 0.0013 at x = 10, 20, 40). Hence
Φ∞ = 0 for any pressure-supported source, the standingwell condition of Step 2 fails, and by Step 1 no third-peak
lift results. The deep-MOND gravity enhancement, the
cs (ψ) envelope, the impedance/feedback gain, and the
forced-echo device all fall in this class or in Step 4 and are
therefore closed by this step (they modify the transient
driving or the background, not the surviving well).
Step 4 (time-only background modifications cancel in the height; achromatic driving boosts all
peaks equally). Suppose instead the proposed lift acts
only through the homogeneous background — a modified
H(η), a time-dependent sound speed cs (η) with no spatial
structure, or an achromatic (k-independent) rescaling of
the driving, Φ → g(η) Φ. Any such modification multiplies the forced amplitude of every mode by a common, kindependent factor. For an achromatic boost Φ → g Φ the
peak heights scale as Hn → g 2 Hn , so every ratio Hn /H1
is left exactly invariant (verified to machine precision in
the forced oscillator: H2 /H1 and H3 /H1 unchanged under g = 2.7). A time-only cs (η) or H(η) enters rs and the
damping envelope but, being independent of k̂ · ĝ and of k
at fixed krs , cancels in Hn /H1 at fixed peak location by
the same argument that makes the 1/µ optical enhancement cancel in the peak ratio (§J 4). The only way a
background device can change H3 /H1 is by shifting zeq —
but zeq is a statement about which species dominates the
perturbation source (J30), i.e. exactly the cold-clustering
abundance of Step 2, not a free background knob. This
closes the time-only class.
Steps 3 and 4 exhaust the two ways a non-cold mechanism

could act (through the transient/pressure-supported driving, or through the homogeneous background); both fail.
By Steps 1–2 the lift exists only with a pressureless standing well of abundance (J31), sharpened to Ωc h2 ≈ 0.10–
0.12 by the observed height.
Remark J.2 (What the theorem does and does not say).
The theorem is a statement about the perturbation source,
not about the ontology of the clustering species. It does
not say “DFD needs a postulated WIMP.” It says: any
theory matching the Planck third-peak height must place
an Ωh2 ≈ 0.10–0.12 pressureless, cold-clustering charge
in the gravitational source (J30) at recombination, with
equality before last scattering. In ΛCDM that charge is
a postulated cold particle of fitted density; in DFD it is
the derived √χ-matter field (App. AV), whose mass uses
the stated 158 multiplicity. The required abundance
Ωχ h2 ≃ 0.12 is the kernel input a cold component of
that density supplies; DFD derives that normalization
(post-inflation, App. AV Step 5b: the finite SU (2)60 CSvacuum Casimir expectation gives Ωχ h2 = 0.1193, −0.6σ,
conditional-derived (relic bridge exact via κrad ; named
open inputs the two stated conventions, the stated count
158, the flux-period identification and the asserted premise
P4′ , Lemma AV.27); a ∼9–44× overshoot is a units (4π 2 )
artefact). The wall is therefore not a defeat for DFD but
the precise specification the χ field is shown to meet: it
enters (J30) identically to a cold component of the same
Ωh2 (Theorem AV.31), so it sits on the right side of the
wall by construction.
Remark J.3 (Why this consolidates the no-CDM constructions). Every DFD-native no-CDM third-peak mechanism
the corpus has tested is closed by a named step of Theorem J.1 or by amplitude: the chromatic-Jacobian lineof-sight ψ-screen, the deep-MOND gravity enhancement,
the nonlinear AQUAL injection, and the forced soundecho are pressure-supported / transient devices closed by
Step 3 (they cannot leave a standing Φ∞ ̸= 0); the cs (ψ)
sound-speed envelope and any modified-H(η) background
are time-only devices closed by Step 4 (they cancel in
Hn /H1 or merely rescale all peaks). The rectified-DC
channel of Prop. NL.29 (an even-harmonic drive on the
odd deep response) is closed separately, by amplitude:
≤ 0.0166 of the baryon potential against νreq = 6.364327,
short by ≥ 380× (Status NL.31). The heavy-mode and
topological-defect routes are closed upstream (no free cold
a−3 channel in the 4D spectrum). The single remaining
carrier of a pressureless standing well is a cold a−3 matter field with equality before recombination — which is
precisely the derived χ field. This is the theorem-grade
form of the empirical convergence recorded across the
no-CDM investigation: the third-peak height is a coldclustering charge, and no time-only or pressure-supported
DFD device can carry it.

174
Appendix K: Microsector Physics: Complete
Derivations

This appendix provides complete derivations for the
DFD microsector results presented in Section XVII. These
results connect the fine-structure constant, fermion mass
spectrum, and quark mixing to the topological structure
of the gauge emergence framework on CP 2 × S 3 .

b.

Twist bundle.

Choose:

E = O(9) ⊕ O⊕5 .

(K5)

The holomorphic 
 Euler characteristic satisfies
χ(CP 2 , O(m)) = m+2
for m ≥ 0. Therefore:
2
 
11
χ(O(9)) =
= 55,
χ(O) = 1,
(K6)
2
and
kmax = χ(E) = χ(O(9)) + 5χ(O) = 55 + 5 = 60 (K7)

1.

Closed Form for α = 1/137 from Chern-Simons
Theory
a.

Setup: Chern-Simons on S 3

The S 3 factor in the internal manifold M7 = CP 2 × S 3
supports Chern-Simons gauge theory. For U(1) gauge
fields, the action is:
Z
k
A ∧ dA,
(K1)
SCS =
4π S 3
where k ∈ Z is the quantized level (gauge invariance under
large gauge transformations requires integer k).

b.

The Level Sum and Fine-Structure Constant

The effective coupling receives contributions from all
Chern-Simons levels. The effective coupling βU (1) = ⟨k +
2⟩ is computed from a weighted sum:
Pkmax −1
(k + 2) w(k)
βU (1) = k=0
,
(K2)
Pkmax −1
w(k)
k=0
2
π
where w(k) = k+2
sin2 k+2
are the SU(2) Chern–Simons
weights.

c.

Heat Kernel on S 3

The heat kernel on S 3 with radius R has the spectral
expansion:
∞
X
2
K(t; S 3 ) =
(n + 1)2 e−n(n+2)t/R .
(K3)

c. Physical selection. The value kmax = 60 is consistent with the microsector physics. The effective coupling
βU (1) ≡ ⟨k + 2⟩, computed from the SU(2) Chern–Simons
weights
w(k) =

2
π
sin2
,
k+2
k+2

(K8)

is the lattice input βU (1) ≈ 3.80 for UV truncation at
kmax = 60. Here levels run k = 0, 1, . . . , kmax − 1 (standard SU(2) WZW/CS convention), giving:
P59
(k + 2) w(k)
⟨k + 2⟩kmax =60 = k=0
= 3.7969 ≈ 3.80.
P59
k=0 w(k)
(K9)
Bridge Lemma (Final Form)
Index: kmax = χ(CP 2 , E) = 55 + 5 = 60 [Spinc HRR]
Physics: βU (1) = ⟨k + 2⟩ = 3.797 at kmax = 60; lattice
comparison with α−1 = 137 to about 1%
Icosahedral: kmax = 60 = |A5 |
E8 echo: roots(E8 )/4 = 240/4 = 60 ✓

2.

Lattice Comparison for α = 1/137

The analytical closed form for α is compared with
lattice Monte Carlo simulations. This section presents
the logic in a way that explicitly avoids circularity: all
inputs are fixed before comparing to α = 1/137.

a.

Inputs (Independent of α)

n=0
2

The (n + 1) factor is the degeneracy of the n-th eigenvalue λn = n(n + 2)/R2 .
d.

Determination of kmax : Closed Spinc Index

The maximum Chern-Simons level is defined as a
closed Spinc index on CP 2 .
a. Setup. For the canonical Spinc structure on CP 2
(determinant line
the Spinc Dirac operator
√Ldet = O(3)),
∗
¯
¯
identifies with 2(∂ + ∂ ). By Hirzebruch–Riemann–
Roch:
kmax := Index(DCP 2 ⊗ E) = χ(CP 2 , E).

(K4)

The following quantities are fixed by geometry and
topology, with no reference to the observed value of α:
a. (1) UV cutoff from topology. The maximum
Chern-Simons level is derived from the closed Spinc index
on CP 2 :
kmax = χ(CP 2 , E) = χ(O(9)) + 5χ(O) = 55 + 5 = 60.
(K10)
See Bridge Lemma (Sec. K 4) for the derivation.
b. (2) Chern-Simons expectation value. With the
2
standard CS weight function w(k) = k+2
sin2 (π/(k + 2)):
βU (1) = ⟨k + 2⟩kmax =60 = 3.7969 ≈ 3.80.
This is a calculable number once kmax is fixed.

(K11)

175
c. (3) Stiffness ratio from Ricci curvature. From
Theorem F.25:
κU (1)
n1
1
=
= .
(K12)
κSU (2)
n2
2
d. (4) Wilson ratio from topology. The Wilson action ratio follows from the stiffness ratio and generation
number:
βSU (2)
n2
=
× Ngen = 2 × 3 = 6.
(K13)
βU (1)
n1
The factor of Ngen = 3 enters because all three generations
contribute equally to the effective lattice coupling.
e. (5) Derived lattice parameters. Combining these
inputs:
βU (1) = 3.80,

(K14)

βSU (2) = 6 × 3.80 = 22.80.

(K15)

These values are predictions, not fits.

b.

the plaquette couplings by w(k)/w(1); it relaxes to k = 0
on every site, where the weight is 2, so both couplings are
doubled and kmax enters only through βU (1) . The SU(2)
stiffness is read with a background of fixed color direction;
averaged over gauge orientations, this estimator returns
one third of the tree-level stiffness. In this normalization
1/αW = 4π(2βU (1) +βSU (2) /6) up to a correction of about
−6, so the lattice checks the chain of inputs and does not
determine α independently.

d.

The prediction is falsifiable at multiple points:
TABLE LXXXI. Sensitivity to the inputs. The kmax = ∞
row’s α is the lattice run at βU (1) = 3.95; the two Wilson rows
ran at βU (1) = 3.77.
Input changed
kmax = ∞
Wilson = 5
Wilson = 7

The Prediction

With these parameters the lattice coupling is compared
with α:
αW ≃ α =

Falsifiability: What Would Have Failed

Value
βU (1) = 3.94
βSU (2) = 18.85
βSU (2) = 26.39

Result α
1/147 (−6.8%)
1/125 (+10%)
1/147 (−7%)

Status
Excluded
Excluded
Excluded

The theory would have failed if:

1
137.036

(K16)

• kmax → ∞ (no cutoff) from the topological index

No continuous fit parameters. Given the discrete
topological sector (twist bundle E, generation number
Ngen ), the inputs (kmax , stiffness ratio) are fixed by geometry. Wilson ratios 5 and 7 give the wrong α (Table LXXXI); for kmax the lattice excludes only the infinite
sum, since values from about 45 to 100 give βU (1) inside
the bracket of Table LXXXV.

• Wilson ratio ̸= 6 from the topological derivation

c.

e.

TABLE LXXX. Lattice results at βU (1) = 3.80: within 0.9%
of α = 1/137 for L6–L12, +1.13% at L16. The L8, L10, L12
and L16 means include one, two, one and three runs whose
U(1) stiffness kept hot-start relics.
L nused αW (mean)
5
5
4
3
9

• Lattice measurement ̸= 1/137 at βU (1) = 3.80 (measured: within 0.9% for L ≤ 12, +1.1% at L = 16)

Lattice Comparison

The lattice simulations are compared with this prediction at βU (1) = 3.80; the L6–L10 runs used βSU (2) = 23.0
and the L12 and L16 runs 22.80:

6
8
10
12
16

• Stiffness ratio ̸= 1/2 from the Ricci curvature argument

0.007297
0.007322
0.007361
0.007318
0.007380

σα

∆α/α

9.4 × 10−5 −0.00%
9.5 × 10−5 +0.34%
6.8 × 10−5 +0.88%
5.1 × 10−5 +0.28%
1.1 × 10−4 +1.13%

The simulation is a three-dimensional periodic lattice
(L3 sites) with Metropolis updates and separate U(1) and
SU(2) runs. An integer level field k on the sites weights

Finite-Size Scaling

Finite-size effects were tested across lattice sizes L = 6–
16:
TABLE LXXXII. Lattice results at βU (1) = 3.80, with
βSU (2) = 23.0 for L6–L10 and 22.80 for L12 and L16. The L8,
L10, L12 and L16 means include one, two, one and three runs
whose U(1) stiffness kept hot-start relics.
L Therm nused /ntotal αW (mean)
6
6k
8 3k–6k
10
3k
12
8k
16 40k

5/5
5/6
4/5
3/5
9/10

0.007297
0.007322
0.007361
0.007318
0.007380

σα

∆α/α
−5

9.4 × 10
−0.00%
9.5 × 10−5 +0.34%
6.8 × 10−5 +0.88%
5.1 × 10−5 +0.28%
1.1 × 10−4 +1.13%

176
f.

L16 Detailed Results

The L = 16 lattice requires increased thermalization
(40k sweeps) due to longer autocorrelation times. Ten
independent runs give:

TABLE LXXXV. β bracket test. Values 3.75–3.85 all yield
α ≈ 1/137. The 3.80 entry is the L6 mean of Table LXXXII,
run at βSU (2) = 23.0, and the 3.77 entry pools twelve of the
seventeen runs at βSU (2) = 22.62 on L6, L8 and L10, two of
them (both L10) with hot-start relics (all seventeen: +1.5%).
βU (1)

TABLE LXXXIII. L16 individual runs with 40k thermalization.
One outlier (s5) excluded due to incomplete equilibration: its
U(1) run kept relics of the hot start. Seeds s3, s4 and s9
show the same relic effect at smaller size and are kept in the
nine-run mean. The nine-run mean is +1.13 ± 0.50% (2.3
standard errors).
Seed

αW

Deviation κratio

s0
s1
s2
s3
s4
s5
s6
s7
s8
s9

0.007194
0.007553
0.007449
0.007480
0.007421
0.008429
0.007303
0.007298
0.007359
0.007359

−1.42% 0.476
✓
+3.51% 0.552
✓
+2.08% 0.528
✓
+2.51% 0.508
relic
+1.69% 0.444
relic
+15.51% 0.431 × (outlier)
+0.08% 0.496
✓
+0.01% 0.496
✓
+0.85% 0.509
✓
+0.84% 0.499
relic

Wilson Ratio Scan

Ten ratios βSU (2) /βU (1) were tested; of these ten only
ratio 6 is consistent, and two further runs at ratio 5.75
give −0.20% and +2.20% (mean +1.0%; Table LXXXVI).
TABLE LXXXIV. Wilson ratio scan at βU (1) = 3.80, except
the ratio-5 and ratio-7 rows, which ran at βU (1) = 3.77. Of
these ten ratios only 6 yields α = 1/137; the other nine fail.
The ratio-6.25 row averages two of the three runs at that ratio;
all three average 0.007073 (−3.1%).
βSU (2) /βU (1) βSU (2)
3
4
5
5.5
6.05
6.25
6.5
7
8
9

11.40
15.20
18.85
20.90
23.0
23.75
24.70
26.39
30.40
34.20

h.

Deviation

3.75 0.007172
3.77 0.007391
3.80 0.007297
3.85 0.007256
3.95 0.006801

−1.7%
+1.3%
∼ 0%
−0.6%
−6.8%

Status

Mean (9 runs) +1.13% 0.501

g.

αW

αW

Deviation

0.008907
0.008234
0.008005
0.007549
0.00730
0.007091
0.007063
0.006797
0.006400
0.006065

+22.1%
+12.8%
+9.7%
+3.5%
∼ 0%
−2.8%
−3.2%
−6.9%
−12.3%
−16.9%

β Bracket Test

The result is robust across a range of βU (1) values:

i.

Additional Runs

Further runs give:
TABLE LXXXVI. Additional runs. All results within expected
uncertainty.
Run ID

βU (1)

αW

Primary runs
GK 377 L6 s12
GK 377 L6 s13
GK 380 L12 s0
GK 380 L12 s1
GK L8 380 s6

3.77 0.007395
3.77 0.007411
3.80 0.007269
3.80 0.007313
3.80 0.007318

Deviation
+1.34%
+1.56%
−0.38%
+0.21%
+0.28%

k0 independence tests (L=6)
GK k0 4 L6
3.80 0.007217 −1.11%
GK k0 12 L6
3.80 0.007334 +0.51%
GK k0 16 L6
3.80 0.007334 +0.50%
Metropolis step size tests
3.80 0.007235 −0.85%
GK eps025 L6
GK eps045 L6
3.80 0.007141 −2.15%
Wilson ratio scan (L=6)
GK RATIO5p75 L6
3.80 0.007283 −0.20%
GK RATIO5p75 L6 s1 3.80 0.007458 +2.20%

The k0 tests vary the starting value of the level field,
which relaxes to k = 0 from every start. The Metropolis
step size tests confirm algorithmic stability.

177
j.

3.

Summary: Lattice Evidence

The UV Cutoff Consistency Check: kmax = 60
Compared with the Lattice

Lattice Comparison Summary
86 total runs across L = 4, 6, 8, 10, 12 lattice sizes give:
• α = 1/137 within 0.9% at βU (1) = 3.80 for L ≤ 12
• UV cutoff kmax = χ(CP 2 , E) = 60 (from Spinc
index); for kmax → ∞ (βU (1) = 3.94) the run at
βU (1) = 3.95 gives α = 1/147 (−6.8%)
• Wilson ratio = 6 derived from (n2 /n1 ) × Ngen ; of
the 10 scanned ratios only 6 fits; two further runs
at 5.75 give −0.20% and +2.20% (mean +1.0%)
All inputs fixed by topology (given the discrete
bundle choice). α = 1/137 follows with no continuous
fit parameters.

The Chern-Simons level sum requires a UV cutoff at
kmax = 60, fixed α-free by the conditional determinantline count (Rem. F.12, Lemma F.11). As a cross-check,
scanning truncation values against lattice simulations
shows that the counted kmax = 60 reproduces α = 1/137
(the run at 3.95, next to the infinite sum’s βU (1) = 3.94,
gives 1/147) — an a-posteriori consistency test, not the
source of the value.

a.

The Truncation Scan

The expectation value ⟨k+2⟩ depends on the truncation
point:

0.0076

TABLE LXXXVII. Truncation scan: only the truncated sum
yields α = 1/137.

0.0074

kmax ⟨k + 2⟩
0.0072

αW

50
60
∞

Predicted α−1

Status

3.77
135.3 (+1.3%)
Close
3.80
137.0
Counted value; consistent
3.94 147 at β = 3.95 (−6.8%)
Excluded

0.0070
±1% of αphys
αphys = 1/137.036
L = 6, β = 3.77 (two runs)

0.0068

run at β = 3.95 (L = 4): αW = 1/147
L = 6, β = 3.80 (mean ± σ)
L = 8, β = 3.80 (mean ± σ)
L = 10, β = 3.80 (mean ± σ)

0.0066

3.75

3.80

3.85

3.90

3.95

4.00

βU(1) = ⟨k + 2⟩

FIG. 14. Lattice αW against βU (1) , with the line and ±1%
band of the measured value. The single L = 4 run at β = 3.95,
next to the converged value 3.94, gives αW = 1/147 (−6.8%).

0.0095

Wilson ratio scan
phys = 1/137

+22.1%

0.0090
+12.8%

0.0085

b.

W

+3.5%

0.0075

The truncation is not arbitrary. In Chern-Simons theory, the effective coupling scales as g 2 ∼ 1/k:

+0.0%

0.0070
-2.8%

0.0065

-3.2%
-6.9%

0.0060

-12.3%
-16.9%

3

4

Physical Interpretation

+9.7%

0.0080

0.0055

The converged sum (kmax → ∞) gives β = 3.9386; the
nearest lattice run, a single L = 4 run at β = 3.95, yields
α = 1/147 (−6.8%), and equilibrium runs at β = 3.94 give
about 1/142. In the lattice’s normalization this excludes
the infinite sum and is consistent with the counted kmax =
60 as the physical UV cutoff. The value comes from the
conditional determinant-line count (χ(O(9) ⊕ O⊕5 ) =
60, Lemma F.11), and the Bridge Lemma states it as a
closed Spinc index (Sec. K 4); the sparse scan excludes
the infinite sum, and a finer integer-by-integer scan over
kmax ∈ [40, 80] would show which values in that range the
lattice admits.

5

5.5

6

6.25

SU(2)/ U(1)

6.5

7

8

9

FIG. 15. Wilson ratio scan. Ten ratios tested (3–9 including
fractional values). Of the ten, only ratio 6 yields α = 1/137;
the other nine miss it by 2.8% or more. The ratio-6 bar is the
L = 6 mean at βSU (2) = 23.0 (ratio 6.05), the ratio-6.25 bar
averages two of the three runs at that ratio (all three: −3.1%),
and the ratio-5 and ratio-7 bars ran at βU (1) = 3.77.

• Low-k sectors (k ≲ 60): Strongly quantum, large
fluctuations—“loud” modes that dominate vacuum
stiffness.
• High-k sectors (k > 60): Weakly coupled, nearly
classical—“quiet” modes that are frozen out of relevant physics.
This is analogous to UV regularization in effective field
theory: high-energy/high-k modes exist mathematically
but decouple from low-energy observables. In DFD the
counted kmax = 60 is the physical cutoff for the ChernSimons vacuum.

178
c.

4.

Why This Is Not Fine-Tuning

The β bracket test (Table LXXXV) demonstrates that
values 3.75–3.85 all yield α ≈ 1/137 within ∼ 2%:

The Bridge Lemma

The Bridge Lemma identifies kmax = 60 as a closed
Spinc index on CP 2 .

• β = 3.75: α = 1/139.4 (−1.7%) — acceptable
• β = 3.80: α = 1/137.0 (∼ 0%) — best
• β = 3.85: α = 1/137.8 (−0.6%) — acceptable
• β = 3.95: α = 1/147 (−6.8%)
d.

Systematic Independence Tests

To address potential concerns about simulation parameter dependence, we tested independence from two key
algorithmic choices:
a. Initial level (k0 ). The level field starts at k = k0
on every site and relaxes to k = 0:

a.

Statement

Theorem K.1 (Bridge Lemma (Closed Index Form)).
For the canonical Spinc structure on CP 2 with twist bundle E = O(9) ⊕ O⊕5 :
kmax = Index(DCP 2 ⊗ E) = χ(CP 2 , E) = 60.
b.

Proof

For the canonical Spinc structure
on CP 2 , the Spinc
√
¯
Dirac operator identifies with 2(∂ + ∂¯∗ ). Twisting by a
holomorphic bundle E gives:
Index(DCP 2 ⊗ E) = χ(CP 2 , E)

TABLE LXXXVIII. Independence from the initial level k0 .
k0

αW

Deviation

4
0.007217 −1.11%
8 (default) 0.00730
∼ 0%
12
0.007334 +0.51%
16
0.007334 +0.50%

All values agree within 1.1%.
b. Metropolis step size (ε). The SU(2) simulation
uses Metropolis updates with proposal size ε:

(K17)

(K18)

c

by the Spin version of Hirzebruch–Riemann–Roch.
The holomorphic Euler characteristic on CP 2 satisfies:


m+2
2
0
2
χ(CP , O(m)) = h (CP , O(m)) =
for m ≥ 0.
2
(K19)
(Higher cohomology vanishes.) Therefore:
 
11
χ(O(9)) =
= 55,
(K20)
2
χ(O) = 1,

(K21)

and
TABLE LXXXIX. Independence from the Metropolis step size.
ε

αW

kmax = χ(E) = χ(O(9)) + 5χ(O) = 55 + 5 = 60.
(K22)

Deviation

0.25
0.007235 −0.85%
0.35 (default) 0.00730
∼ 0%
0.45
0.007141 −2.15%

All values agree within 2.2%, confirming algorithmic
stability.
UV Cutoff Finding
The counted value kmax = 60 passes these lattice checks:
• The truncated sum (kmax = 60) yields α = 1/137
within 0.9% for L ≤ 12 (+1.1% at L = 16)
• The converged sum (kmax → ∞, βU (1) = 3.94) is
excluded by the run at βU (1) = 3.95, which yields
α = 1/147 (−6.8%)
• Ten Wilson ratios tested (3–9 incl. fractional): of
the ten, only ratio 6 works (Table LXXXIV); two
further runs at 5.75 give −0.20% and +2.20%
(mean +1.0%; Table LXXXVI)
• Five βU (1) values tested: 3.75–3.85 lie within 2% of
1/137, and the run at 3.95 next to the converged
value is 6.8% low (Table LXXXV)
• The result is independent of simulation parameters
(k0 , ε)

c.

Physical Selection

The value kmax = 60 is consistent with the microsector
physics. The effective coupling βU (1) = ⟨k + 2⟩, computed
2
π
from the CS weights w(k) = k+2
sin2 k+2
, is the lattice
input βU (1) ≈ 3.80 for kmax = 60. Here levels run k =
0, 1, . . . , kmax − 1:
P59
(k + 2) w(k)
⟨k + 2⟩kmax =60 = k=0
= 3.7969 ≈ 3.80.
P59
k=0 w(k)
(K23)

d.

Quantity

Consistency Checks

Derivation

Echo

kmax = 60 χ(O(9)) + 5χ(O) roots(E8 )/4 = 240/4
kmax = 60 CS weight selection |A5 | (icosahedral)

179
5.

• 3rd gen τ (n = 1.0): Lepton gauge path introduces
one power of α

Charged Fermion Mass Derivation
a.

The Mass Formula

All nine charged fermion masses follow the unified formula [145]:
v
mf = Af · αnf · √ ,
2

(K24)

where:
• α = 1/137.036 is the fine-structure constant (from
kmax = 60; bracket coefficient open)
√
• v/ 2 = 174.1 GeV is the Yukawa normalization
scale
• nf is a sector-dependent exponent determined
by the fermion’s coupling path on CP 2
• Af is a prefactor, data-selected except for the top
and the charm (App. FM)

b.

Sector-Dependent Exponent Assignment

The three fermion generations are localized at the three
vertices of CP 2 (the fixed points of the (Z/3Z)2 action).
The Higgs field is localized near the third-generation
vertex.
The exponents nf are sector-dependent, not uniform
across leptons and quarks. This arises from the different
Yukawa coupling paths:
• Up-type quarks couple to the conjugate Higgs
H̃ = iσ2 H ∗
• Down-type quarks couple directly to H
• Leptons couple through a gauge path with an additional step

2

TABLE XC. Sector-dependent exponents nf from CP localization.
1st gen
2.5
2.5
2.5

2nd gen
1.5
1.0
1.5

• 2nd gen down/leptons (n = 1.5): Standard intermediate distance
a. Quantitative origin of the exponents. The
geodesic-distance picture above is the heuristic for a
derived formula [145]: the bare exponents arise from
spinc line-bundle degrees on CP 2 ,
kf + k H
nfbare =
,
nf = nfbare + ∆nf ,
(K25)
2
with kf ∈ Z the fermion bundle degree, kH = +1 for
e
H-coupling (leptons, down-type) and kH = −1 for Hc
coupling (up-type); the factor 1/2 is the spin signature
(keff = kf + c1 (Ldet )/2). The degree assignments are
(kτ , kµ , ke ) = (1, 2, 4), (kt , kc , ku ) = (1, 3, 6), (kb , ks , kd ) =
(1, 2, 4), reproducing every exponent in the table above.
The single correction ∆nb = −1 (all other ∆nf = 0)
arises from color-vertex 3saturation on S 3 at the bottomquark vertex, where α3S = 1/(k3 + h∨
3 ) = 1/4; the shift
is quantized by the integer representation dimension plus
Chern–Simons level [145].

c.

Prefactor Structure

The prefactors Af are assembled from the operator
algebra constructed below: Af = G[g, g] × (sector factor),
where G = diag(2/3, 1, 1) is the generation operator from the primed
microsector trace (Theorem K.4),
√
Dℓ = diag(1, 1, 2) is the Dirac normalization, Ru =
Tr(I4 ) = 4 and Rd = 9 are the sector factors, and
Qd = diag(1, Nf /b0 , 1/(Nf b0 )) = diag(1, 6/7, 1/42) with
the exact QCD integers b0 = 7, Nf = 6 (see the explicit
Yukawa-operator construction below and [145]).
TABLE XCI. Prefactors Af in closed form.

The resulting exponent structure is:

Leptons
Up-type quarks
Down-type quarks

• 2nd gen charm (n = 1.0): Conjugate Higgs H̃
coupling shortens the path

3rd gen
1.0
0
0

The physical interpretation:
• 1st generation (n = 2.5): Maximum geodesic
distance from Higgs vertex
• 3rd gen quarks (n = 0): Direct coupling at the
Higgs vertex, no α suppression

1st gen.
2/3
8/3
6

Leptons
Up-type quarks
Down-type quarks

2nd gen.
1
1
6/7

d.

Complete Mass Table

e.

Statistical Summary

3rd√gen.
2
1
1/42

• Mean absolute error: 1.42% at leading order (a
refinement to 0.61% by generation-1 priming is
claimed in the companion volume; the 0.082% after
two-loop running does not stand, since the running
factors it uses are not Standard-Model running)

180
f.

TABLE XCII. Charged fermion mass predictions from
Eq. (K24).
Fermion nf
e
µ
τ
u
c
t
d
s
b

Af

Predicted
Observed
Error
Charged Leptons
2.5 2/3 0.528 MeV
0.511 MeV
+3.32%
1.5 √1 108.5 MeV
105.66 MeV
+2.72%
1.0
2 1.797 GeV
1.777 GeV
+1.12%
Up-Type Quarks
2.5 8/3 2.11 MeV
2.16+0.49
−2.23%
−0.26 MeV
1.0 1 1.270 GeV 1.27 ± 0.02 GeV +0.04%
0
1 174.1 GeV 172.76 ± 0.30 GeV +0.78%
Down-Type Quarks
2.5 6
4.75 MeV
4.67+0.48
+1.75%
−0.17 MeV
1.5 6/7 93.0 MeV
93+11
+0.03%
−5 MeV
0 1/42 4.15 GeV
4.18+0.03
−0.83%
−0.02 GeV

Structural Ratios

The selected prefactors have the ratios:
Ad
6
18
=
=
= 2.25,
Au
8/3
8
1
At
=
= 42,
Ab
1/42
√
Aτ
2 √
=
= 2.
Aµ
1
g.

a.

• All nine predictions within 3.4% of observed central
values
√
• One universal normalization v/ 2 = 174.1 GeV
for all nine fermions
√ (leading order; the top is
dressed to (1 − α)v/ 2 = 172.83 GeV by the (1 − α)
EW/QED factor in the refined prediction tables,
App. AT)
a. Derivation
status. The mass formula mf = Af ·
√
αnf · v/ 2 is a self-consistent computational formula, not
a mnemonic. The sector-dependent exponents arise from
the different Yukawa coupling geometries on CP 2 :
• Up quarks couple via H̃ = iσ2 H ∗ (modified vertex)
• Down quarks couple via H directly
• Leptons couple via H through a different gauge
path
The prefactors Af arise from:
Af = (gauge CG) × (A5 class factor) × (generation weight),

(K26)
with explicit values {2/3, 1, 2, 8/3, 6, 6/7, 1/42}; seven
of the nine species rest on data-selected prefactors (caveat
below; App. FM).
What is derived:
√

• The Higgs localization width εH = 3/60 = 0.05
(Theorem H.4)
• The sector-dependent exponent pattern (Table XC)
• The hierarchy pattern m(1) : m(2) : m(3) ∼ α2.5 :
αn2 : αn3

(K28)
(K29)

Explicit Finite Yukawa Operator

Hilbert space.

The finite Yukawa space is:

HF = Hspecies ⊗ Hchirality ⊗ Hgen ⊗ Haux
• Maximum error: 3.32% (electron, leading order)

(K27)

(K30)

where Hgen = span{|1⟩, |2⟩, |3⟩} is the 3-dimensional generation space.
b. Generation operator.
on Hgen

G = diag(2/3, 1, 1)
c.

(K31)

QCD running operator (down-type).

Qd = diag(1, Nf /b0 , 1/(Nf b0 )) = diag(1, 6/7, 1/42)
(K32)
where b0 = (11Nc − 2Nf )/3 = 7 is the 1-loop QCD beta
function coefficient.
d. Dirac normalization (leptons).
√
Dℓ = diag(1, 1, 2) on Hgen
(K33)
Lemma K.2 (Localization–Symmetry Kernel Uniqueness on CP 2 ). Assume (i) chiral modes localized on three
sites P = {p0 , p1 , p2 } ⊂ CP 2 , (ii) S3 symmetry permuting sites, andP(iii) symmetry-respecting quadrature
R
F dµF S = κ i F (pi ). Then, with democratic couCP 2
pling (no diagonal preference), the induced kernel on
V = span{|pi ⟩} ∼
= C3 is unique up to scale:
Kd = λd J3 ,

where J3 =

2
X

|pi ⟩⟨pj |.

(K34)

i,j=0

Proof. S3 invariance requires πKπ −1 = K for all π ∈ S3 .
The commutant of S3 on C3 is span{I3 , J3 }. Democratic
coupling (no diagonal preference) gives K ∝ J3 .
Corollary K.3 (Up-type tangent kernel). If the H̃ channel couples through real tangent T with dimR (T ) = 4 and
residual isotropy O(4), then Ku = λu I4 by Schur’s lemma.
e. Absorbed normalization. The quadrature constant
κ combines with gY εH into a single global scale:
λ = gY εH κ

(K35)

Any rescaling κ 7→ cκ affects all Yukawas uniformly (λ 7→
cλ), so there are no flavor-dependent knobs.
f. Yukawa operator.
X
Πf,R (G ⊗ Kf )Πf,L
(K36)
Y =
f

181
where Kf depends on sector: Kf = Dℓ (leptons), Kf =
Ku (up quarks), Kf = Kd · Qd (down quarks).
g. Computed overlaps. The prefactor is:
Af = ⟨gf |(generation operators)|gf ⟩ × (CP2 factor)
(K37)
f Gen ⟨g|G|g⟩
e
µ
τ
u
c
t
d
s
b

h.

1
2
3
1
2
3
1
2
3

2/3
1
1
2/3
1
1
2/3
1
1

Sector factor

Proof. The diagonal entry r(C3 ; 0, 0) = 8/3 represents the
“same-phase” coupling channel (LH and RH both in generation 1). The off-diagonal sum r(C3 ; 0, 1) + r(C3 ; 0, 2) = 4
represents “different-phase” channels. The ratio equals
the complementary projector fraction (Ngen − 1)/Ngen =
2/3, verifying consistency with Route A.

Af

Dℓ [1, 1] = 1
2/3
Dℓ [2, 2] =√1
√1
Dℓ [3, 3] = 2
2
Ru = 4
8/3
1
1
1
1
Qd [1, 1] × Rd = 1 × 9 6
Qd [2, 2] = 6/7
6/7
Qd [3, 3] = 1/42
1/42

Derivation of G[1, 1] = 2/3 from Primed Microsector
Trace

The entry G[1, 1] = 2/3 of the generation operator
G = diag(2/3, 1, 1) is derived from the microsector trace
structure. We present two equivalent derivations.
a. Route A: Primed trace on the 9D generation block
(primary derivation).
Theorem K.4 (Generation Suppression from Primed
Trace). Let Π be the 9-dimensional isotypic block carrying the generation structure (Proposition Y.12), and
let Mr (r = 0, 1, 2) be the generation-r projector with
rank(Mr ) = 3. Under the primed microsector trace prescription (removal of the generation-specific channel), the
first-generation suppression factor is:
G[1, 1] =

Tr(Π − M0 )
9−3
2
=
=
Tr(Π)
9
3

Then the generation suppression equals the diagonal-tooff-diagonal ratio:
W [0, 0]
8/3
8/3
2
G[1, 1] = P
=
=
= . (K41)
2+2
4
3
s̸=0 W [0, s]

(K38)

Proof. The generation projectors {M0 , M1 , M2 } are orthogonal idempotents summing to Π, each with rank 3
(Proposition Y.12). The primed microsector trace removes the “self-generation” channel. For generation 1
(index r = 0), the surviving weight is the complementary
projector fraction:
Tr(Π − M0 )
9−3
2
G[1, 1] =
=
= .
(K39)
Tr(Π)
9
3
By normalization convention, G[2, 2] = G[3, 3] = 1 (generations 2 and 3 at the Higgs vertex).
b. Route B: Bin-overlap matrix (corollary via
Lemma Y.16). The same factor emerges from the Z3 ×Z3
bin-overlap structure:
Corollary K.5 (Bin-Overlap Realization). Let
W = [r(C3 ; r, s)]2r,s=0 be the bin-overlap matrix from
Lemma Y.16:


8/3 2
2
W =  2 8/3 2  .
(K40)
2
2 8/3

c. Structural identity. Both derivations give
G[1, 1] = 2/3 = (Ngen − 1)/Ngen . This is not a coincidence: the primed trace removes a rank-3 channel
from a 9D block, and the bin-overlap matrix has
diagonal/off-diagonal ratio 8/3 : 4 = 2 : 3. Both encode
the same topological invariant.
G[1, 1]: Derived
G[1, 1] = 2/3 is derived from the primed microsector
trace (Theorem K.4):
G[1, 1] =

Tr(Π − M0 )
9−3
2
=
=
Tr(Π)
9
3

Status: The Yukawa sector has zero continuous free
Yukawa couplings: the nine fermion masses (1.42%
leading-order mean error) are written with α, v and the
operators below; the top and charm are derived, and the
other seven species rest on data-selected prefactors
(caveat).
• α−1 = 137.036 (closed form, kmax = 60; bracket
coefficient √
open)
• v = MP α8 2π (exponent derived, prefactor fitted;
Theorem Z.3)
• Kd = J3 , Ku = I4 (derived with democratic
coupling, Lemma K.2)
• Qd , Dℓ (derived, QCD/γ-matrix normalization)
• G[1, 1] = 2/3 (derived, Theorem K.4)
Caveat (this is not full forcing). “Zero free
parameters” means zero continuous knobs, not a
complete first-principles derivation of the spectrum. The
absolute masses retain ∼4 discrete selection bits that are
data-selected, not derived: (i) the generation↔sector
assignment, (ii) the Higgs up/down conjugation bit, (iii)
the G-operator trace branch, (iv) the degree ordering kf .
Independent computation confirms the reason (App. FM):
the forced non-normal A5 family operator splits the three
generations only by ∼2×, while the down sector needs
∼252×; the large hierarchy is carried by the α-power
ladder (the diagonal/normal layer), and the
within-multiplet near-degeneracy of the geometric
operator is a theorem-grade obstruction, not a fitted
residual. Read the result as “zero continuous knobs”: the
seven species other than the top and the charm rest on
seven data-selected prefactors, one per species
(App. FM), and the four discrete labels listed here are
the choices that deploy them.

182
6.

CKM Matrix from CP 2 Geometry
a.

Wolfenstein Parameterization

The CKM matrix in Wolfenstein form is:

2
1 − λ2
λ
A λ3 (ρ − iη)
2
 + O(λ4 ).
VCKM = 
−λ
1 − λ2
A λ2
3
2
A λ (1 − ρ − iη) −A λ
1


determinant-orthogonal Berry offset (Postulate Y.10),
which leaves θ̄ = 0 at tree level; the resulting unitaritytriangle angle is γ = arctan(98/43) = 66.3093◦ (with
J = 2.9731×10−5 ), theorem-grade inside the strengthened
DFD–SD branch. The conjectural relation γ = 4π/11 is
not DFD’s prediction.

d.

Predictions and Comparison

(K42)

b.

Geometric Origin of λ

TABLE XCIII. CKM parameters: CP-sector predictions
(App. AO) vs. PDG 2024.

The Cabibbo angle λ ≈ 0.225 arises from the overlap
between first and second generation quarks:
λ = |Vus | = e−d12 /σH ,

(K43)

2

where d12 is the CP geodesic distance between the first
and second generation vertices, and σH is the Higgs localization width.
For the equilateral configuration of the three vertices
on CP 2 :
d12 = d23 = d31 = d0 ≈ 1.49σH ,

(K44)

λ = e−1.49 ≈ 0.225.

(K45)

giving:

c.

Higher-Order Parameters

• A: The ratio of up-type to down-type localization
widths
• ρ, η: The complex phase from the Kähler structure
of CP 2
Explicitly:
(u)

A=

(d)
σH
iδCP

ρ + iη = e

◦

r
·

mt
· fgeom ≈ 0.81,
mb

· ggeom ,

a The per-channel 2.5σ slightly overstates the independent miss:
A ≡ |Vcb |/λ2 is definitionally correlated with λ, so approximately
1.1 of the 4.6 percentage points of the A deviation double-books
the λ information already counted in the first row. The
independent reading is λ at +1.8σ and |Vcb | at −1.75σ.

e.

The parameters A, ρ, η arise from:

σH

Parameter Predicted (App. AO) Observed (PDG 2024) Deviation
λ
0.2262
0.22501 ± 0.00068
1.8σ
A
0.788
0.826+0.016
2.5σ a
−0.015
ρ̄
0.157
0.1591 ± 0.0094
0.2σ
η̄
0.358
0.3523+0.0073
0.7σ
−0.0071
Derived Predictions
|Vub /Vcb |
0.091
0.086 ± 0.006
0.8σ
|Vtd /Vts |
0.211
0.211 ± 0.007
0.0σ
−5
JCP
2.97 × 10−5
(3.12+0.13
1.2σ
−0.12 ) × 10

(K46)

Key Prediction: |Vub /Vcb | = λ

p

ρ̄2 + η̄ 2

p
In Wolfenstein form |Vub | = Aλ3 ρ̄2 + η̄ 2 and |Vcb | =
Aλ2 , so the apex-sensitive ratio is parameter-free once
the integer pattern fixes (λ, ρ̄, η̄):
p
|Vub |
= λ ρ̄2 + η̄ 2 = 0.2262 × 0.391 = 0.088. (K48)
|Vcb |
Observed: |Vub /Vcb | = 0.086 ± 0.006 (PDG 2024). The
value 0.088 is the leading-order barred-Wolfenstein expression; the exact standard-parameterization matrix
(App. AO) gives |Vub /Vcb | = 0.0906, agreement at 0.8σ.

(K47)

where δCP ≈ 68 is the CP-violating phase from the
complex structure of CP 2 .
a. The CP-sector result (Appendix AO). The
localization-overlap estimates above are sharpened to the
cohomological magnitude skeleton λ = 31α, A = 108α,
η̄ = 49α (line-bundle dimensions on CP 2 together with
D = dimR (CP 2 × S 3 ) = 7), with the CP-even apex
ρ̄ = 43
2 α fixed by the Euler-projection postulate. The
estimate δCP ≈ 68◦ does not hold: with real (Hermitian)
Yukawa kernels the literal flavor configuration lies on
the Lagrangian real locus RP 2 ⊂ CP 2 , the CKM matrix
is real orthogonal, and the Jarlskog invariant vanishes,
J = 0, at theorem grade (Theorem AO.7). The observed
CP violation is supplied by a single conjugation-odd,

7.

Summary: Microsector Consistency

The microsector results form a self-consistent framework:

183
b.

Microsector Summary
Inputs:
• Topology: M7 = CP 2 × S 3
• One scale: Planck mass MP = 1.22 × 1019 GeV
Derived:
• Fine-structure constant: α−1 = 137.036 (from
kmax = 60 on CP 2 ; bracket coefficient open; the
identification of the hypercharge-trace coefficient
with 1/e2 is not derived)
• Bridge Lemma: kmax = χ(CP 2 , E) = 60 connects
α to mass tower
√
• Higgs VEV: v = MP α8 2π = 246.09 GeV (0.05%
error; exponent derived, prefactor fitted)
• 9 fermion masses: 1.42% leading-order mean error;
the top and charm derived, the other seven on
seven data-selected prefactors (see the caveat
above and App. FM)
• CKM matrix: λ = 31α = 0.2262 (integer pattern,
App. AO); vertex-separation motivation
• PMNS matrix: trimaximal TM1 (App. X);
sin2 θ13 = 3α a stated conjecture
• Strong CP: θ̄ = 0 at tree level, loop-bounded, with
θYM = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields
(Theorem L.3)
• Koide relation: Qℓ = 2/3 is not derived (DFD’s
α-power lepton spectrum gives Qℓ ≈ 0.665; the
measured 2/3 holds to ∼0.001%, beyond DFD’s
∼1% mass precision — so DFD neither predicts
nor sharply contradicts Koide)
Consistency checks:
• CKM unitarity:
|Vud |2 + |Vus |2 + |Vub |2 = 1.000 ± 0.001
• PMNS angles: trimaximal TM1, θ23 = 45◦ in soft
∼2σ tension with NuFIT 6.0 (octant unresolved)
• JCP prediction agrees with observation at 1.2σ

Physical Origin of Factors

• Factor α8 : Represents the loop structure connecting Planck to electroweak: α8 = (α2 )4 is four 2-loop
factors.
√
• Factor 2π: an asserted O(1) Gaussian-measure
normalization — not derived. (A “geometric mean
of loop normalizations / same as kα = α2 /(2π)” rationale does not hold: loop measures carry (2π)−n ,
the wrong sign, and DFD’s own per-mode determinant ledger (Lemma O.1) cancels 2π factors in
ratios.) The best simple constant√near the best-fit
prefactor v/(MP α8 ) = 2.508 is 2π = 2.5066 (a
0.054% gap);
√ a band ∼ [2.50, 2.52] fits the VEV to
< 0.5%, so 2π is the best-looking but not uniquelyforced choice. The derived content of this result
is the α8 hierarchy; the O(1) prefactor is fitted
(cf. App. FK), and no printed rule fixes it: it is
a property of the Higgs potential’s loop-corrected
minimum, which this volume does not compute
(App. Z 3).
9.

Strong CP: Tree Level and Loop Bound
a.

θ̄ = 0 at tree level from the det-orthogonal tree-level
protection together with the η-invariant argument, with
θYM = 0 fixed by the spectral action given that nothing
is added to it for the gauge fields (Theorem L.3).
b.

8.

The Higgs Scale Hierarchy

The bare-parameter form of the hierarchy problem is
addressed by the relation:
√
v = MP × α8 × 2π.
(K49)
a.

α = 1/137.035999
8

√

α = 8.0412 × 10
2π = 2.5066

vpred = MP × α8 ×

√

(K50)
(K51)

−18

(K52)
(K53)

2π = 246.09 GeV

Loop Level

a. Quark mass phases.
emergence:
Z
Yij = gY

Yukawa couplings from gauge
ψ̄i ϕH ψj dµFS .

(K54)

Observed: v = 246.22 GeV. Agreement: 99.95%.

(K55)

CP 2

Det-orthogonal mass matrices, positive on the branch of
Postulate Y.10, give
arg(det Yu × det Yd ) = 0

Numerical Verification

MP = 1.220890 × 1019 GeV

Tree Level

(K56)

at tree level (App. AO).
b. Electroweak contributions. The (3, 2, 1) partition
separates SU(3)c (on C3 ) from SU(2)L (on C2 ) topologically. The CKM phase (Postulate Y.10) is a weak-sector
effect that reaches θQCD only through the degree-thirteen
Jarlskog structure (a multi-loop effect), at the StandardModel level.
c. Summary of protection mechanism.
1. Determinant protection: det-orthogonal mass
matrices, positive on the branch of Postulate Y.10,
keep arg det(Mu Md ) = 0 (App. AO)
2. Topological separation: (3, 2, 1) partition walls
off QCD from weak CP violation at tree level

184
3. Anomaly: the η-invariant of the even-dimensional
CP mapping torus vanishes (Theorem L.3)
Result: θ̄ = 0 at tree level, with θYM = 0 fixed by
the spectral action given that nothing is added to it for
the gauge fields; the anomaly channel is closed iff CP is
non-anomalous (Appendix L), and loop corrections are
bounded at the Standard-Model level (≲ 10−16 ).

10.

PMNS Matrix Derivation
a.

Physical Picture

• Charged leptons localized at CP 2 vertices (hierarchical)
• Neutrino R-H sector at the center (democratic)
• Result: Large mixing (tribimaximal base)
b.

Tribimaximal Mixing

When neutrinos at center have equal overlap with all
vertices:
p
p

p2/3 p1/3 p0
(K57)
UTBM = −p 1/6 p1/3 p1/2
1/6 − 1/3
1/2

c.

Corrections from Charged Lepton Masses (localization
estimate; the adopted branch is TM1, App. X)

θ13 ≈

q

me /mµ × 1.2 ≈ 4.8◦

θ23 = 45◦

(K58)

(maximal; the µ-τ reflection
forces θ23 = π/4 and δCP = −π/2)

θ12 ≈ 35.3◦ − 2◦ ≈ 33◦

Summary: DFD Unified Framework

DFD: Unified Framework
Single topology: CP 2 × S 3
Scale structure: The counted integer kmax = 60 and
the stated input Ngen = 3 + the measured G, the one
dimensionful scale, + one measured continuous number,
the early-clock coefficient C (App. FK; C = 1 on
bound-matter clocks, Postulate ZS.11)
Theorem-grade:
• µ(x) = x/(1 + x), the weak-response form of the
derived√exponential response (Thm. N.8; App. BS)
• a0 = 2 α cH0 derived from stationarity
(Thm. N.14)
• Dust branch under K ′ = µ: w → 0, c2s → 0
(Thm. Q.7)
Derived quantities:
• α−1 : closed form exact under two stated choices;
last 21 ppm on the open bracket; identification
with 1/e2 not derived (§X)
• (H0 tP )2 = Cα57 ≈ 10−122 (exponent forced; C
measured on the early clock, C = 1 on
bound-matter
√ clocks, Postulate ZS.11)
• v = MP α8 2π (Higgs scale; exponent derived,
prefactor fitted)
• Strong CP: θ̄ = 0 at tree level, loop-bounded, with
θYM = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields, and
given Postulate Y.10 (Thm. L.3)
• Fermion masses (1.42% LO; seven prefactors
data-selected), CKM, PMNS (Ngen = 3 a discrete
input)
• Proton stable (if baryon number is violated only by
the electroweak anomaly)
Falsifiable predictions:
• Channel-resolved clock structure of the extended
non-metric channel (Sec. XI); co-located
cavity–atom null (Thm. XI.2)
• No QCD axion; No 4th generation; No proton
decay (conditional)

(K59)
(K60)

The solar correction is within ∼5% of observed and the
reactor estimate falls short of the observed 8.5◦ ; θ23 = 45◦
is the maximal µ-τ prediction, a sharp falsifiable result
in soft ∼2σ tension with NuFIT 6.0 (octant unresolved).

d.

11.

Why PMNS ̸= CKM

Appendix L: Strong CP: Tree-Level Closure,
Loop-Bounded
1.

What must be shown

In any 4D gauge theory with quarks, the physical strongCP parameter is
θ̄ = θbare + arg det Mu + arg det Md .

Matrix

Localization

Result

CKM
PMNS

Both at vertices
Leptons at vertices, ν
at center

Small mixing (hierarchical)
Large mixing (TBM)

(L1)

The statement “θ̄ = 0 to all orders” would be equivalent to the statement that the full quantum effective
action respects an exact CP symmetry; this appendix
establishes the tree-level statement and the absence of
the CP anomaly, with loop corrections bounded at the
Standard-Model level. Since the operator
1
Tr(F ∧ F )
(L2)
Oθ ≡
32π 2

185
changes sign under CP, any CP-invariant quantum effective action forbids a generated coefficient for Oθ . Thus
the claim reduces to two conditions:
1. Classical CP invariance: the microscopic action
is CP invariant at θbare = 0.
2. No CP anomaly: the fermion measure (determinant/Pfaffian) is invariant under CP.
If both hold, then θbare = 0 is protected as a selection rule
at tree level and the anomaly generates no effective θ term;
loop corrections through the CKM sector are bounded at
the Standard-Model level (main theorem below).
2.

Tree-level inputs

The DFD microsector on M = CP 2 × S 3 with gauge
bundle E = O(9) ⊕ O⊕5 produces:
• The Standard Model gauge group GSM = SU(3)C ×
SU(2)L × U(1)Y , with its electroweak factor resting
on one stated identification and colour an input
(App. F),
• Real Yukawa eigenvalues from the Kähler structure,
• arg det(Mu Md ) = 0 at tree level for det-orthogonal
mass matrices positive on the branch of Postulate Y.10 (App. AO),
• Nonzero CKM CP violation (J ̸= 0) from the detorthogonal Berry offset of Postulate Y.10 (App. AO);
the minimal real kernel gives J = 0 (Theorem AO.7).
The CKM phase violates CP, so Condition (1) holds here
only in the weaker form the main theorem uses: the strong
sector carries no tree-level phase, arg det(Mu Md ) = 0.
Closing the anomaly channel requires establishing Condition (2): CP non-anomaly.
3.

The Dai–Freed anomaly formula

For a discrete symmetry σ (here σ = CP), the anomaly
is a U(1) phase given by the holonomy of the Pfaffian/determinant line bundle over background fields. The
Dai–Freed theorem [146, 147] expresses this holonomy as
an exponentiated η-invariant on the mapping torus.
Let M = CP 2 × S 3 be the microsector manifold with
its gauge bundle E and fermion bundle W = S ⊗ E ⊕
S̄ ⊗ Ē (a Z2 -graded Clifford module; the antilinear CP
lift exchanges the two summands). Define the mapping
torus:
TCP ≡ (M × [0, 1]) / (x, 0) ∼ (CP(x), 1) .
The CP anomaly phase is then:


iπ
ACP = exp
η(DTCP ) ,
2

(L3)

(L4)

where DTCP is the Dirac operator of the Clifford-module
bundle induced by W on TCP , and η(·) is the APS ηinvariant [146].
Criterion. CP is non-anomalous iff ACP = 1, i.e. iff
η(DTCP ) ∈ 4Z.
4.

Theorem: η vanishes automatically in even
dimensions

Theorem L.1 (Automatic vanishing of η in even dimensions). Let X be a closed even-dimensional oriented
Riemannian manifold, and let DE denote the Dirac operator of a Z2 -graded Clifford-module bundle over X with
compatible connection (for a Spinc manifold, the Spinc
Dirac operator twisted by a Hermitian vector bundle E
with unitary connection; in general no Spinc structure is
needed). Then the spectrum of DE is symmetric about 0,
hence
η(DE ) = 0,


and therefore exp iπ
2 η(DE ) = 1.

(L5)

Proof. Because dim X is even, the Clifford volume element
supplies a Z2 grading S = S + ⊕ S − of the module with
chirality operator Γ = diag(+1, −1). The twisted Dirac
operator is odd with respect to this grading:
ΓDE Γ−1 = − DE .

(L6)

Consequently, if DE ψ = λψ with λ ̸= 0, then DE (Γψ) =
−λ(Γψ), and the multiplicities of ±λ match exactly. For
Re s > dim X the series below converges absolutely by
Weyl’s law, so the pairing may be carried out term by
term. Thus the η-function, defined initially for Re(s) ≫ 0
by
X
sign(λ) |λ|−s ,
(L7)
η(DE , s) =
λ̸=0

vanishes identically term-by-term (each +λ cancels a −λ),
and by analytic continuation η(DE ) = η(DE , 0) = 0.
Corollary L.2 (Trivial CP anomaly in DFD). The mapping torus TCP has dimension
dim TCP = dim M + 1 = 7 + 1 = 8
2

(even).

(L8)

The CP involution on CP (complex conjugation in homogeneous coordinates), combined with the identity on
S 3 , is an orientation-preserving isometry of M , so TCP
is a closed oriented Riemannian 8-manifold. It admits
no Spinc structure (H 2 (TCP ; Z) = 0 while w2 (TCP ) ̸= 0),
and the lift of CP maps E to Ē; the fermion bundle
W = S ⊗ E ⊕ S̄ ⊗ Ē with the antilinear CP lift nevertheless descends to a Z2 -graded Clifford-module bundle over
TCP (a Spin ×Z2 O(2) structure exists, with LR the real
rank-two bundle obtained from det E by the conjugation
gluing: w2 (TCP ) = w2 (LR ) and w1 (LR )2 = 0), so Theorem L.1 applies in its Clifford-module form. Moreover, for
the product metric with the untwisted round S 3 of radius

186
2
2
2
r3 and CP 2 -only twisting, DT2 CP = DCP
2 ,E + DS 3 − ∂t ≥
9/(4r32 ) > 0, so DTCP has no zero modes and η = h = 0
for either a determinant or a Pfaffian fermion measure.
Hence, by Theorem L.1:


iπ
η(DTCP ) = 0 ∈ 4Z,
ACP = exp
· 0 = 1. (L9)
2

Remark. This result does not depend on a delicate
explicit evaluation of η; it uses only the structural fact
that the operator in Eq. (L4) is a twisted Dirac operator
on an even-dimensional closed manifold, hence has exact
±λ spectral pairing by Eq. (L6).

5.

Main theorem: tree-level closure, loop-bounded

Theorem L.3 (Strong CP closure: tree-level θ̄ = 0,
loop-bounded). In the DFD microsector on M = CP 2 ×
S 3 with the Standard Model fermion content:
1. Given that nothing is added to the spectral action
for the Standard-Model gauge fields, colour included
(that is, each topological sector is weighted by that
action alone), the microscopic theory is CP invariant in the strong sector at tree level: the spectral
action contains no Tr(F ∧ F ) term, so θbare = 0,
and on the branch of Postulate Y.10 connected to
ξ = 0, det(Mu Md ) > 0.

through Yukawa degree eleven for Hermitian tree-level
mass matrices with the Standard-Model chiral couplings;
the onset is the multi-loop effect of the cited references),
and the induced angle is ≲ 10−16 [149, 150].
6.

A conditional alternative: quaternionic structure

A conditional alternative route, requiring the CP lift
to commute with the Dirac operator, comes from the
quaternionic structure on the S 3 factor.
Lemma L.4 (3D charge conjugation). Let σ a be Pauli
matrices and consider the 3D Euclidean Dirac operator
D3 = iσ a ∇a . Define the antiunitary charge conjugation
C3 ≡ σ 2 ◦ K (with K complex conjugation). Then
C32 = −1,

2

Proof. Condition (1) holds at tree level. The spectral
action Tr f (D2 /Λ2 ) of Proposition WS.5, D = DM ⊗ 1 +
γ5 ⊗DF , is real, and reversing the orientation of spacetime
takes D to −(γ5 ⊗ 1) D (γ5 ⊗R1), whose square has the
same spectrum as D2 , while Tr(F ∧ F ) changes sign;
so the spectral action contains no Tr(F ∧ F ) term [148],
and the internal trace that fixes α (premise G2) contains
no four-dimensional field. With nothing added to the
spectral action for the gauge fields, no topological sector
carries a phase of its own, and θbare = 0 in the basis in
which DF carries the mass matrices of Postulate Y.10.
With the positive-definite Sf of that postulate and the
offset on its branch connected to ξ = 0 (Theorem AT.16),
det(Mu Md ) > 0, so arg det(Mu Md ) = 0 at tree level and
the vacuum angle is 0, not π. Condition (2) follows from
Corollary L.2: the mapping torus has even dimension (8),
so the twisted Dirac operator has symmetric spectrum
and η = 0 automatically.
Since both conditions hold, the tree-level effective action
contains no θ term. Loop corrections with the StandardModel chiral Yukawa structure regenerate a coefficient
of Tr(F ∧ F ) only through the degree-thirteen Jarlskog
structure (the shift of arg det(Mu Md ) vanishes identically

a ∗ 2

(L10)
a

Proof. The Pauli identity σ (σ ) σ = −σ implies
C3 σ a C3−1 = −σ a , while antiunitarity gives C3 i C3−1 =
−i. Therefore C3 (iσ a )C3−1 = iσ a , proving commutation
with D3 . Finally C32 = σ 2 (σ 2 )∗ = −⊮.
When the CP lift commutes with the Dirac operator,
the quaternionic structure (J 2 = −1) forces the fermion
determinant to be real and nonnegative [147, 151], giving
an independent route to ACP = 1 on the product background; the argument of Corollary L.2 does not rely on
it.

2. The CP anomaly phase is trivial: ACP = 1 (Corollary L.2).
Therefore θ̄ = 0 at tree level, and its loop-induced value
is at the Standard-Model level (≲ 10−16 ; App. AO). No
axion is required.

C3 D3 C3−1 = D3 .

7.

Falsifiable prediction

Theorem L.3 implies:
• No QCD axion exists. Axion searches (ADMX,
ABRACADABRA, CASPEr, etc.) will find nothing.
• Any observed |θ̄| > 10−15 would falsify this
mechanism.
This is a sharp, experiment-confrontable prediction distinguishing DFD from Peccei–Quinn solutions.
8.

Summary: why the S 3 factor does triple duty

With θbare = 0 fixed by the spectral action given
that nothing is added to it for the gauge fields, the
Strong CP problem is closed at tree level in DFD by
the det-orthogonal tree-level protection together with the
η-invariant argument (θ̄ = 0 at tree level, loop-bounded
below 10−16 ), not by introducing new particles. The mechanism is dimensional: the microsector M = CP 2 × S 3
has dim M = 7, so the mapping torus has dim TCP = 8
(even), forcing η = 0 by spectral symmetry.
The same S 3 factor that:
1. Carries baryon number as π3 (S 3 ) = Z winding (a
stated identification; proton stability, Theorem G.4,
does not use it),

187
2. Provides gauge emergence: π3 (SU(3)) = Z,
also contributes the “+1” to make dim TCP = 8 even,
which closes the anomaly channel of Strong CP. The
generation count, Ngen = 3, is a stated input, which no
index on CP 2 × S 3 supplies (App. F).
Appendix M: Scattered and Locally Emitted Lines in
a Static ψ

This appendix gives DFD’s value of the enhancement
factor Γ in the Ly-α/O VI asymmetry ratio of Sec. XIV G.
1.

Definitions and Setup

Let ψ(x) be the DFD scalar field with refractive index
n = eψ and one-way light speed c1 = c e−ψ . Consider two
UV lines observed by UVCS:
• H Ly-α: Dominated by resonant scattering of chromospheric radiation in the corona.
• O VI: Dominated by local (collisional) emission in
the corona.
Let A denote the measured asymmetry amplitude statistic, and define:

2
ALyα
σOVI
R≡
=Γ
.
(M1)
AOVI
σLyα
2.

Gaussian Detuning Scaling

For a symmetric line profile with thermal width σ and
small detuning δ ≪ σ, a Taylor expansion of the Gaussian
gives:
 2
∆I
δ
A=
∝
(M2)
I
σ
to leading order (the linear term vanishes by symmetry). This scaling follows from the sensitivity of resonant
absorption/scattering to wavelength mismatch.
3.

The Value of Γ

In a static ψ a photon’s frequency is conserved along
its path, so the outgoing leg cannot change the integrated
scattered flux: resonant scattering responds only to the
detuning at the scatterer, as in standard coronal physics.
DFD’s optics therefore adds no enhancement, and Γ takes
its standard value.
4.

Observational Constraint on Γ

The 2007–2009 UVCS data do not measure R, since
their two annual amplitudes are different statistics and

neither shows a significant annual modulation with independent units. The width factor (σOVI /σLyα )2 = 9.0 of
Sec. XIV G holds for a Ly-α temperature of 104 K; the
coronal H I that UVCS observes is at about 1–3 × 106 K,
so Γ needs the widths measured in the spectra that give
R, as in the matched dim/bright slope described there. In
the same data the velocity shift of the intensity-contrast
groups is null (Cohen’s d = −0.03; Sec. XIV).

188
Appendix N: First-Principles Derivation of µ(x) and
a0

This appendix derives both the MOND crossover function
√ µ(x) = x/(1 + x) and the acceleration scale a0 =
2 α cH0 from the S 3 Chern-Simons microsector with
explicit, minimal assumptions. (Section XVI A 6 distinguishes the cosmological
acceleration cH0 from the galac√
tic crossover a0 = 2 α cH0 , whose value on bound-matter
2
clocks is a0 = 1.1966 × 10−10 m/s (Postulate ZS.11);
throughout, a⋆ = 2a0 /c2 denotes the Lagrangian gradient
scale, not an acceleration.) The derivation proceeds in
two stages:
1. Stage I (asymptotics theorem-grade): The two
asymptotic limits of µ(s) are forced by microsector
multiplicativity and the composition law (Theorem N.8); the specific transition form µ(s) = s/(1 +
s) is the weak-response form (Assumption N.3) of
the exponential response forced by the composition law (Lemma N.6); it tracks the derived law of
App. BS (RAR form) to ≲ 0.02 dex, below SPARC
precision.
2. Stage II (Derived): The crossover invariant
Ξ∗ = 3/2 is selected by scaling
stationarity (Theo√
rem N.12), yielding a0 = 2 α cH0 (Theorem N.14).
1.

The S 3 Partition Function (Exact Result)

Lemma N.1 (S 3 partition function exponent). For
SU (2) Chern-Simons theory on S 3 at integer level k ≥ 1,
the exact Witten partition function is [132]:
r


2
π
sin
.
(N1)
ZS 3 (k) =
k+2
k+2
In the large-k regime, sin(π/(k + 2)) ∼ π/(k + 2), hence:


ZS 3 (k) = const · (k + 2)−3/2 1 + O(k −2 ) , (N2)
3
log ZS 3 (k) = const − log(k + 2) + O(k −2 ).
2
The exponent 3/2 = dim SU (2)/2 is fixed by the gauge
group; since SU (2) ∼
= S 3 it equals dim(S 3 )/2.
2.

Microsector-to-ψ Map and Level Response

Assumption N.2 (Microsector multiplicative weight defines eψ ). The DFD scalar ψ is defined (up to an additive
constant) by the ratio of microsector weights:
eψ(s) :=

ZS 3 (k0 )
,
ZS 3 (keff (s))

(N3)

where k0 is the background level and keff (s) is the effective
level in an environment parameterized by a dimensionless
s ≥ 0.

Assumption N.3 (Minimal weak-field level response).
In the weak-response regime, the effective level scales as:
keff (s) = k0 (1 + s),

(N4)

with k0 ≫ 1 so that k0 ± O(1) corrections are negligible
in logarithms.
Proposition N.4 (ψ inherits the 3/2 coefficient). Under
Assumptions N.2–N.3 and using Lemma N.1:
3
ψ(s) = log(1 + s) + O(k0−1 ).
(N5)
2
Proof. From Eqs. (N3) and (N2):
ψ(s) = log ZS 3 (k0 ) − log ZS 3 (keff (s))
3
= [log(keff (s) + 2) − log(k0 + 2)] + O(k0−1 ).
2
Insert keff (s) = k0 (1 + s) and expand:
log(k0 (1 + s) + 2) − log(k0 + 2) = log(1 + s) + O(k0−1 ).

3.

The Key Theorem: µ is Fixed by a Composition
Law

The exponential form of µ follows from the composition
law of Assumption N.5.
Assumption N.5 (Independent segments compose by
saturation union). If two independent contributions add
in ψ (because microsector weights multiply), then the
effective response µ satisfies the saturation-union law:




µ(ψ1 + ψ2 ) = 1 − 1 − µ(ψ1 ) 1 − µ(ψ2 ) ,
(N6)
µ(0) = 0,

0 ≤ µ < 1.

Lemma N.6 (Composition ⇒ exponential). Under Assumption N.5 on ψ ≥ 0, continuity of µ, and µ ̸≡ 0, there
exists a constant c > 0 such that, for ψ ≥ 0:
µ(ψ) = 1 − e−cψ .

(N7)

Proof. Define g(ψ) := 1 − µ(ψ). Then Eq. (N6) becomes
g(ψ1 + ψ2 ) = g(ψ1 )g(ψ2 ) with g(0) = 1 and g(ψ) ∈
(0, 1]. By the standard Cauchy functional equation for
multiplicative g under continuity, g(ψ) = e−cψ for some
c ≥ 0. Since µ ̸≡ 0, c > 0.
Assumption N.7 (Deep-field limit fixes the slope). In
the small-s regime, the desired MOND closure has µ(s) =
s+O(s2 ) when expressed in terms of the same s appearing
in the level response (N4).
Theorem N.8 (Weak-response µ(s) from the S 3 power
law). Assume Assumptions N.2, N.3, N.5, and N.7. Then,
in the large-k0 regime and within the weak-response domain of Assumption N.3:
s
µ(s) =
+ O(k0−1 )
(N8)
1+s

189
Proof. By Lemma N.6, µ(ψ) = 1 − e−cψ . Using Proposition N.4, ψ(s) = 23 log(1 + s) + O(k0−1 ). Thus:


3
µ(s) = 1 − exp −c · log(1 + s) + O(k0−1 )
2
= 1 − (1 + s)−3c/2 + O(k0−1 ).
Assumption N.7 requires µ(s) = s + O(s2 ) as s → 0, i.e.,
(1 + s)−3c/2 = 1 − s + O(s2 ), which forces 3c/2 = 1, hence
c = 2/3. Substituting yields µ(s) = 1 − (1 + s)−1 =
s/(1 + s).
Microsector Result: exponential response; µ(x) =
x/(1 + x) is its weak-response form
The exponential response (Lemma N.6) and its
weak-response form µ(s) = s/(1 + s) (Theorem N.8)
follow from:
1. The S 3 partition function exponent
3/2 = dim SU (2)/2
2. Microsector multiplicativity (weights multiply ⇒ ψ
adds)
3. Saturation-union composition law
(Assumption N.5)
4. Weak-response level response and deep-field limit
slope (Assumptions N.3, N.7)
These uniquely fix the two asymptotic limits (µ → 1,
µ ∼ s) and the exponential form; the transition-region
shape follows from the saturation-union closure
(Assumption N.5) together with what is assumed for the
exponent (Remark BS.11): the weak-response form tracks
the derived response of App. BS (RAR form) to ≲ 0.02
dex on the SPARC range, and the variational closure
(§N 6) is a cross-check of the asymptotics only.

Remark N.9 (Two-vertex QED count).
A vertex count
√
in QED gives the number de = 2 α heuristically.
√ Each
photon-fermion vertex contributes amplitude e ∝ α. For
a neutral atom with two charged
constituents (electron
√
and nucleus), counting one α per constituent gives:
√
√
√
datom
= α + α = 2 α ≈ 0.171.
(N9)
e
√
This gives a0 = de · cH0 = 2 α · cH0 .
a. Physical interpretation. Photons couple directly
to the optical metric with dγ = 1. Electrons do not couple
directly to ψ; they √
interact through QED vertices. Each
vertex contributes α < 1. Matter couples less strongly
than light because its interaction is mediated.
b. Status of the count. This is a heuristic vertex
count: for independent subsystems the energies add,
Eatom = Ee + EN , so their fractional
√ responses average rather than add. The prefactor 2 α of a0 is fixed by
the variational derivation below (Theorem N.14).

4.

The Acceleration Scale a0 : Variational Derivation

√
We now derive a0 = 2 α cH0 from a variational principle that selects the crossover point using the S 3 micro-

sector scaling charge.

a.

The Unique IR Control Parameter

Given DFD postulates (flat R3 , scalar ψ, a = (c2 /2)∇ψ
in the weak static field) and a single global µ-closure, the
onset of non-Newtonian response can depend only on
the unique dimensionless scalar built from |a| and the
cosmological scale cH0 :

2
|a|
Ξ := ka
,
(N10)
cH0
where the coefficient ka = 3/(8α) is fixed by the microsector (Section VIII B).

b.

Microsector Scaling Charge

Lemma N.10 (Scaling charge from S 3 ). For SU (2)
Chern-Simons on S 3 , the partition function satisfies
log ZS 3 (k) = const − 23 log(k + 2) + O(k −2 ). The dimensionless scaling charge is:
3
∂ log ZS 3
= .
(N11)
qS 3 := −
∂ log(k + 2)
2
This is the same topological coefficient that appears in
the µ(x) derivation (Theorem N.8).

c.

The Spacetime Functional

We now show that the crossover point Ξ∗ = 3/2 is
selected by an explicit spacetime integral functional built
only from the DFD field ψ and the cosmic scale cH0 .
a. Local dimensionless invariant. Under DFD postulates, the local dimensionless invariant, with |a| the
proper acceleration of static observers (Eq. (AP12) in the
weak static field), is:

2
|a|
k a c2
Ξ(x) = ka
= β |∇ψ|2 ,
β :=
. (N12)
cH0
4H02
b. The minimal spacetime functional. Define the dimensionless functional:
Z


3
S[ψ] :=
d3 x Ξ(x) − qS 3 log Ξ(x) ,
qS 3 = .
2
Ω
(N13)
No additional scale has been introduced: the logarithm is
well-defined because Ξ is dimensionless.
c. Interpretation. S is not asserted to be the full
dynamical action of DFD. It is the minimal coarse-grained
IR functional whose only nontrivial coefficient is the S 3
scaling charge qS 3 , and whose stationary point fixes the
crossover invariant.

190
d.

√
Derived: a0 = 2 α cH0

Homogeneous-Limit Theorem

Definition N.11 (Homogeneous-gradient sector). Fix
a bounded region Ω of volume V and a reference profile
ψ0 . Consider the one-parameter family ψλ := λ ψ0 with
λ > 0. Then ∇ψλ = λ ∇ψ0 and:
Ξλ (x) = λ2 Ξ0 (x).

Status: The dimensionless law and prefactor are derived.
On bound-matter clocks the physical normalization is the
C = 1 value, with no measured coefficient
(Postulate ZS.11); it is not a galactic fit.
The derivation chain:

(N14)

1. ka = 3/(8α) from gauge emergence
(Section VIII B)

Theorem N.12 (Scaling stationarity selects the mean
crossover invariant). Let ψλ = λψ0 and define the mean
invariant:
Z
1
Ξ0 :=
d3 x Ξ0 (x).
V Ω
Then stationarity of S[ψλ ] with respect to λ occurs at:
Z
3
1
qS 3
2
Ξ∗ :=
d3 x Ξλ∗ (x) = qS 3 = .
,
λ∗ =
V
2
Ξ0
Ω
(N15)
Proof. Insert Eq. (N14) into Eq. (N13):
Z



S[ψλ ] =
d3 x λ2 Ξ0 − qS 3 log λ2 Ξ0

2. qS 3 = 3/2 from S 3 partition function
(Lemma N.10)
R
3. S[ψ] = (Ξ − qS 3 log Ξ) d3 x — explicit spacetime
functional (N13)
4. Ξ∗ = 3/2 from scaling stationarity (Theorem N.12)
√
5. a0 = 2 α cH0 from algebra (Theorem N.14)
What is derived vs. postulated:
• Derived: qS 3 = 3/2 from the S 3 partition
function; stationarity of an explicit spacetime
functional then sets Ξ∗ = qS 3 .
• Postulated: the functional form (N13) (a
coarse-grained functional, not the full action).
Comparison:
on bound-matter clocks (Postulate ZS.11)
√
a0 = 2 α cH0loc = 1.1966 × 10−10 m/s2 vs. the
comparator a0 = (1.20 ± 0.02stat ± 0.24sys ) × 10−10
m/s2 [10], within the systematic error.

Ω

Z


= λ V Ξ0 − qS 3 2V log λ +
d3 x log Ξ0 .
2

Ω

Differentiate with respect to λ and set to zero:
2q 3 V
q 3
dS
= 2λV Ξ0 − S
= 0 ⇒ λ2∗ = S .
dλ
λ
Ξ0

5.

Summary and Falsifiable Predictions

Then Ξ∗ = λ2∗ Ξ0 = qS 3 = 3/2.
Corollary N.13 (Local homogeneous limit). If Ξ0 (x) is
approximately spatially constant in Ω, then Ξ0 = Ξ0 and
the stationarity condition becomes the pointwise statement:
Ξ∗ =

e.

3
2

(N16)

Result
Status
Key Input
µ(s) = s/(1 + s) Thm. N.8 Composition + level response
ψ = 32 log(1 + s) Prop. N.4
Witten partition function
Ξ∗ = 3/2
Thm.
N.12
Spacetime stationarity
√
a0 = 2 α cH0
Thm. N.14
ka + Ξ∗ (both derived)

a.

The MOND Scale Theorem

Theorem N.14 (MOND scale from spacetime functional).
Combining Corollary N.13 with ka = 3/(8α):
√
a0 = 2 α cH0
(N17)
Proof. From Eq. (N10) at Ξ = Ξ∗ :
s
r
Ξ∗
3/2
a0 = cH0
= cH0
ka
3/(8α)
r
√
√
3 8α
= cH0
×
= cH0 4α = 2 α cH0 .
2
3

TABLE XCIV. Status of MOND derivation from microsector.

(N18)

Falsifiable predictions.

1. Exponential µ-response: The interpolation must
be the exponential response of Lemma N.6 and
App. BS, whose
weak-response form is µ(x) = x/(1+
√
x), not x/ 1 + x2 or other forms. (Already favored
by SPARC data, Section VII.)
2. Locked a0 value on bound-matter clocks: Precision measurements of a0 from
large galaxy sam√
ples should converge to 2 α cH0loc = 1.1966 ×
2
10−10 m/s on distances computed with H0loc (Postulate ZS.11); convergence to the early-clock value
2
1.1208 × 10−10 m/s on distances computed with
early
H0
falsifies the postulate.
3. Epoch-consistency scaling: Since a0 is fixed
by topology plus H0 , the present-epoch value at
z = 0 is derived given the clock (Postulate ZS.11).

191
A companion note [112] argues (program-grade)
that under the same epoch-consistency rule that
promotes the current closure GℏH02 /c5 = Cα57
to G(t)ℏH(t)2 /c5 = Cα57 (Section XIX), the acceleration√scale should be promoted similarly to
a0 (z) = 2 α cH(z). High-redshift kinematics test
it, and the first tests disagree: low-mass discs at
0.33 < z < 1.44 show a rise on the published bins,
which counted galaxy by galaxy is about 1σ within
the survey and about 3σ anchored to SPARC’s z = 0
value, with DFD’s law fitting on the bound-matter
clock, and depends on how the molecular gas is
added, while massive discs at 0.6 < z < 2.5 give an
a0 close to the local value at z ≃ 2 (App. A0Z). The
promotion is contested; direct molecular-gas masses,
kinematic masses not limited by a halo-model prior,
galaxy-level errors and the environments of the highredshift discs decide it.

6.

Alternative Derivation: Variational Approach

The S 3 composition law derivation above gives µ(x) =
x/(1 + x). Here we present an independent variational
derivation that yields a closely related result, providing a
cross-check on the functional form.

a.

(N19)

a2
c2
Lψ = 0 U (s) − ψ(ρ − ρ̄),
(N20)
8πG
2
where U (s) is a priori unknown. Variation gives:
8πG
∂i [U ′ (s) ∂i ψ] = − 2 (ρ − ρ̄).
(N21)
c
Identifying the constitutive law:
µ(u) ≡ U (s)

2

(s = u )

(N22)

yields the nonlinear Poisson equation ∇ · [µ(u)∇ψ] =
−(8πG/c2 )(ρ − ρ̄).

b.

Asymptotic Constraints

Two physical limits constrain U (s):
a. Strong field (u ≫ 1). In the Newtonian limit, we
require µ(u) → 1, hence:
U (s) ∼ s

as s → ∞.

(N23)

b. Deep field (u ≪ 1). For flat rotation curves, we
require µ(u) ∼ u, hence:
U (s) ∼ s3/2

as s → 0.

Closed-Form Solution

A minimal convex interpolant satisfying these asymptotics can be obtained via Legendre construction. The
result is:
√
1 + 2u − 1 + 4u
, u > 0.
(N25)
µ(u) =
2u
a.

Asymptotic checks.
√
u≪1:
1 + 4u = 1 + 2u − 2u2 + · · ·

⇒ µ(u) = u + O(u2 ) ✓
√
√
u≫1:
1 + 4u = 2 u(1 + O(u−1/2 ))
1
⇒ µ(u) = 1 − √ + · · · ✓
u
b. Monotonicity and ellipticity.


1 √
2u
µ′ (u) = 2
1 + 4u − 1 − √
2u
1 + 4u
> 0 (∀u > 0),

dµ
µ′ (u)
=
> 0,
ds
2u
establishing global convexity.
U ′′ (s) =

Write the dimensionless gradient invariants:

′

c.

(N26)

(N27)

(N28)

so the operator is strictly elliptic.
c. Convexity. Since µ = U ′ (s) with s = u2 :

Setup: Auxiliary-Field Action

|∇ψ|
|∇ψ|2
u≡
,
s ≡ u2 =
.
a⋆
a2⋆
Consider the static sector with action density:

Any admissible U must interpolate between s3/2
(deep field) and s (strong field) while remaining convex
(U ′′ (s) > 0) to ensure a strictly monotone constitutive
law and an elliptic operator.

(N24)

d.

(N29)

Comparison with S3 Result

The variational result (N25) and the S 3 composition
law result µ(x) = x/(1 + x) are not identical, but share
the same asymptotic structure:
Variational S 3 Composition
u≪1
u≫1
Monotone
Convex U

µ∼u
µ→1
✓
✓

µ∼x
µ→1
✓
✓

The two derivations yield non-identical interpolation
functions—µ(x) √
= x/(1 + x) (composition law) versus
µ(u) = (1 + 2u − 1 + 4u)/(2u) (variational)—that share
the same two asymptotic limits but are not the same
function: the transition-region shape is fixed only up to
this closure choice.
a. Physical interpretation. The variational approach
treats µ as the derivative of a convex energy density—the
standard EFT perspective; the S 3 composition law derives
µ from microsector multiplicativity. What is uniquely
forced is the pair of asymptotic limits (µ → 1 Newtonian,

192
µ ∼ x deep-MOND); the intermediate-u transition shape
is closure-dependent. The variational route is best read as
a cross-check confirming the asymptotic structure, not an
independent derivation of the same transition function.

7.

The Complete Picture: MOND from S3 Topology

MOND Crossover: Complete Derivation Summary
Input: S 3 Chern-Simons microsector with partition
function ZS 3 (k) ∝ (k + 2)−3/2
Outputs:
x
µ(x) =
(Thm. N.8; asymptotics)
(N30)
1+x
3
(Thm. N.12)
(N31)
Ξ∗ =
2
√
a0 = 2 α cH0
(Thm. N.14)

(N32)

The two asymptotic limits of µ are forced; the µ
transition-region shape is closure-fixed (below SPARC
precision).
Consequence: Galaxy rotation curves follow from the
topology of S 3 —the same manifold that carries baryon
number, and gives α = 1/137.

The Galactic Missing-Mass Problem: the µ-Law Account
The galactic “missing mass” — flat rotation curves and
the RAR — needs no fitted halo. It is a geometric
effect from the S 3 microsector vacuum-weight response
to matter density (the universal µ-law). The cosmological
cold dark matter (CMB third peak, clusters) is a
separate, derived particle, the χ-matter field of App. AV.
It is present in galaxies too, and its pull composes with
the µ-law there by the floor law, which is stated, not
derived (App. AV).
The same topology that:
• Carries the generation label (Ngen = 3, a discrete
input; π3 (S 3 ) = Z supplies the winding lattice, not
the count — App. F)
• Carries baryon number (a stated identification;
proton stability, Theorem G.4, does not use it)
• Gives α = 1/137 (from kmax = 60 on CP 2 )
• Closes the anomaly channel of Strong CP
(dim(TCP ) = 8 even)
• Fixes the clock–α constraint GℏH02 /c5 = C α57
(exponent a theorem; on bound-matter clocks
C = 1 and H0loc = α28.5 /tP = 72.09 km/s/Mpc; on
the early clock C = 0.878 ± 0.003 measured,
H0early = 67.52 km/s/Mpc; Postulate ZS.11,
App. ZS)
also produces:
• Flat rotation curves with µ(x) = x/(1 + x)
• MOND scale a0 = 1.1966 × 10−10 m/s2 on
bound-matter clocks
• The radial acceleration relation
• The baryonic Tully-Fisher relation
Rotation curves from geometry — no fitted halo required.
(The cosmological cold dark matter — CMB third-peak
height and cluster mass, the latter at the amplitude its
abundance gives under ΛCDM-like assembly (estimate;
App. GR) — is the derived χ-matter particle of App. AV;
it composes with the µ-law inside galaxies by the floor
law.)

193
Appendix O: The α57 Mode-Count Exponent and the
G–H0 –α Invariant
1.

Mathematical core: primed-determinant scaling
fixes the exponent

Let H be a finite-dimensional complex Hilbert space of
dimension kmax , and let K : H → H be a self-adjoint, positive semidefinite operator with dim ker(K) = Ngen . Denote by det′ (K) the primed determinant over the nonzero
spectrum of K.
Lemma O.1 (Primed determinant scaling). For any
g > 0,
′
′


det g K = g kmax −Ngen det(K) .
(O1)
max
. ExProof. Diagonalize K on H with eigenvalues {λi }ki=1
actly Ngen of these are zero; the remaining N := kmax −
QN
Ngen satisfy λi > 0. Then by definition det′ (K) = i=1 λi
′
(product over the nonzero spectrum), and det (gK) =
QN
QN
N
i=1 (gλi ) = g
i=1 λi .

Definition O.2 (Microsector hierarchy factor as a determinant ratio). Define
ε(g) :=

det′ (K)
.
det′ (g K)

(O2)

Corollary O.3 (Forced exponent). If kmax = 60 and
Ngen = 3, then
ε(g) = g −57

and in particular

ε(α−1 ) = α57 .
(O3)

Proof. Immediate from Lemma O.1 and Definition O.2
with N = kmax − Ngen = 57.
a. Provenance of the two integers. The two integers
in the exponent kmax − Ngen = 57 have distinct provenance:
• Ngen = 3 is a discrete input (the observed family
number; App. F 5, Rem. F.21). It is numerically
coincident with χtop (CP 2 ) = 1 + 1 + 1 = 3 (the
Euler/de Rham index), but that identification is not
a Dirac index theorem: the untwisted spinc index on
CP 2 is Td(CP 2 ) = 1, and the twisted per-multiplet
indices do not share a factor of 3. We do not claim
Ngen is derived.
• kmax = χ(CP 2 , E) = 60 for the Spinc twisting
bundle E = O(9) ⊕ O⊕5 , the total microsector
mode count via Spinc –HRR — exact arithmetic
given the inputs Ngen = 3 and n = 5 and the
q1 = 3 derived under the electroweak identification
(Rem. F.10); within that reading the count does
not use α, and the α match (21 ppm from the integers) is what excludes the reading with line-bundle
doublets (App. F 5, Rem. F.12).
The Wilsonian integration in Sec. O 2 then integrates
out the kmax − Ngen = 57 nonzero modes, giving the α57
suppression.

2.

Gaussian mode-integration realization

The ratio ε(g) admits a concrete physical realization
as the partition-function ratio obtained by Gaussian integration over the nonzero-mode sector. Let K+ denote K
restricted to the nonzero spectrum, and define for g > 0
the Gaussian normalization integral over N = 57 complex
modes:
Z


Z(g) :=
exp − ⟨ϕ, (gK+ )ϕ⟩ d2N ϕ
CN

=

N
Y

πN
π
.
= N
g λi
g det′ (K)
i=1

(O4)

The ratio to the reference (g = 1) partition function is:
Z(g)
= g −N = g −57
Z(1)

(O5)

so the suppression factor is ε(g) ≡ Z(g)/Z(1) = g −57 .
a. Per-mode eigenvalue cancellation. Each complex
mode ϕi contributes independently. At coupling g = α−1
(gauge-normalized; see Lemma O.5 below):


R 2
d ϕi exp −(λi /α) |ϕi |2
πα/λi


R
=
= α.
(O6)
π/λi
d2 ϕi exp −λi |ϕi |2
The eigenvalue λi cancels exactly in the ratio. The permode suppression factor is α regardless of the detailed
spectrum of K; the exponent depends only on the mode
count N = 57, not on the eigenvalues.

3.

From determinant ratio to physical hierarchy:
derivation

The scaling of the measured invariant I = GℏH02 /c5 as
ε(α−1 ) = α57 is established by three lemmas; its coefficient is measured.
Lemma O.4 (KK reduction). The internal Dirac operator DK on K = CP 2 × S 3 , in the Toeplitz truncation at
level kmax = 60 with the generation sector fixed at the
input value Ngen = 3, has exactly 3 protected (zero-sector)
eigenvalues and 57 nonzero eigenvalues. In the Wilsonian
effective theory at energies below the KK scale, the 57
nonzero modes are integrated out by Gaussian approximation, leaving the effective action for the ψ-field zero
mode.
Proof. Two clarifications fix the correct statement. First,
since K is a closed odd-dimensional manifold, the elliptic index of DK vanishes identically, and the round-S 3
factor has no harmonic spinors (Lichnerowicz). The rigorous index-theoretic content on the S 3 factor is the APS
spectral flow IS 3 (k2 ) = k2 (App. F 5, Theorem F.14 and
Rem. F.15), not a static kernel dimension. Second, the
3-dimensional protected sector is the generation space
G, whose dimension Ngen = 3 enters as a discrete input
(Rem. F.21); within the Toeplitz truncation the microsector Hilbert space has dimension kmax = 60, fixed α-free

194
by the determinant-line chain given the selected discrete
inputs (Rem. F.12: SM → q1 = 3 → a = 9 → kmax = 60).
That this same kmax = 60 reproduces α−1 = 137.036 is
an a-posteriori consistency check, not the source of the
value. The nonzero-mode count is N = 60 − 3 = 57.
These modes acquire KK masses mi ∝ |λi | and are integrated out at energies E ≪ mi by the standard Wilsonian
procedure.
Lemma O.5 (Uniform gauge normalization). Each of the
57 nonzero modes contributes exactly one factor of α to
the partition-function ratio, giving Z(α−1 )/Z(1) = α57 .
Proof. Three facts combine:
1. Uniform normalization. The spectral action
Tr f (D2 /Λ2 ) determines α through the a4 Seeley–
DeWitt coefficient: 1/(4α) = f0 Λd−4 TrK (T 2 ),
where TrK (T 2 ) is a single trace over all modes
of K simultaneously. The coupling α is a single
number for the entire gauge sector, not a per-mode
quantity. The gauge-normalized kinetic operator
is therefore Kphys = Kgeom /α, with the factor 1/α
uniform across all modes.
2. Complex mode structure. The Chern–Simons
theory on S 3 is quantized via holomorphic quantization [132], giving a state space with Kähler structure.
In the Toeplitz truncation, the modes are naturally
complex, so the Gaussian integral uses the complex
measure d2 ϕi .
3. Eigenvalue cancellation. The ratio of the gaugenormalized integral to the reference integral is
(πα/λi )/(π/λi ) = α, independent of λi (Eq. O6).
For 57 independent complex modes the product
gives α57 .
With α fixed by the exact construction of §X C, the
ζ(0) of the CP 2 charge trace times the terminating heat
trace of the ρ-shifted Casimir on S 3 with the heat-kernel
regulator, the normalization carries no a2k remainder,
and the eigenvalue cancellation is exact, not approximate.
The plateau rule, f (n) (0) = 0 for all n ≥ 1, removes
the a2k with k ≥ 3 only in even total dimension; on
the seven-dimensional K it does not, so it is not used
here.
a.

Notes on the Gaussian integration.

• Integration measure. The integration measure on
the Toeplitz-truncated Hilbert space CN is the flat
Lebesgue measure d2N ϕ, which coincides with the
Liouville measure of the Toeplitz-truncated Kähler
space under the constant Kähler metric on the finitedimensional fiber.
• Mass gap. The 57 nonzero eigenvalues {λi } are
bounded below by the Kaluza–Klein mass gap
√
(̸=0)
λmin ∼
α MP , ensuring that the Gaussian
(quadratic) approximation for the Wilsonian integration is controlled.

• Coupling normalization. The convention g 2 = α
(rather than g 2 = 4πα) follows from absorbing the
4π into the heat-kernel coefficient normalization f0 .
Lemma O.6 (Hierarchy identification). The dimensionless invariant I = GℏH02 /c5 scales as the partitionfunction ratio ε(α−1 ) = α57 : I = C ε(α−1 ), with C
measured (Theorem O.7).
Proof. The invariant I can be rewritten as a squared scale
ratio:


2
2
ℓP H0
EHubble
GℏH02
I =
=
=
(O7)
c5
c
EPlanck
p
where ℓP = ℏG/c3 , EHubble = ℏH0 , and EPlanck =
MP c2 . This is the squared ratio of the cosmological IR
scale to the Planck UV scale.
The partition-function ratio ε(α−1 ) computes the same
hierarchy: integrating out the 57 massive microsector
modes from the UV (Planck) theory yields the effective IR
(Hubble) theory, with suppression factor α57 (Lemmas O.4
and O.5).
The DFD microsector is finite-dimensional (dim H =
60). Unlike standard QFT, where the cosmologicalconstant calculation is quartically UV-divergent and
scheme-dependent, the microsector partition function (O4) is a finite product with no UV divergence, no
cutoff dependence, no renormalization ambiguity, and no
scheme dependence. The exponent of the identification
therefore inherits the exactness of the finite-dimensional
computation, free of the ambiguities that make the standard cosmological-constant problem intractable, for the
microsector piece only (Rem. AU.3).
4.

The derived invariant

Define the observed dimensionless invariant
G ℏ H02
I :=
.
(O8)
c5
As shown in the main text (critical density vs. Planck
density algebra),
ρc
3
ρΛ
3
=
I
and
= ΩΛ
I.
(O9)
ρPl
8π
ρPl
8π
Theorem O.7 (G–H0 –α invariant (spectral-action-derived)). Let K = CP 2 × S 3 with Chern–Simons
truncation at kmax = 60 and Ngen = 3 (Appendix F).
Within the DFD spectral action, the exact partition
function of the finite-dimensional microsector (60 modes,
3 zero, 57 nonzero) with gauge-normalized kinetic operator K/α gives the hierarchy suppression ε(α−1 ) = α57
(Lemmas O.4–O.6). Identifying this with the UV/IR
hierarchy fixes the exponent but leaves one dimensionless
normalization, measured in the acoustic sector (C = 1 on
bound-matter clocks under Postulate ZS.11, Eq. (479)):
G ℏ H02
= Cα57 ,
c5

C = 0.87784 ± 0.00266 (measured, early clock).

(O10)

195
Consequently,
3
ρc
=
Cα57 ,
ρPl
8π

3
ρΛ
= ΩΛ
Cα57 .
ρPl
8π

(O11)

Proof. By Lemma O.4, the 57 nonzero internal modes
are integrated out in the Wilsonian effective theory.
By Lemma O.5, the Gaussian integration over 57 complex modes with uniform gauge normalization 1/α gives
ε(α−1 ) = α57 , with the per-mode factor α independent of the eigenvalues. By Lemma O.6, the partitionfunction ratio fixes the α57 scaling of the physical hierarchy I = GℏH02 /c5 ; the printed spectral action does not
determine its overall coefficient C, as established by the
eight-route no-go in App. ZS. The density relations follow
from (O9).
a. Derivation status. Lemmas O.4 and O.5 are
theorem-grade: the mode count is 60 − 3 given the discrete inputs, the gauge normalization is from the a4
spectral coefficient, and the eigenvalue cancellation is
exact algebra. Lemma O.6 uses the Wilsonian effectivefield-theory framework applied to the finite-dimensional
DFD spectral action—the same level of rigour as standard QFT derivations, with the additional advantage
that the finite dimensionality eliminates all UV ambiguities. The identification is falsifiable. It is written GℏH02 /c5 = C α57 : the exponent 57 is a theorem,
while the dimensionless coefficient is a measured input,
C = 0.87784 ± 0.00266 (BAO+θ∗ ; plik-lite 0.8772), with
the cross-channel H0 = 67.52 ± 0.09 km/s/Mpc concordant across five CMB-anchored channels; the C=1 value
(H0 = 72.09) is excluded at ∼46σ as today’s early clock by
the CMB-anchored channels (eight-route non-derivability
no-go and measurement in App. ZS), and it is the boundmatter clock H0loc (Postulate ZS.11, Eq. (479)).
b. The cosmological-constant hierarchy. The hierarchy ρc /ρPl = (3/8π)Cα57 spans 57 × log10 (137) +
log10 (8π/3) ≈ 122.7 orders of magnitude. Each of the
57 frozen KK modes contributes one factor of 1/137 suppression. The mode count is 60 − 3 (Ngen = 3 an input);
the suppression factor is the gauge coupling from the
same topology. This is the microsector piece; the mattersector vacuum is assumed to cancel, and so is the spectral
action’s Planckian vacuum term (Rem. AU.3). Independent recent work likewise locates late-time acceleration
in a quantum-gravitational correction to the Friedmann
equation with no new fields [152]; there the deformation
scale and exponent are free inputs where DFD prints the
closed form (H0 tP )2 = C α57 , and its w > −1 prediction
is adjudicated against DFD’s forced w = −1 by the same
DESI/Euclid w(z) measurements.
5.

Connection to the Einstein Product Condition

The internal geometry enters the spectral action
through the volume and the curvature of K. The α
constraint does not fix the shape ratio τ ≡ R2 /R1 : the
CP 2 factor of α’s trace does not depend on its radius,

R
since CP 2 F ∧ F = 8π 2 q 2 is topological (§X C), and in
a joint reading 1/G (from a2 ) and 1/α (from a4 ) both
scale with Vol(K). TheR shape enters through the internalcurvature part of a2 , K RK dvol = π 8/7 Vol(K)5/7 Φ(τ )
with Φ(τ ) = 24τ 6/7 + 6τ −8/7 , which belongs to the fourdimensional vacuum
√ term. At fixed volume it is stationary at τ∗ = 1/ 3, the Einstein product condition
6/R12 = 2/R22 , with Φ′′ /Φ = 2.94, a curvature ratio rather
than a mass; this is a minimum of the potential for the
connection Laplacian that fixes α (§X B) and a maximum
for an internal Dirac operator, and the overall volume
is not fixed at this order. On the truncated microsector,
which the closed form for α and the α57 derivation use,
the internal space is finite and carries no size or shape
modulus.

196
Appendix P: Clock Coupling and Majorana Scale
1.

Scope and Convention Lock

This appendix derives two relations used in the microsector framework, each conditional on an explicitly stated
assumption:
α2
kα =
,
2π

(P1)

MR = MP α3 .

(P2)

The derivations follow the same “no hidden knobs”
methodology used in Appendix O (the α57 hierarchy): all
dimensionless outputs must be built from (i) the unique
dimensionless coupling α (given by the Chern-Simons
microsector closed form at kmax = 60) and (ii) topological
integers established in the paper (in particular the discrete
input Ngen = 3; App. F generation-count remark).
2.

Theorem P.1: Schwinger Coefficient ae = α/(2π)

Theorem P.1 (Schwinger one-loop anomalous magnetic
moment). In QED with one charged Dirac fermion of
charge e and mass m, the one-loop correction to the onshell vertex yields
ge − 2
α
ae :=
= F2 (0) =
+ O(α2 ),
(P3)
2
2π
where α = e2 /(4π) in ℏ = c = 1 units and F2 (q 2 ) is the
Pauli form factor.
Proof. Write the renormalized on-shell vertex as


iσ µν qν
F2 (q 2 ) u(p),
ū(p′ )Γµ (p′ , p)u(p) = ū(p′ ) γ µ F1 (q 2 ) +
2m
(P4)
with q = p′ − p and F1 (0) = 1 by charge renormalization.
The one-loop vertex graph gives (in Feynman gauge)
Z 4
(̸ p′ − ̸ k) + m
d k
γα ′
Γµ(1) = (−ie)3
4
(2π)
(p − k)2 − m2
(P5)
(̸ p− ̸ k) + m α 1
×γ µ
γ
.
(p − k)2 − m2 k 2
Projecting onto the Pauli structure and taking q 2 → 0
on-shell, standard Feynman-parameter reduction yields
Z 1
α
α
F2 (0) =
dz 2z =
.
(P6)
2π 0
2π
(Any UV divergence resides in F1 and cancels after renormalization; F2 (0) is finite.)
3.

Theorem P.2: Clock Coupling kα = α2 /(2π)

a. Microsector axiom (already used in the paper).
The “clock coupling” is defined operationally by the fractional shift of a purely electromagnetic atomic transition

under a small static DFD potential ψ:
δν
= kα ψ + O(ψ 2 ).
(P7)
ν
b. Key microsector input. In the DFD microsector,
α is topologically fixed (Appendix K) and therefore does
not vary with ψ at tree level.
Theorem P.2 (Clock coupling constant). Assume the
microsector “no hidden knobs” principle: in the weak-field
regime, the leading EM-sensitive ψ insertion is a single
gauge vertex and therefore carries one factor of α. Then
the coefficient kα in (P7) is forced to be
kα = α a e =

α2
2π

(P8)

Proof. By hypothesis, the leading ψ insertion into the EM
sector is a single gauge vertex, hence contributes a factor
α. The only universal, gauge-invariant, dimensionless oneloop EM correction that couples to atomic spin/magnetic
structure and is independent of atomic details is the Pauli
form factor at zero momentum, F2 (0) = ae (Theorem P.1).
Therefore the leading dimensionless coefficient multiplying
ψ in the EM sector is the product α ae . Using Theorem P.1
gives kα = α2 /(2π).
c. Remark (what is and is not a new assumption).
The only nontrivial input beyond QED is the microsector
rule that the leading ψ →EM insertion is a single gauge
vertex (“one α”), rather than an arbitrary analytic function of α. This is exactly the same kind of admissible
“no hidden knobs” restriction used in Appendix O to turn
the α57 hierarchy into a theorem.

a.

Observational Test: Fine-Structure Constant Variation

The clock coupling kα = α2 /(2π) predicts that the
fine-structure constant varies with the cosmological background potential ψ(z) (Sec. IX F, Eq. (253)):
∆α
ξ res
(z) = kα [ψ(z) − ψ0 ],
ψ(z) = LPI Ωm (z). (P9)
α
2
With Ωm,0 = 0.312 and Ωm (z=1) = 0.784:
2
∆α
res α
res
× 0.472 = ξLPI
× (+2.0 × 10−6 ),
= ξLPI
α z=1
4π
(P10)
res
with ξLPI
an undetermined screened residual (Sec. IX F),
so no definite value is predicted. The value +2.3 ×
10−6 , which follows from the excluded ψ-screen value
∆ψ(z=1) = 0.27, is not a DFD prediction.
a. ESPRESSO comparison. The ESPRESSO spectrograph at the VLT has measured ∆α/α in quasar absorption systems. The 2022 ESPRESSO collaboration
analysis reports:
∆α
= (+1.3 ± 1.3) × 10−6 .
(P11)
α z∼1

197
and, under the identification below, the Majorana mass
scale is

α(z) Prediction vs. ESPRESSO
res
DFD prediction: ∆α/α = ξLPI
× (+2.0 × 10−6 ) at
z = 1 (conditional; the value +2.3 × 10−6 follows from
the excluded screen)
ESPRESSO (2022): (+1.3 ± 1.3) × 10−6
res
Agreement: none to state until ξLPI
is determined; not
a validated prediction

b.

MR = MP ενR (α−1 ) = MP α3

Proof. Because HνR is finite-dimensional (non-extensive
microsector) with dim HνR = Ngen , constant scaling multiplies every eigenvalue by g and therefore multiplies the
determinant by g Ngen :

Key features.

det(gKνR ) = g Ngen det(KνR ).
res
ξLPI

> 0): DFD predicts α
1. Positive sign (for
increases at higher redshift (larger ψ). ESPRESSO
data prefer positive ∆α/α.
res
2. Magnitude: At ξLPI
= 1 the ∼ 10−6 level matches
current sensitivity.

3. z-dependence: ∆α/α ∝ Ωm (z) − Ωm,0 gives the
res
redshift shape, with the overall scale ξLPI
undetermined.
c. Predictions for ELT. The Extremely Large Telescope will improve sensitivity to ∼ 10−7 . DFD predictions:
res
z Ωm (z) − Ωm,0 ∆α/α (×10−6 × ξLPI
)

0.5
1.0
1.5
2.0
3.0

0.29
0.47
0.56
0.61
0.65

4.

+1.2
+2.0
+2.4
+2.6
+2.8

a. Setup. The right-handed neutrinos are gauge singlets (see Appendix H). Let HνR denote the internal
Hilbert subspace supporting the νR degrees of freedom.

Hence ενR (g) = g
. By the “no hidden knobs” principle, the Majorana scale can only be the unique fundamental mass MP multiplied by a dimensionless singletsector factor built from g and Ngen ; the determinant ratio
above is the unique such factor with the correct scaling
behavior. Substituting g = α−1 and Ngen = 3 gives
MR = MP α3 .
a.

Parallel Structure with Appendix O

The MR = MP α3 derivation parallels Appendix O
exactly:

(P12)

the discrete input of App. F (Rem. F.21); no index on
CP 2 × S 3 supplies it.
b. Toeplitz scaling input (same mechanism as Appendix O). Let KνR be the positive operator controlling the singlet-sector quadratic form in the Toeplitzquantized microsector. The microsector coupling parameter is g = α−1 , and constant-symbol scaling acts by
KνR 7→ g KνR .
Theorem P.4 (Majorana scale from determinant scaling). Assume (i) the singlet-sector quadratic form is nonextensive and Toeplitz-quantized on HνR , (ii) the only
dimensionless knob is g = α−1 , and (iii) dim HνR = Ngen
(Lemma P.3). Then the unique dimensionless singletsector suppression factor is
det(KνR )
= g −Ngen = αNgen = α3 , (P13)
ενR (g) :=
det(gKνR )

Appendix P (α3 )

State space HUV , dim = kmax = 60 HνR , dim = Ngen = 3
Operator Kinetic K, dim ker = 3 Majorana M, no kernel
Exponent kmax − Ngen = 57
Ngen = 3
Dictionary GℏH02 /c5 := C ε(α−1 ), MR /MP := ενR (α−1 )
C measured on the early
clock (C = 1 on
bound-matter clocks,
Postulate ZS.11)
Result
GℏH02 /c5 = Cα57
MR /MP = α3

Both use the same “no hidden knobs” principle.

Lemma P.3 (Generation multiplicity). The singlet sector
has one state per generation:
dim(HνR ) = Ngen = 3,

(P15)

−Ngen

Appendix O (α57 )

Majorana Scale MR = MP α3

(P14)

b.

Neutrino Mass Predictions

√
With v = MP α8 2π = 246.09 GeV (exponent derived,
prefactor fitted; Section XVII, App. Z) and the see-saw
formula mν ∼ m2D /MR :
a. Numerical result.
MR = MP × α3
= 1.22 × 1019 GeV × (137)−3 = 4.74 × 1012 GeV.
(P16)
b. Mass hierarchy. The neutrino mass ratios are
those of App. X’s Branch B:
m1 : m2 : m3 = 1 : α−3/11 : α−3/11−7/20 .

(P17)

Quantity

Prediction

m3 /m2

−7/20

Observed

α
= 5.60
(App. X, Branch B)

Σmν
Status

≈ 60 meV
< 120 meV (Planck+BAO)
Consistent, testable by DESI + Simons Observatory

model-dependent

198
c. Absolute scale. The Dirac Yukawa sits on a stated
2
rung, yD
= Fν α, one discrete input; with that rung the
priming factor Fν = 14/13 fixes it (Proposition AT.1,
Appendix AT). The Type-I seesaw mν3 = m2D /MR with
14
the locked scales m3 = 14
and MR = MP α3
13 πMP α
forces
2 m3 MR
14
2
yD
=
=
α,
2
v
13
q
(P18)
p
v
14
α
=
15.43
GeV,
m D = m 3 MR = √
2 13
√
where √the π of m3 cancels against the 2π in v =
2
2
1
MP α8 2π, and 14
13 = 1 + Ngen sin θW with sin θW =
3/13, Ngen = 3 (Prop. AT.1). This returns the absolute
scale
m2
14
mν3 = D =
πMP α14 = 50.16 meV
(P19)
MR
13
(the rung restated, not an independent check).

5.

Summary

Appendix Q: Temporal Completion: Dust Branch
under K ′ = µ

This appendix derives the temporal sector from the
same S 3 microsector that gives the weak-response µ(x)
of Appendix N. The key results are:
1. The temporal deviation invariance follows from the
saturation-union law (Assumption N.5)
2. The unique temporal segment variable is ∆ =
(c/a0 )|ψ̇ − ψ̇0 |
3. With K ′ (∆) = µ(∆), the dust branch emerges:
w → 0, c2s → 0
We also include a no-go lemma showing
that the naive
√
quadratic identification K ′ (Qt ) = µ( Qt ) gives w → 1/2
(not dust). This shows the dust branch is not automatic:
it follows from adopting the deviation-invariant ∆ closure.

1.

Temporal Deviation Invariance from
Saturation-Union

Appendix P: Theorem Status
kα = α2 /(2π): conditional on the “one gauge vertex”
axiom, a postulated non-metric operator outside the
minimal action (Thm. XI.2).
• Theorem P.1: ae = α/(2π) (Schwinger, QED —
fully proven)
• Theorem P.2: kα = α × ae (no hidden knobs
axiom)
• Observational test: ESPRESSO — conditional on
res
; not validated
the undetermined ξLPI
MR = MP α3 : an identification (the dictionary below),
listed among the inputs (App. FK).
• Input: Ngen = 3 (discrete input; numerically
χtop (CP 2 ) — App. F)
• Theorem P.4: det(gM) = g
linear algebra)

Ngen

det(M) (pure

• Dictionary: MR /MP := ενR (α−1 ) (explicit
identification)
• Predictions: the Branch-B spectrum of App. X;
Σmν = 61.5 meV

Both derivations follow the Appendix O protocol:
theorem-grade mathematics plus explicit “no hidden
knobs” axiom or dictionary identification. The exponents
(2 for kα , 3 for MR ) are not fitted—the first follows from
the one-gauge-vertex axiom together with the Schwinger
coefficient, the second is the generation number Ngen = 3,
a discrete input (App. F).

Theorem Q.1 (Temporal deviation invariance). Assume
the saturation-union composition law (Assumption N.5):




µ(ψ1 + ψ2 ) = 1 − 1 − µ(ψ1 ) 1 − µ(ψ2 ) ,
(Q1)
µ(0) = 0,

0 ≤ µ < 1.

Then for any background ψ0 and deviation ∆ψ,
µ(ψ0 + ∆ψ) − µ(ψ0 ) = (1 − µ(ψ0 )) µ(∆ψ)

(Q2)

Equivalently, the normalized incremental response depends
only on the deviation:
µ(ψ0 + ∆ψ) − µ(ψ0 )
= µ(∆ψ).
(Q3)
1 − µ(ψ0 )
Proof. Insert ψ1 = ψ0 and ψ2 = ∆ψ into Eq. (Q1):
µ(ψ0 + ∆ψ) = 1 − (1 − µ(ψ0 ))(1 − µ(∆ψ))
= µ(ψ0 ) + (1 − µ(ψ0 ))µ(∆ψ).
Rearrange to obtain (Q2).

2.

Unique Local Temporal Invariant

We identify the unique local scalar that represents the
microsector “increment” induced by time evolution along
a chosen screen flow.
a. Setup (DFD observer dictionary). Let uµ be the
unit timelike 4-velocity field of the cosmological screen
flow (comoving congruence in the dictionary), and let
ψ(x) be the DFD scalar. The screen-background field ψ0
is the ψ-screen solution already present in the cosmology
section (Sec. XVI).

199
Definition Q.2 (Local temporal increment density).
c
ψ̇ − ψ̇0 .
ψ̇ := uµ ∇µ ψ,
ψ̇0 := uµ ∇µ ψ0 ,
∆ :=
a0
(Q4)
√
Here a0 = 2 α cH0 is the MOND acceleration scale; the
combination c/a0 has units of time, so ∆ is dimensionless.
Theorem Q.3 (Temporal segment identification).
Among all local scalars built from ∇ψ and the screen
flow uµ , the quantity ∆ in Eq. (Q4) is the unique choice
(up to a constant factor) that satisfies:
1. Reparameterization covariance: invariance under reparameterizations of the flow parameter along
uµ .

For the k-essence stress-energy with p = K and ρ =
2Qt K ′ (Qt ) − K:
√
Qt
2
√ − Q3/2
ρ = 2Qt ·
+ ···
3 t
1 + Qt
4 3/2
= Qt + O(Q2t ).
3
3/2 
 4 3/2 

Thus w = p/ρ = 23 Qt
= 1/2.
3 Qt
Remark Q.5 (Role of the no-go lemma). Lemma Q.4
proves we did not cherry-pick the dust result. The S 3
composition law alone, with a naive quadratic identification, gives w = 1/2, not dust. The dust branch requires
the deviation-invariant closure below.

2. Segment additivity: for concatenated microsector
segments along the flow, the total “increment” equals
the sum of segment increments.

4.

3. Reference invariance: the amplitude vanishes
when ψ = ψ0 (the background).

a. Microsector-to-EFT identification (deviationinvariant). The temporal analog of the spatial AQUAL
closure, consistent with Theorem Q.1, uses the linear
deviation ∆:

Proof. A local scalar depending on ∇ψ and uµ at firstderivative order must be of the form f (uµ ∇µ ψ). Segment
R µ additivity applies to the integrated increment
u ∇µ ψ dλ, so the deviation from the background flow is
µ
u ∇µ (ψ − ψ0 ) = ψ̇ − ψ̇0 . Reference invariance forces subtraction of ψ̇0 . Dimensionlessness requires normalization
by a0 /c, yielding ∆.
3.

No-Go Lemma: Quadratic Invariant Gives
w → 1/2

Before proving the dust branch, we establish why the
naive k-essence identification fails.
Lemma Q.4 (No-go: quadratic invariant). Define the
quadratic temporal invariant Qt := (uµ ∇µ ψ)2 and suppose
the constitutive law is
√
p
Qt
′
√ .
K (Qt ) = µ( Qt ) =
(Q5)
1 + Qt
Then near Qt → 0:
2 3/2
(Q6)
K(Qt ) = Qt + O(Q2t ),
3
and the effective equation of state satisfies
p
1
w := →
(Qt → 0).
(Q7)
ρ
2
This is not dust.
√
Proof. Integrating (Q5) with q := Qt :
Z Qt
Z q
√
q ′2
K(Qt ) =
µ( s) ds = 2
dq ′
′
1
+
q
0
0
= q 2 − 2q + 2 ln(1 + q).
3/2

Taylor expanding at q → 0: K = 23 q 3 + O(q 4 ) = 23 Qt
O(Q2t ).

+

Dust Branch from Deviation-Invariant Closure

a20
K(∆),
8πG

∆
1+∆
(Q8)
where ∆ is the deviation invariant (Q4). This uses the
weak-response form µ(∆) = ∆/(1 + ∆) of App. N. Scope:
the deviation-invariant temporal EFE structure (Theorem Q.1) is forced by the composition law, but the specific
kinetic identification K ′ (∆) = µ(∆) is the minimal natural choice (the temporal analog of the spatial AQUAL
closure), not a uniquely forced identity: Lemma Q.4 exhibits a distinct,
√ mathematically admissible identification
K ′ (Qt ) = µ( Qt ) (which yields w → 12 , not dust). The
dust branch (Theorem Q.7) is therefore a conditional consequence of adopting K ′ = µ, not of the composition law
alone.
Ltemp =

K ′ (∆) = µ(∆) =

Lemma Q.6 (Shift symmetry current). Because Ltemp
depends on ψ only through ψ̇ (via ∆), it is invariant under
ψ 7→ ψ + const and yields a conserved current:
∇µ J µ = 0,

Jµ =

a20
c
K ′ (∆)
sgn(ψ̇ − ψ̇0 ) uµ .
8πG
a0
(Q9)

Theorem Q.7 (Dust branch). In a homogeneous FRW
dictionary with uµ = (1, 0, 0, 0) and ψ̇0 ̸= 0, solutions
near the screen background satisfy:
a3 a0 µ(∆) = const,

∆ ∝ a−3 /a0

(∆ ≪ 1), (Q10)

and their effective equation of state and sound speed obey
p
w := → 0,
c2s → 0 as ∆ → 0.
(Q11)
ρ
Proof. From (Q9) and ∇µ J µ = 0, homogeneity gives
d
3 0
3
′
′
dt (a J ) = 0, i.e. a a0 K (∆) = const. Using K (∆) =
2
µ(∆) yields (Q10). For ∆ ≪ 1, µ(∆) = ∆+O(∆ ), hence
∆ ∝ a−3 /a0 .

200
a2

0
For the stress-energy, take p = Ltemp = 8πG
K(∆) and
∂Ltemp
′
ρ = ψ̇ ∂ ψ̇ − Ltemp . Near ∆ = 0: K (∆) = ∆ + O(∆2 )

and K(∆) = 12 ∆2 + O(∆3 ). Thus:


c
a20
2
ψ̇0 sgn(ψ̇ − ψ̇0 ) ∆ + O(∆ ) ,
ρ=
8πG a0


a20 1 2
3
p=
∆ + O(∆ ) .
8πG 2
Therefore w = p/ρ = O(∆) → 0 as ∆ → 0. The adiabatic
sound speed c2s = dp/dρ satisfies dp/d∆ = O(∆) and
dρ/d∆ = const + O(∆), hence c2s → 0.
5.

Summary: What is Theorem-Grade vs. Program

Theorem-Grade Results
Proved from S 3 composition law + deviation
invariance:
1. Temporal deviation invariance (Theorem Q.1)
2. Unique temporal segment scalar
∆ = (c/a0 )|ψ̇ − ψ̇0 | (Theorem Q.3)
3. The kinetic identification K ′ (∆) = µ(∆) (same µ
as the spatial sector) is the minimal natural choice
consistent with the composition law, not uniquely
forced — Lemma Q.4 shows an alternative
identification is mathematically available
(natural/program-grade, not theorem-grade);
only the deviation-invariant EFE structure (item 1)
and, conditionally on this choice, the dust branch
(item 4) are proved
4. Dust branch: w → 0, c2s → 0 as ∆ → 0
(Theorem Q.7)
√
5. No-go: Quadratic K ′ (Qt ) = µ( Qt ) gives
w → 1/2 (Lemma Q.4)

Program-Level (Not Claimed as Theorem)
Requires further work:
• Full P (k) shape matching ΛCDM (linear
perturbation analysis)
• Transfer function derivation in DFD dictionary
• Quantitative confrontation with survey data
(noting GR-sandbox / fiducial-processing issues)
The dust branch (w → 0, c2s → 0) describes homogeneous
deviations only (Axiom WS.1); proving the full P (k)
match is a program item.

Remark Q.8 (Linear versus quadratic deviation). The
dust branch emerges because the microsector responds
to the linear deviation ∆ = |ψ̇ − ψ̇0 |, not the quadratic
Qt = (ψ̇ − ψ̇0 )2 . It is the adopted identification (item 3
of the Theorem-Grade Results box above), not a forced
one.

6.

Primordial optical engine: the background screen
flow

The temporal kinetic functional K(∆) is not the primordial engine. By construction ∆ = (c/a0 )|ψ̇ − ψ̇0 | is the
deviation from the background screen flow ψ̇0 (reference
invariance, §Q), so on the homogeneous FRW background
ψ̇ = ψ̇0 identically and ∆ = 0 (App. AE, AE.1). Hence
K(∆) governs homogeneous deviations δψ(t) only — the
dust branch (Axiom WS.1) — and vanishes on the background; it cannot drive a horizon engine. (The engine
is not a high-∆ branch of K.) The engine is instead the
background screen flow ψ0 (t) itself. The scalar action
alone is flat along it (Theorem AE.1), since its source
is the density contrast; the full action, with matter on
the matter-coupling metric, sources the homogeneous
field with (ρ + 3p)/2, and the Friedmann law of §XIX E
governs it. Because n = eψ0 and the optical speed is
ceff = c e−ψ0 , an early phase with ψ0 < 0 (fast light)
relaxing to ψ0 (t0 ) = 0 enlarges the optical causal horizon while the comoving optical Hubble radius shrinks,
if it runs fast enough (Theorem Q.9). The early fastlight era follows from the theory, since in the static arena
ψ̄ = 2 ln A and light moves at c/A2 on the arena clock,
but at the rate the Friedmann law gives it opens no horizon, and accelerating the observable scale factor needs a
source of negative pressure and a stage that reheats the
universe after it (Rem. Q.11).
Theorem Q.9 (Screen-flow horizon engine). Let a ∝ tp
(0 < p < 1, H = p/t) and let the background screen
flow be ψ0 (t) = β ln(t/t⋆ ), so ceff = c e−ψ0 ∝ t−β . The
opt
comoving optical Hubble radius is RH
= ceff /(aH) ∝
1−p−β
t
and the accumulated optical horizon is dopt
hor =
R t −ψ
R t ′−β−p ′
opt
′
0
.
Both
the
shrinking
of
RH
c
e
/a
dt
∝
t
dt
0
0
opt
and the divergence of dhor hold iff
β >1−p

=⇒

β > 12 (radiation, p = 12 ),
β > 13 (matter, p = 23 ).
(Q12)

opt
Proof. e−ψ0 ∝ t−β and aH ∝ tp−1 give RH
∝ t1−p−β ,
decreasing iff 1 − p − β < 0. The horizon integrand t′−β−p
diverges at the lower limit iff β + p ≥ 1. Both conditions
are β > 1 − p. (A power-law screen flow ψ0 = ψ∞ − At−s ,
A > 0, gives the same conclusion with an even stronger
−s
early fast-light phase ceff ∝ eAt → ∞.)

Theorem Q.10 (Branch selection by the optical arrow). The present normalization ψ0 (t0 ) = 0 (n = 1 today) admits three flat-direction branches: ψ̇0 = 0 (static,
ceff =const, no engine); ψ̇0 < 0 (then ψ0 > 0 and ceff < c
in the past, worsening the horizon); and ψ̇0 > 0 (then
ψ0 < 0, ceff > c in the past; it opens the horizon only
at the rate β > 1 − p). A fast-light primordial boundary
condition ψ0 (0+ ) → −∞ therefore selects ψ̇0 > 0 — the
engine branch.

201
Remark Q.11 (Status: the engine is not free, and it
does not make the primordial perturbations). The flatdirection status of ψ0 (Thm. AE.1) holds for the scalar
action alone, whose source is the density contrast: that
action supplies no force selecting the fast-light flow or
its rate β > 1 − p. The flow is not free of energy on
the observable scale factor. In the static arena of §II F
the homogeneous field is the expansion itself, ψ̄ = 2 ln A;
with an arena factor a(t), as written below, only the
split of A = a eψ̄/2 between a and ψ̄ is unobservable
(Theorem ZS.3), and the engine changes A itself: it is
accelerated expansion of the matter-coupling metric (the
physical payoff below). Matter on that metric sources the
homogeneous field with (ρ + 3p)/2, a term the contrast
source of Theorem AE.1 removes by construction and the
Friedmann law of §XIX E restores, so under that law it
needs a source with ρ + 3p < 0 while it runs, up to about
(4 × 1014 GeV)4 at the rate the χ relic allows (Eq. (Q13)),
and a stage that reheats the universe after it; the print
supplies neither (computed). The early fast-light era itself
follows from the theory: in the static arena light moves
at c/A2 on the arena clock, and the Friedmann law fixes
the rate of the flow at β = 4/(3w + 5), 2/3 in radiation,
below the engine condition β > 1 of that arena, so the
fast-light past is the ordinary expansion and opens no
horizon; the boundary data of Axiom PO below add only
the expanding branch, and the arrow of time rests on
the smooth beginning of Postulate Q.15. The screen flow
is not a varying-speed-of-light source for the primordial
spectrum either. On the matter-coupling metric a falling
optical speed is the expansion read on the arena’s clock
and changes no light-cone observable, and the local scalar
sector, an instantaneous constraint (Axiom WS.1), adds
no second cone. Even read as waves timed by absolute
time, ψ disturbances would move slower than light by the
ratio eψ̄ in the fast-light past, the reverse of what a soundspeed mechanism needs (computed). DFD therefore has
no mechanism for the primordial perturbations: they
enter through the smooth beginning of Postulate Q.15,
As is one calibrated input (App. FK), and the origin
of the perturbations is the theory’s main open problem
(§XVIII A). Any accelerating stage is bounded by the
χ relic, Hopt < 4 × 1010 GeV, which keeps the tensor
ratio below 10−7 . Consistency check: the primordial
background ψ̄ = ψ0 < 0 (n < 1, fast light) is the cosmological mean and is distinct from — and compatible with
— the local ψ > 0 (n > 1) of bound structures, which are
overdensities on that background.
Axiom PO (Primordial Optical State)
The universe’s Wheeler–DeWitt state is the no-boundary
optical state, specified by three clauses acting on disjoint
summands of the configuration space:
(i) Branch: the Euclidean ground-state branch
Ψ0 ∝ e−SE /ℏ (the Hartle–Hawking-type choice; the
rival e+SE /tunneling branch is thereby excluded);
(ii) Internal preparation: the empty-core WRT surgery

state on the closed internal S 3 (App. AV, Brick 1),
which, with the asserted premise P4′ (Lemma AV.27),
fixes the sector measure pj = |S0j |2 ;
(iii) Flat direction: ψ0 (0+ ) → −∞ with the fast-light
branch ψ̇0 > 0 on the homogeneous flat direction.

Theorem Q.12 (Consistency of Axiom PO). Axiom
PO is consistent. The Wheeler–DeWitt fluctuation-sector
state is already unique (Perron–Frobenius nondegeneracy
on the coercive Sobolev sector) and the internal vacuum
is rigid, a stated premise (layer 7 of the single-vacuum
theorem, Extended Derivations): the internal Dirac operator DF is held at its printed spectrum, and the physical
inner fluctuations are those of End(E) for the microsector bundle E = O(9) ⊕ O⊕5 (Lemma F.11), which carry
the gauge fields (their group is selected, not forced: its
electroweak factor rests on one stated identification and
colour is an input, App. F), not those of the full matrix
algebra M60 (C), under which its 60 eigenvalues would
be dynamical and the spectral action would drive them
off the printed spectrum (Prop. FK.3). So Axiom PO’s
only genuine freedom is the one homogeneous function
ψ0 (t) (the branch is fixed by normalizability, Rem. Q.14).
The three clauses constrain disjoint direct summands of
the configuration space (Euclidean branch / internal S 3
/ homogeneous flat direction), and the Einstein–Hilbert
term, which is not flat along the scale factor, fixes the
expansion of the homogeneous direction without fixing its
starting data, so clause (iii) stays independent of the Λ
anchor; hence no clause constrains another’s sector and
no contradiction can arise. Jointly the axiom yields: the
expanding branch (branch selection, Thm. Q.10 context;
the arrow of time rests on Postulate Q.15), the χ sector
measure pj = |S0j |2 (Brick 1–3, with the asserted premise
P4′ ), and the E0 = α57 MP4 anchor (eigenvalue step asserted, not derived, and under the lock of the lapse to
the scale factor a Hamiltonian eigenvalue acts as curvature, not as vacuum energy; see the status note to the
Wheeler–DeWitt ground-state theorem in the Extended
Derivations). Fast-light sign: under the Friedmann law
of §XIX E the homogeneous field in the static arena is
ψ̄ = 2 ln A, so an expanding universe has ψ̇0 > 0, the
fast-light branch, with ψ0 = β ln t and β = 4/(3w + 5)
for a fluid of equation of state w (β = 2/3 in radiation);
the engine condition β > 1 on the static arena then needs
w < −1/3, a source with ρ + 3p < 0.
Anti-Theorem Q.13 (Axiom PO is not derivable from the
written dynamics). The Wheeler–DeWitt operator is a
real operator; any unique normalizable state selected by
Perron–Frobenius positivity is therefore real and T -even,
and a T -even state can neither select a time orientation
nor supply the smooth beginning that the arrow of time
rests on. Moreover the Friedmann law that governs the homogeneous mode is invariant under time reversal, so both
branches are present symmetrically there: normalizability
and essential self-adjointness select no branch. Hence
no refinement of the written dynamics can derive clause

202
(iii) or the arrow: they rest on a Past-Hypothesis-class
input, the smooth beginning of Postulate Q.15, of the
same epistemic class as thermodynamics’ low-entropy past
and a low gravitational entropy, while the no-boundary
internal state is near-maximal entropy (≃ 93% of ln 61),
a cosmogenic selection, not a low-entropy postulate.
Remark Q.14 (Two inputs, not three). The early universe
rests on two inputs. Clause (ii) is the no-boundary prescription of clause (i) read on the internal S 3 : the empty
core is the absence of any insertion (Eq. (AV19)). It cannot come from energy minimization, since the abundance
operator is the Sugawara weight and its ground state,
j = 0, holds no dark matter; the data select it instead,
since a Wilson line of any spin from 12 to 59
2 in the core
gives Ωχ h2 = 0.135 to 0.141 and the Cardy boundary
weight gives 0.1255, against Planck’s 0.1200 ± 0.0012 (the
simple current j = 30 leaves the weights unchanged; computed). Clause (i) adds nothing beyond normalizability:
on the coercive fluctuation sector the branch e+SE /ℏ is
not normalizable, and on the internal sector the weight is
a phase. Clause (iii) is the expanding big bang read on the
arena clock: in the static arena ψ̄ = 2 ln A → −∞ at the
start, so it adds only the expanding branch (Rem. Q.11),
the direction of time away from the smooth beginning of
Postulate Q.15. The inputs are therefore the no-boundary
state, with nothing inserted, and the smooth beginning,
which no quantum state supplies (Antitheorem Q.13).
Postulate Q.15 (Smooth beginning). At the bang the
background is smooth, homogeneous and isotropic on the
slices of absolute time, and the early ripples are small and,
on every scale, in the regular class, whose Weyl curvature
vanishes at the bang, with the calibrated amplitude As and
the stated tilt ns = 1 − 2/60 (App. FK).
The postulate is stated, not derived: it is a premise of
the class of thermodynamics’ low-entropy past, and it
carries three things. It carries the arrow of time: at a
smooth bang the Weyl curvature vanishes, a state of low
gravitational entropy, and the clumping of matter grows
away from it (computed). It carries the uniformity of the
sky, which it states rather than explains, so the horizon
problem is not solved. And it carries the primordial
ripples, whose amplitude and tilt it takes as numbers and
whose origin stays open: the nine derived routes to an
early stage or a source of the ripples close (Prop. FK.3). It
is a statement about the classical ripples: every quantum
state of the printed theory carries the singular class as
well, and the conformal vacuum that J⋆ selects carries
it at equal weight (computed), so the no-boundary state
of Axiom PO, which fixes the ψ fluctuation sector and
the internal S 3 , does not supply it. The tilt tests it, and
so does the absence of primordial B-modes: the χ relic
bounds the tensor ratio at r ≲ 10−7 (App. SM).
Physical payoff (used in App. SM). The optical
metric g̃ = diag(−c2 e−ψ0 , a2 eψ0 δij ) is itself a flat FRW
metric, with scale factor ã = a eψ0 /2 and proper time dt̃ =
e−ψ0 /2 dt; it accelerates exactly when β > 1−p (the engine
condition (Q12)) and is de Sitter for ψ0 = 2 ln(t/t⋆ ).

Every species that couples to g̃ therefore sees ordinary
horizon-exit physics at the rate Hopt ≡ eψ0 /2 (H + ψ̇0 /2),
and a light χ gains (Hopt /2πfχ )2 of period-1 variance
per e-fold. Independently of how ψ0 (t) is selected, that
variance must not push Ωχ h2 past Planck’s two-sigma
ceiling (σθ2 < 1.9 × 10−3 ) over the N ≃ 36 e-folds of scale
between today’s horizon and the horizon at χ’s oscillation
onset, which gives
q
2
Hopt < 2πfχ σθ,max
/N ≃ 4 × 1010 GeV,
(Q13)
about 550× below the de Sitter-dictionary value H⋆ =
8πfχ = 2.4 × 1013 GeV (Thm. SM.3). The count of 36
e-folds is right for a stage that lasts only as long as it
must; each earlier e-fold adds a uniform displacement of
χ across our patch, so a longer stage tightens the bound
as N −1/2 , to about 8 × 109 GeV at 1000 e-folds, and a
stage without end leaves the starting angle free, which
must then lie within |θ0 | ≤ 0.044 of a period (computed).
The engine could inflate g̃ only with a source of negative
pressure and a stage that reheats (Rem. Q.11), and then
only at this low scale. This forces the primordial tensor
ratio below 10−7 , the tensor reclassification of App. SM.
Remark Q.16 (Scope: the engine is pre-BBN ; the validated late-time cosmology is unchanged). The fast-light
screen flow (ψ0 → −∞) is confined to the deep pre-BBN
primordial era; in the reading of this subsection, with
an expanding arena, ψ0 has relaxed to ψ0 ≃ 0 by bigbang nucleosynthesis, and whatever stage ends the engine
must reheat the universe before then, which the print
does not supply (Rem. Q.11). In the static arena of
the formalism the homogeneous field is the expansion
itself, ψ̄ = 2 ln A = −2 ln(1 + z), and keeps running to
the present. This opens no second cone for the perturbations: the local scalar sector is an instantaneous
constraint in absolute time, whose c-speed wave form is
only a numerical relaxation (Axiom WS.1), and the dust
branch of this appendix governs homogeneous deviations
only, so no ψ-wave speed enters the potential on CMB
scales, whichever clock is used. The engine therefore does
not rescale post-BBN redshifts and leaves the validated
DFD cosmology intact: real BBN and recombination
(z∗ ≃ 1100), the measured TCMB (z) = (1 + z) T0 , the
cobaya-validated acoustic scale, and the Etherington duality DL = (1 + z)2 DA (App. AG). The late-time dark
energy, the constant α57 vacuum term (App. AU), is not
the primordial engine. Caveat: a global optical redshift
factor that modified post-BBN redshifts — changing the
inferred age, or making 1 + z partly optical at recombination — would contradict all of the above and the
measured TCMB (z) = (1 + z); the engine makes no such
claim, and any reading of the CMB redshift as “partly
optical” (the Robitaille-type non-cosmic-CMB position)
is excluded here, not adopted.

203
Appendix R: EM–ψ Back-Reaction Coupling

This appendix develops the framework for electromagnetic back-reaction on the scalar field ψ, introducing a
single dimensionless parameter λ that controls whether
EM fields can source ψ oscillations. The resonance and
parametric analysis below assumes a dynamical ψ mode
with a natural frequency; under Axiom WS.1 the local ψ
is an instantaneous constraint with no such mode, so the
bounds and forecasts that rest on that analysis hold only
for a sector that carries one. We derive both “accidental”
constraints from existing cavity stability and “intentional”
search protocols that could reach |λ − 1| ∼ 3 × 10−9 .

1.

Physical Interpretation of λ

The parameter λ toggles the EM–ψ interaction:
• λ = 1: EM probes the optical metric n = eψ but
does not source ψ
• |λ − 1| ̸= 0: EM can pump ψ modes (laboratory
generation possible)
a. Intuitive picture.
a paddle:

Think of ψ as water and EM as

• λ = 1: The paddle slides across without making
waves
• |λ − 1| ̸= 0: The paddle makes waves; pump with
the right rhythm and they grow
b. Relation to core postulates. The core DFD postulates (Sec. I B) specify how ψ affects EM propagation
(n = eψ , c1 = ce−ψ ). The parameter λ addresses the
inverse question: can EM fields actively modify ψ? This
is a distinct physical degree of freedom not constrained
by the forward propagation relations.

2.

Mode Equation and Pumping Channels
a.

• U (t) = U0 [1 + m cos(2ωt)]: stored EM energy with
modulation depth m
• α: parametric coupling coefficient
The invariant Ξ carries a 2ω component for a cavity
driven at frequency ω, providing two pumping channels.

b.

Channel 1: Driven Resonance (2ω = Ωψ )

When twice the EM drive frequency matches the ψmode frequency, direct resonant driving occurs. The
steady-state amplitude is:
|λ − 1||G|
|q|res ≃
,
(R2)
2Mψ Ωψ γψ
where the geometry overlap is:
Z
G ≡ u(r) Ξ̂2ω (r) d3 r,
(R3)
with Ξ̂2ω the 2ω Fourier component of Ξ.
c.

Channel 2: Parametric Amplification (2ω ≃ 2Ωψ )

The stiffness modulation from U (t) creates parametric
gain. The Mathieu gain parameter is:
U0
h = (λ − 1)
Hm,
(R4)
Mψ Ω2ψ
with instability growth rate:
1
(R5)
Γ ≃ hΩψ − γψ .
2
The overlap H is:
Z
1
µ0 2
ε0
H=
H . (R6)
u2 (r) w(r) d3 r, w = E 2 +
U0
4
4
a. Instability threshold. Parametric instability occurs
when Γ > 0:
2γψ Mψ Ω2ψ
|λ − 1|min =
.
(R7)
Ωψ U0 Hm

Single Lab-Mode Reduction
3.

Reduce the ψ field to a single laboratory mode q(t)
with natural frequency Ωψ and damping γψ :
Z
(λ − 1)
u(r) Ξ(r, t) d3 r + αU (t)q
q̈ + 2γψ q̇ + Ω2ψ q =
Mψ
(R1)
where:
• u(r): normalized spatial profile of the ψ mode
• Mψ : effective mass of the mode


2
• Ξ(r, t) ≡ − 12 e−2ψ0 B 2 − Ec2 : EM Lagrangian invariant (∝ Fµν F µν )

a.

Geometry Transparency

When the Driven Overlap Cancels

For a single, symmetric pillbox cavity driven in a pure
eigenmode (TM010 or TE011 ), Bessel identities and timeaveraged equipartition make:

Z 
E2
2
B − 2 d3 r ≈ 0 ⇒ G ≈ 0.
(R8)
c
The driven channel is geometrically transparent for
symmetric cavities in pure eigenmodes.

204
b.

b.

How to Restore the Overlap

1. TE+TM superposition: Co-phased modes with
matched radii give G = u(z0 )e−2ψ0 η× U0 cos ϕ,
where η× = O(0.1–1).
2. Asymmetric geometry: Small irises or nearcutoff asymmetries break equipartition.
3. Mode beating: Two nearby modes at frequencies
ω1 , ω2 produce 2ω = ω1 + ω2 components.
Parametric Overlap: Area-Ratio Law

For a ψ-mode “tube” of height L and cross-section Aψ ,
with N compact cavities of total aperture Acav,tot placed
at antinodes:
2
Acav,tot
H ≈ κeff
,
L
Aψ

(R9)

where κeff = O(1) captures mode-shape details.
Combining with (R7) and using Mψ ≃ Aψ L/(2πcs ) for
a 1D standing mode:
|λ − 1|min =

A2ψ
πγψ
cs U0 m κeff Acav,tot

(R10)

As written, Eq. (R10) carries units of m/J (Mψ above
has units of m2 s, not those of a mode mass), so it fixes
the ratio of the accidental and intentional values below,
about 9 × 103 , but not their absolute size.

4.
a.

Constraints on |λ − 1|

Accidental Constraint from Cavity Stability

The mere stability of existing high-Q cavities—the absence of observed parametric instability near twice the
drive frequency—provides a conservative bound.
a. Conservative parameters.
• Stored energy: U0 ∼ 100 kJ
• Modulation depth: m ∼ 0.01 (ambient amplitude/PLL dither)
• Loss ratio: γψ /Ωψ ∼ 10−3
• Tube area: Aψ ∼ 0.8 m2
• Cavity aperture: Acav,tot ∼ 3 × 10−3 m2
• κeff ∼ 1, cs ≤ c

Using Eq. (R10):
|λ − 1| ≲ 3 × 10−5

Three methods restore G ̸= 0:

c.

Result.

(R11)

Any substantially larger coupling would have produced
obvious parametric instability in normal cavity operation—and it has not.
c. Companion note: minimal-sector baseline. A
companion note [116] analyzes the minimal tree-level
optical-metric
the gauge-invariant action
R √ EM sector:
−(1/4µ0 )
−g̃ Fµν F µν d4 x in the DFD optical metric
n = eψ , with no additional operators and no dual-sector
constitutive splitting. For an ideal symmetric single-mode
standing-wave cavity with exact equipartition, the barecoupling value of λ equals unity at tree level, λbare = 1.
This establishes a minimal-sector null baseline; it does not
claim that the effective laboratory inference λeff equals
unity. In DFD the two are related by
λeff − 1 = δQ + δgeom + δκ + δξ ,

(R12)

where δQ ∼ 1/Q is a finite-Q equipartition-violation
mimic, δgeom captures the geometry-restoration channels
of Sec. R 3 (TE+TM superposition, asymmetric cavity
geometries, mode beating), δκ captures any dual-sector
constitutive splitting, and δξ captures a beyond-baseline
dimension-5 operator ξ ψ Fµν F µν . The accidental bound
Eq. (R11) and the projected intentional-search reach
(Eq. (R15) below) therefore constrain the sum |λeff − 1|,
not a single intrinsic DFD parameter. Separating the
channels in a real experiment requires Q-scaling series,
cavity-geometry variation, and polarization-sensitive protocols.
d. λ-channel protection: structural origin of the tree
baseline. The companion-note baseline λbare = 1 is
structural, not accidental. Direct
variation
R
√ of theµνminimal gauge-invariant action −(1/4µ0 )
−g̃ Fµν F d4 x
−ψ
+ψ
in
metric (g̃00 = −e , g̃ij = e δij , for which
√ the optical
−g̃ Fµν F µν ∝ e−ψ B 2 − e+ψ E 2 /c2 ) gives
δSEM
E2
ψ→0
∝ e+ψ 2 + e−ψ B 2 −−−−→ 2µ0 uEM , (R13)
δψ
c
the opposite-sign metric exponents differentiating to a
common sign: minimal coupling sources ψ only through
the EM energy density uEM — the (ε + 3P ) channel of
the Optical Source Theorem (Theorem II.1; for radiation
ε + 3P = 2u, hence the factor 2) — so light gravitates,
exactly as the forward theory requires. The pumping invariant Ξ ∝ (B 2 − E 2 /c2 ) of Eq. (R1) is the opposite-sign
combination, a different operator that minimal coupling
cannot generate at tree level for any field strength or cavity geometry. The λ-drive coefficient therefore vanishes
at tree level as a matter of coupling structure — a protection argument, not merely the ideal-cavity equipartition
computation of the companion note.
e. Intrinsic radiative floor and a positive-signal prediction (sketch grade). [Status: identification at sketch
grade; the O(1) operator matching between the ξ ψFµν F µν
vertex and the transition-frequency response that defines
kα is stated, not yet computed.] The radiative response of

205
EM observables to the local optical scale ψ is the same
physical object that App. P derives as the clock coupling
kα = α ae = α2 /(2π) ≃ 8.5 × 10−6 (Theorem P.2, resting on the same “one gauge vertex” microsector rule).
Identifying the induced δξ with this coupling places the
intrinsic floor at
|λeff − 1|intrinsic ∼ kα ≈ 10−5–6 ,

(R14)

a factor ∼ 3.5 below the accidental bound Eq. (R11) —
consistent with the absence of observed parametric instability to date — and ∼ 3×103 above the intentional-search
reach (Eq. (R15) below). Two consequences follow. (i) A
conditional discriminator (sketch grade): a generic gravitationally coupled dilaton carries the anomaly-strength
coefficient ξ ∼ α/(3π) ≈ 7.7 × 10−4 , which the accidental bound Eq. (R11) excludes by a factor ∼ 25 —
existing cavity-stability data discriminate in favor of
the microsector-suppressed DFD value over the genericdilaton alternative. (ii) A two-tier falsifiable forecast:
conditional on the kα identification, DFD predicts the
intentional search below finds a positive signal at the kα
level, not a null. A signal at ∼ 10−5–6 would confirm
the microsector clock coupling in a second, independent
channel; a null at 10−7 falsifies the kα identification (while
leaving the structural tree baseline λbare = 1 of Eq. (R13)
intact, testable down to the 3 × 10−9 reach floor).

b.

Intentional Search: Projected Reach

Why λ ̸= 1 Has Not Been Detected

5.

Three factors explain the null result in existing metrology:
1. Pure eigenmodes suppress the driven channel.
Symmetric cavities in pure modes have G ≈ 0 by
Bessel-function orthogonality and equipartition.
2. Parametric pumping needs deliberate 2ω.
Routine metrology avoids such tones and heavily
filters them to suppress amplitude-modulation sidebands.
3. 2ω features treated as technical noise. Any
residual 2ω response is interpreted as technical AM
sidebands and actively suppressed, not investigated
as a potential signal.
a.

To detect |λ − 1| ̸= 0:

• Use TE+TM superposition (restores G ̸= 0)
• Deliberately apply 2ω modulation
• Preserve (not suppress) 2ω response
• Monitor for resonant growth at Ωψ
6.

With deliberate optimization using the same physics:
• U0 → 1 MJ (factor 10 increase)

Intentional Detection Protocol

Intentional ψ-Pump Detection: Required Capabilities
1. High-Q resonator (Q ≳ 104 ) with stored energy
U0 ≳ 1 MJ (pulsed acceptable)

• m → 0.1 (factor 10 increase)
• Array apertures at all antinodes: Acav,tot → 3×10−2
m2 (factor 10)
• Shrink tube area: Aψ → 0.27 m (factor ∼3 reduction)
2

• Maintain γψ /Ωψ ∼ 10−3

2. Phase-stable amplitude modulation at 2ω
with depth m ∼ 0.1 on stored energy
3. Placement of cavity apertures at ψ
antinodes (maximize H; use multiple irises)
4. Phase-sensitive readout near Ωψ ; preserve 2ω
tones (do not auto-suppress)
5. Null sensitivity target: |λ − 1| ≲ 3 × 10−9

The design law (R10) then gives:
|λ − 1| ∼ 3 × 10−9

(accessible reach)

(R15)

a. Orthogonal cross-check: Driven amplitude. With
a TE+TM superposition (η× ̸= 0, phase ϕ = 0):
∆ψ ≡ u(z0 )|q|res ≈

TABLE XCV. Accidental vs. intentional search parameters.
Parameter
Stored energy U0 (J)
Modulation depth m
Cavity aperture Acav,tot (m2 )
Tube area Aψ (m2 )
Loss ratio γψ /Ωψ
Projected |λ − 1|min

Accidental
105
0.01
3 × 10−3
0.8
10−3
≲ 3 × 10−5

Intentional
106
0.10
3 × 10−2
0.27
10−3
∼ 3 × 10−9

|λ − 1|η× U0 cs
.
πAψ γψ

(R16)

For η× ∼ 0.3, U0 = 100 kJ, Aψ = 0.8 m2 , γψ = 0.03 s−1 :
∆ψ ∼ 1.2 × 10−3 |λ − 1|,

(R17)

which crosses cavity-atom sensitivity (Sec. XII) in the
10−12 –10−15 range for |λ − 1| in 10−9 –10−12 . As written,
Eq. (R16) carries units of J/m (through the same Mψ as
Eq. (R10)) and, with cs = c, evaluates to 1.2 × 1014 |λ − 1|,
so it does not fix the coefficient 1.2 × 10−3 .

206
7.

Relation to Core DFD Framework

a. Consistency with postulates. The parameter λ
does not modify the core postulates:
• Refractive index: n = eψ (unchanged)
• One-way light speed: c1 = ce

−ψ

a.

• Matter acceleration: a = (c /2)∇ψ (unchanged)
Eq.

(21)

(unchanged

for

The vacuum permittivity and permeability can respond
asymmetrically to ψ:
ϵ(ψ) = ϵ0 n e+κψ ,

Summary

where n = e and κ is the split parameter.
The product is preserved:
1
c
ϵ(ψ)µ(ψ) = ϵ0 µ0 n2 ⇒ vph = √ = .
ϵµ
n

(R20)

• κ = 0: Electric and magnetic sectors respond identically to ψ (symmetric case)
• κ ̸= 0: Sector-differential response; electric and
magnetic energies couple differently

2. Existing cavity stability provides an accidental
bound:
|λ − 1| ≲ 3 × 10−5 .

(R18)

3. Deliberate optimization enables an intentional
search reaching:
|λ − 1| ∼ 3 × 10−9 .

(R21)

Thus the optical metric phase speed is unchanged by the
split.
a. Physical interpretation.

1. The parameter λ controls EM back-reaction on ψ:
λ = 1 means EM probes but doesn’t pump; |λ−1| ̸=
0 enables laboratory ψ-generation.

(R19)

4. The null detection so far is explained by geometry
transparency and suppression of 2ω components in
standard metrology.
5. A dedicated search protocol with TE+TM superposition and preserved 2ω response could either
discover λ ̸= 1 or constrain it below 3 × 10−9 with
upgraded apparatus.
Key Result
Unoptimized cavities accidentally constrain
|λ − 1| ≲ 3 × 10−5 . An intentional 2ω modulation test
with upgraded hardware (Table XCV) pushes about
four orders of magnitude tighter. A dedicated
measurement could either discover λ ̸= 1 or constrain it
below 3 × 10−9 .

9.

µ(ψ) = µ0 n e−κψ ,

ψ

The λ parameter describes a dynamic EM–ψ interaction
orthogonal to the static field relations. It affects how
rapidly oscillating EM fields can pump ψ modes, not how
ψ affects light propagation.
b. Default value. Without additional physics, λ = 1
(no back-reaction) is the natural default. Any |λ − 1| ̸= 0
indicates additional EM–gravity coupling beyond metric
propagation effects.
8.

Constitutive Split Preserving vph = c/n

(unchanged)

2

• Field equation:
static/quasi-static)

a. Status of the κ parameter. At tree level, the Gordon optical metric gives κ = 0, i.e. no electric–magnetic
constitutive split.

Dual-Sector Extension: The κ Parameter

Beyond the λ parameter controlling EM back-reaction,
a second parameter κ controls the differential response
of electric and magnetic sectors to ψ.

b.

The Unified Bracket

With the split (R20), a single bracket governs energy
exchange, body force, and ψ sourcing:
B2
− ϵE 2 .
(R22)
µ
a. Energy exchange. The Poynting theorem acquires:
κ
(R23)
∂t u + ∇ · S = −J · E − ψ̇ B.
2
b. Body force. Fields exert force on the medium:
κ
fψ = − B ∇ψ.
(R24)
2
c. ψ sourcing. EM fields can source ψ:
δLψ
κ
= Smass + B.
(R25)
δψ
2
B≡

c.

Standing-Wave Energy Equality

For a lossless, steady-state standing wave in a linear
medium, the cycle-averaged integrated energies are equal:
Z
Z
ϵE 2 dV =
B 2 /µ dV,
(R26)
V
V
R
so V B dV = 0. The integrated bracket vanishes for ideal
standing waves.
a. Local imbalance. Nonzero local bracket arises at
O(θ2 ) due to longitudinal fields in paraxial Gaussian
modes. For a TEM00 cavity mode with waist w0 :
λ
ϵE 2 − B 2 /µ ∼ θ2 ϵ|E0 |2 ,
θ=
.
(R27)
πw0

207
For λ = 1064 nm, w0 = 300 µm: θ2 ≃ 1.3 × 10−6 .
d.

Appendix S: Standard Model Extension Dictionary

Experimental Tests

a. Accidental bound from cavity stability. Absence
of 2ω parametric instabilities in extreme-Q resonators
constrains unintended EM↔ ψ pumping.
b. LPI residual as κ test. After the constitutive-chain
cancellation of Sec. XII A, the cavity–atom observable
is a screened residual rather than an order-unity slope.
Nevertheless, the sector-resolved residual still depends on
κ via:
(M )

res
ξLPI
(κ) = (screened residual of) 1 − αL

(S)

− αat (κ),

(S)

αat (κ) = Kϵ(S) κ + O(κ2 ).
(R28)
(S)
where Kϵ is the atomic EM-energy sensitivity.
(S)
c. Order-of-magnitude for Kϵ . Atomic

optical
transition energies scale with the effective Rydberg R∞ ∝
1/ϵ2 , giving:
δE
δϵ
≃ −2
E gross
ϵ

⇒

Kϵ(S) ∼ O(1–3).
(S)

For Sr and Yb clock transitions, Kϵ
unity.

e.

(R29)

is plausibly order

This appendix maps DFD parameters onto the language of the Standard-Model Extension (SME) [153], enabling direct comparison with published experimental
constraints.

1.

SME Framework Overview

The SME provides a phenomenological framework for
parameterizing possible violations of Lorentz invariance
and the Einstein Equivalence Principle. For gravitational
tests with atomic clocks, the relevant observable is:
δ(fA /fB )
∆U
(S1)
= (βA − βB ) 2 ,
(fA /fB )
c
where βA , βB encode gravitational redshift anomalies for
species A and B.

2.

DFD↔SME Correspondence

In DFD, the same observable is:
δ(fA /fB )
∆Φ
= (ξA − ξB ) 2 ,
(S2)
(fA /fB )
c
where the effective coupling ξA includes both matter and
photon sector contributions:

Experimental Discrimination

ξA ≡ KA + δA,γ ,

(S3)

2. Dual-wavelength measurements:
κ is
wavelength-independent; dispersion effects are not.

with KA twice the magnitude of the per-unit-ψ channelresolved coupling of Eq. (291) (whose pure-α leading term
α
is kα · SA
; Sec. XI B) and δA,γ = 1 if species A involves
a photon-sector reference. After the constitutive-chain
cancellation of Sec. XII, the photon-sector contribution
δA,γ is absorbed into the tree-level cancellation and the
surviving observable is a screened residual.
Identifying ∆U ↔ ∆Φ, the direct correspondence is:

3. Multi-species clock comparisons: Different
(S)
atoms have different Kϵ values.

βA − βB ←→ ξA − ξB = (KA − KB ) + (δA,γ − δB,γ )
(S4)

κ can be tested via:
1. TE/TM polarization swaps: Pure TE (magnetic
dominant) vs pure TM (electric dominant) modes
have opposite bracket signs.

Dual-Sector Extension Summary
The κ parameter:
• Controls differential ϵ/µ response to ψ while
preserving vph = c/n
• Unified bracket B = B 2 /µ − ϵE 2 governs energy,
force, and sourcing

3.

Translation Table

TABLE XCVI. DFD↔SME parameter correspondence.
DFD Quantity SME Analogue Meaning
ψ/2
KA
δA,γ
ξA = KA + δA,γ
kα = α2 /(2π)

U/c2
βA (matter)
βA (photon)
Total βA
—

Background grav. field
Species-dep. coupling
Photon-mode coupling
Composite LPI param.
DFD-specific scale

208
4.

Appendix T: Family and Clock-Type Parametrization
of LPI Tests

Experimental Constraints Reinterpreted

Published SME bounds can be reinterpreted as DFD
constraints:

5.

Cavity-Atom Comparisons in SME Language

For cavity-atom LPI tests (Sec. XII), the SME parameterization becomes:


ξLPI
d
νatom
ξatom − ξcavity
= 2 ,
(S5)
=
dΦ νcavity
c2
c
where the tree-level assignment ξcavity = 1 is not used.
After the constitutive-chain cancellation of Sec. XII, both
cavity and atomic sectors share the universal geometric
redshift at tree level, so only a screened residual mismatch
survives:
res
ξLPI −→ ξLPI
.

(S6)

a. Significance. In SME-style language, DFD does
not predict an order-unity cavity coefficient. Instead it
predicts that any measurable cavity–atom anomaly must
arise from a channel-resolved residual, consistent with the
four-term structure of Eq. (291) and the screening logic
summarized in Secs. XI and XII.

This appendix presents a phenomenological
parametrization organizing clock comparison tests
by chemical family and clock type. The framework
provides a compact way to encode where current data
pull and where future tests should focus.

1.

Two-Parameter Model

Motivated by the pattern of hints and nulls in clock
comparisons, we parameterize the gravitational coupling
coefficient as:
(i)

Ki = kN CN + ke Ce(i) + kα κ(i)
α ,

(T1)

where:
(i)

• CN : Nuclear-sector charge depending on chemical
family
(i)

• Ce : Electronic-sector charge depending on clock
type
(i)

• κα = Siα : Standard α-sensitivity
• kN , ke , kα : Coupling strengths to be fit or constrained
a.

Family charges.

Based on chemical grouping:

Element family

CN

Alkaline earth (Sr, Ca, Mg) 0
Alkali (Cs, Rb, H)
1
Post-transition (Al, Hg, In) 1.5
Lanthanide (Yb, Dy)
2
b.

Clock-type charges. Based on interrogation mode:


optical neutral
0
Ce = 0.5 trapped ion
(T2)

1
microwave hyperfine

These assignments are deliberately coarse; the point
is not that nuclear scalar charges take precisely these
values, but that grouping by family and clock type yields
a testable pattern.

2.

Constraints from Data

For each clock pair (A, B), the observable is:
∆KAB = kN ∆CN + ke ∆Ce + kα ∆κα ,

(T3)

(A)
(B)
where ∆CN = CN − CN and similarly for other quan-

tities.

209
TABLE XCVII. SME bounds reinterpreted in DFD framework.
Experiment

SME Constraint

DFD Interpretation
−7

Ref.

−7

H maser/Cs (14-yr) |βH − βCs | < 2.5 × 10
|KH − KCs | < 2.5 × 10
[154]
Yb+ E3/E2 (PTB)
|βE3 − βE2 | < 10−8
Same-ion: composition cancels [155]
Hg+ /Cs
|βHg − βCs | < 5.8 × 10−6
|KHg − KCs | < 5.8 × 10−6
[156]
+
+
−7
Al /Hg
|βAl − βHg | < 5.3 × 10
|KAl − KHg | < 5.3 × 10−7
[157]

a. E3/E2 constraint on kα . The Yb+ E3/E2 sameion comparison has ∆CN = ∆Ce = 0 but ∆κα = −6.95.
The PTB bound |∆KE3/E2 | < 10−8 thus constrains:
|kα | < 1.4 × 10−9 .

(T4)

3. Consistency across untested channels validates the
two-parameter structure.

4.

Relation to DFD Microsector

This bounds the unscreened pure-α coefficient of this
phenomenological model; it is not a bound on the commonsector kα of Appendix P (Sec. XI).
b. Cross-species constraints. With kα = 0 fixed, the
two-parameter model (kN , ke ) is constrained by:

The phenomenological charges (CN , Ce ) are not derived
from first principles in this appendix. However, they are
compatible with the DFD microsector in the following
sense:

• H/Cs null (∆CN = ∆Ce = 0): automatically satisfied

• The overall scale kN , ke ∼ 10−5 –10−6 is consistent
with kα = α2 /(2π) structure.

• Hg+ /Cs: ∆CN = 0.5, ∆Ce = −0.5, bound |∆K| <
5.8 × 10−6

• The family grouping (alkaline earth vs. lanthanide)
suggests coupling to properties correlated with
atomic structure, not just α.

• Dy/Cs: ∆CN = 1, ∆Ce = −1, bound |∆K| < 10−5
• Cs/Sr hint: ∆CN = 1, ∆Ce = 1, suggests ∆K ∼
3 × 10−5

• The clock-type structure (ion vs. neutral) aligns
with the sector-coupling hierarchy in Sec. XI G.

A joint fit yields kN ∼ 6 × 10−6 , ke ∼ 1.5 × 10−5 with
χ2ν ≈ 1.

A full derivation of (CN , Ce ) from the CP2 × S 3 microsector remains an open problem. The present appendix
establishes the empirical pattern that such a derivation
must reproduce.

3.

Predictions for Untested Channels

The model predicts specific ∆K values for channels
where high-precision ratios exist but LPI analyses have
not been performed:

5.

Summary

The family+clock parametrization provides:
1. A compact organization of existing LPI constraints

TABLE XCVIII. Family+clock model predictions for untested
LPI channels.
Channel
Sr+ /Sr

∆CN ∆Ce Predicted ∆K

Test type

7.5 × 10−6

0 +0.5
Pure electronic
Ca+ /Sr
0 +0.5
7.5 × 10−6
Pure electronic
H-maser/Sr +1 +1 equal to Cs/Sr Same charges as Cs/Sr
Hg/Sr
+1.5 0
9 × 10−6
Pure nuclear-family
+
+
Yb /Sr
+2
0
1.2 × 10−5
Pure nuclear-family
Hg/Yb
−0.5 0
−3 × 10−6
Partial cancellation
Ca/Sr
0
0
0
Null prediction

a.

2. Specific predictions for channels where analyses are
actionable

Falsification criteria.

1. An observed Ca/Sr LPI signal at ∼ 10−5 would
falsify the family structure.
2. Hg/Sr or Ca+ /Sr showing null results at < 10−6
would severely constrain both kN and ke .

3. Clear falsification criteria
4. A target pattern for microsector derivation
The decisive tests are Hg/Sr (pure nuclear-family) and
Sr+ /Sr (pure electronic), both of which can be performed
with existing clock technology.

210
Appendix U: Mathematical Well-Posedness of the
DFD Field Equations

This appendix establishes the mathematical foundations of DFD as a well-posed partial differential equation
system. We treat both the static (elliptic) boundary value
problem relevant for equilibrium configurations and the
dynamic (hyperbolic) Cauchy problem of its numerical
relaxation (under Axiom WS.1 the scalar is an instantaneous constraint). The analysis follows standard methods
from monotone operator theory [158, 159] and quasilinear
elliptic theory [32, 33].
For DFD to stand alongside General Relativity as a
viable relativistic gravity theory, it is not enough to match
phenomenology. The underlying PDE must be mathematically well posed: given appropriate initial (and, when
relevant, boundary) data, there should exist a unique solution in a suitable Sobolev class, depending continuously
on the data.

1.

The Static Field Equation: Elliptic Theory

The DFD static field equation is:

 8πG
(U1)
−∇ · µ(|∇ψ|)∇ψ = 2 ρ,
c
where µ : [0, ∞) → [0, ∞) is the interpolation function
satisfying µ(x) = x/(1 + x).

F (x) ≥ x − e−1 (TOTAL). For (A4), define a(ξ) = µ(|ξ|)ξ
and compute:
∂ai
ξi ξj
= µ(|ξ|)δij + µ′ (|ξ|)
.
(U5)
∂ξj
|ξ|
Since µ′ (s) = 1/(1 + s)2 > 0, this matrix is positive
semi-definite, establishing monotonicity.
Remark U.2 (Catalog of admissible µ-families). Other
functions satisfying (A1)–(A4) include:
• p-Laplacian: µ(s) = sp−2
• Saturating: µ(s) = (1 + s2 )(p−2)/2
p
• Regularized MOND-like: µ(s) = s/ s2 + s2a
The weak-response approximant µ(x) = x/(1 + x) of the
DFD-derived response is distinguished by its topological
origin (Appendix N).

b.

Weak Formulation and Variational Structure

Define the flux map a(ξ) := µ(|ξ|)ξ. For ψ ∈ W 1,p (Ω)
with boundary data ψ = ψD on ∂Ω, the weak formulation
is:
Z
Z
a(∇ψ) · ∇v dx =
f v dx, ∀ v ∈ W01,p (Ω), (U6)
Ω

Ω

where f = (8πG/c2 )ρ.
Define the energy density:
Z 1
H(ξ) :=
a(tξ) · ξ dt,

(U7)

0

a.

so that a(ξ) = ∇ξ H(ξ). Then the energy functional
Z
Z
E[ψ] :=
H(∇ψ) dx −
f ψ dx
(U8)

Structural Assumptions on µ

We impose the following conditions (all satisfied by
µ(x) = x/(1 + x)):
• (A1) Continuity: µ is continuous on [0, ∞).
• (A2) Coercivity: ∃ α > 0, β ≥ 0 such that (p = 2)
µ(|ξ|)|ξ|2 ≥ α|ξ|2 − β

∀ ξ ∈ R3 .

• (A4) Monotonicity: For all ξ, η ∈ R3 ,


µ(|ξ|)ξ − µ(|η|)η · (ξ − η) ≥ 0.

Ω

c.

Main Existence and Regularity Theorems

(U2)

• (A3) Growth: ∃ β > 0 such that
|µ(|ξ|)ξ| ≤ β(1 + |ξ|)p−1 .

Ω

is convex and coercive under (A1)–(A4). Critical points
of E are weak solutions of Eq. (U1).

(U3)

(U4)

If strict, uniqueness follows.
Lemma U.1 (DFD µ satisfies (A1)–(A4)). The weakresponse approximant µ(x) = x/(1 + x) derived in Appendix N satisfies all four structural assumptions with
p = 2.
Proof. (A1) is immediate. For (A2)–(A3), note that
µ(s) ∈ [0, 1) for all s ≥ 0, and µ(s)s2 = s3 /(1 + s) =
s2 − s + s/(1 + s) ≥ s2 − s ≥ 21 s2 − 12 , which is (A2) with
α = β = 12 ; (A3) holds since |µ(s)s| ≤ s. For the exact
law the same holds with F (x) ≥ x−0.648 (SOURCE) and

Theorem U.3 (Existence for Static Problem). Under
(A1)–(A4), for any f ∈ (W01,p (Ω))′ , there exists a weak
solution ψ ∈ W 1,p (Ω) of Eq. (U1) attaining prescribed
Dirichlet boundary data.
Proof. The operator A : W01,p (Ω) → (W01,p (Ω))′ defined
by
Z
⟨Aψ, v⟩ =
a(∇ψ) · ∇v dx
(U9)
Ω

is monotone by (A4), coercive by (A2), and hemicontinuous by (A1). The Browder-Minty theorem [158] then
guarantees existence.
Theorem U.4 (Uniqueness). If a(ξ) = µ(|ξ|)ξ is strictly
monotone (which holds for µ(x) = x/(1 + x)), the weak
solution of Theorem U.3 is unique.

211
• the spatial principal symbol is uniformly positive
definite,

Theorem U.5 (Regularity). If f ∈ Lq (Ω) with q > 3/p′ ,
0,α
then any weak solution satisfies ψ ∈ Cloc
(Ω) for some
1
α > 0. If additionally µ ∈ C and f ∈ C 0,γ , then ψ ∈
1,α
Cloc
(Ω).

λ−1 |ξ|2 ≤ aij ξi ξj ≤ λ |ξ|2 ,
00

Exterior Domains and Optical Boundary Conditions

For astrophysical applications, we consider Ω = R3 \
BR with boundary conditions motivated by DFD optical
phenomenology:
• Asymptotic condition: |∇ψ(x)| → 0 as |x| → ∞.
• Horizon boundary: Ingoing-flux Neumann condition
a(∇ψ) · n̂ = ghor ,

with outgoing flux set to zero.
(U10)

Theorem U.6 (Exterior Well-Posedness). Under (A1)–
(A4) and the above boundary conditions (the horizon condition as stated in §U 5), there exists a weak solution
1,p
ψ ∈ Wloc
(Ω) with the gradient vanishing at infinity. If
the boundary operators are strictly monotone, the solution
is unique.

2.

The Dynamic Field Equation: Hyperbolic Theory

The c-speed form of the field equation, which Axiom WS.1 (App. WS) reads as a numerical relaxation
of the instantaneous constraint and not as a physical
evolution law, takes the form:
aµν (ψ, ∂ψ) ∂µ ∂ν ψ + bµ (ψ, ∂ψ, x) ∂µ ψ + c(ψ, ∂ψ, x) = S(x),
(U11)
where aµν is derived from the optical metric g µν [ψ], Greek
indices run from 0 to 3 with x0 = t, and we adopt the
Minkowski metric ηµν = diag(−1, 1, 1, 1) as background
reference.

a.

Structural Assumptions for Hyperbolic Theory

We impose conditions on the coefficients that capture
the key features of the DFD strong-field equation while remaining within the classical quasilinear hyperbolic framework:
(H1) Uniform hyperbolicity of principal part.
There exists λ ≥ 1 such that for all (t, x) in the region of
interest, all admissible ψ and ∂ψ, and all spatial covectors
ξ = (0, ξ1 , ξ2 , ξ3 ):
• aµν (ψ, ∂ψ)ξµ ξν = aνµ (ψ, ∂ψ)ξµ ξν (symmetry);

0i

where for the DFD equation a = −c , a = 0
and aij = µ(|∇ψ|)δ ij + µ′ (|∇ψ|) ∂i ψ ∂j ψ/|∇ψ| is
the flux Jacobian of Lemma U.1.

The proofs follow standard methods from quasilinear
elliptic regularity theory [32, 33].

d.

(U12)
−2

(H2) Regularity of lower-order terms. For each
multiindex α with |α| ≤ s (for fixed s > 5/2), the derivatives ∂ α bµ and ∂ α c exist and are continuous, bounded by
polynomials in |ψ| and |∂ψ|.
(H3) Regularity of source. S(x) ∈ H s−1 on the relevant spatial domain.
Definition U.7 (Uniform Hyperbolicity). The DFD operator in Eq. (U11) is uniformly hyperbolic in a region
Ω ⊂ R3+1 if (H1) holds with some λ > 0.
Proposition U.8 (Uniform hyperbolicity of the regularised approximant). For the galactic approximant
p


µ(x) = x/(1 + x) regularised as µs0 (x) = µ x2 + s20 ,
s0 > 0, condition (H1) holds on every region with
λ = 1+1/s0 . For the unregularised approximant, µ(0) = 0
and (H1) fails wherever ∇ψ = 0.
For the exact law (Appendix BS), (H1) holds on every


region where |∇ψ| ≥ δ > 0, with λ = max Λ, 1/µ(δ) :
√
on the SOURCE branch, with u = y, F ′ (x) = 2(1 −
−u 2
−u
−u
e ) /[2(1 − e ) − ue ], whose denominator vanishes
at u = 0 and grows at the rate (1 + u)e−u > 0, and on
the TOTAL branch F ′ (x) = 1 − e−x + xe−x ; both are
positive for every x > 0, as is µ.
p
Proof. With r = |∇ψ|2 + s20 ≥ s0 , the eigenvalues of the
flux Jacobian are µ(r) ≥ µ(s0 ) = s0 /(1 + s0 ) (twice) and
µ(r) + µ′ (r)|∇ψ|2 /r ≤ µ(r) + rµ′ (r) = 1 − (1 + r)−2 < 1.
For the unregularised approximant the eigenvalues are
µ(s) and d[sµ(s)]/ds, s = |∇ψ|, both of which vanish at
s = 0.
Remark U.9 (Choice of Sobolev index). We assume s >
n/2 + 1 with n = 3 spatial dimensions, so s > 5/2.
This guarantees that H s (R3 ) is a Banach algebra under
pointwise multiplication and embeds continuously into
C 1 (R3 ). The nonlinear coefficients can then be controlled
by the H s norm of ψ, which is essential for closing energy
estimates.

b.

Reduction to First-Order Symmetric Hyperbolic Form

Introduce variables U = (u0 , u1 , u2 , u3 , u4 )T with:
u0 = ψ,

ui = ∂i ψ (i = 1, 2, 3),

u4 = ∂t ψ.

(U13)

Then Eq. (U11) becomes:
A0 (U ) ∂t U +

3
X
j=1

Aj (U ) ∂j U = F (U, x),

(U14)

212
where the matrices Aµ (U ) are 5 × 5 symmetric and, under
(H1), A0 (U ) is uniformly positive definite.
In the block decomposition (u0 | u1 , u2 , u3 | u4 ), with
Da = (aij ) the flux Jacobian of (H1) and ej the j-th unit
vector,


1 0
0
A0 (U ) = 0 Da 0  ,
(U15)
0 0 c−2


0
0
0
0
Da ej  ,
Aj (U ) = − 0
T
0 (Da ej )
0

(U16)

the row u0 being ∂t u0 = u4 , the rows u1 , u2 , u3 the identities ∂t ui = ∂i u4 multiplied by Da, and the row u4
Eq. (U11); the system is symmetric hyperbolic exactly
when Da is uniformly positive definite, which is (H1).

c.

Local Well-Posedness for the Cauchy Problem

Theorem U.10 (Local Well-Posedness on R3 ). Let s >
5/2 and assume (H1)–(H3). For initial data
(∇ψ0 − g, ψ1 ) ∈ H s−1 (R3 ) × H s−1 (R3 )

(U17)

for a constant vector g, with inf |∇ψ0 | > 0 unless the
response is regularised, and time-independent source S ∈
H s−1 (R3 ), there exists T > 0 (depending on norms of
initial data) such that the Cauchy problem admits a unique
solution


∇ψ − g, ∂t ψ ∈ C 0 [0, T ]; H s−1 (R3 ) .
(U18)
The solution depends continuously on initial data in these
function spaces.
Proof. The reduction to Eq. (U14) produces a symmetric
hyperbolic system. Under (H1)–(H3), standard energy
estimates in Sobolev spaces yield local existence, uniqueness, and continuous dependence. The original field ψ is
recovered as the first component of U .

d.

Initial-Boundary Value Problems

For bounded domains Ω ⊂ R3 with smooth boundary,
we consider the IBVP:


Eq. (U11)
(t, x) ∈ [0, T ] × Ω,


ψ(0, x) = ψ (x),
x ∈ Ω,
0
(U19)

∂
ψ(0,
x)
=
ψ
(x),
x ∈ Ω,
t
1



ψ(t, x) = g(t, x),
(t, x) ∈ [0, T ] × ∂Ω.
a. Compatibility conditions. For solutions in H s (Ω)
with s > 5/2, compatibility conditions between (ψ0 , ψ1 )
and g are required at the corner {t = 0} ∩ ∂Ω:
• Zeroth order: ψ0 |∂Ω = g(·, 0).
• First order: ψ1 |∂Ω = ∂t g(·, 0).

• Higher orders: ∂tk ψ|t=0,∂Ω = ∂tk g(·, 0) for k ≤
⌊s − 1⌋, where higher time derivatives of ψ at t = 0
are determined from the PDE itself.
b.

Energy estimates. Define the Sobolev energy:
X Z


Es (t) =
|∂ α ∂t ψ|2 + |∇∂ α ψ|2 dx.
(U20)
|α|≤s

Ω

Under (H1)–(H3) and compatibility conditions, one obtains a differential inequality of the form:


d
Es (t) ≤ C(M ) Es (t) + ∥S∥2H s−1 + ∥g∥2H s−1/2 (∂Ω) ,
dt
(U21)
where C(M ) depends on L∞ bounds for ψ and ∂ψ. Gronwall’s lemma then yields:

C(M )t
Es (t) ≤ e
Es (0)
 (U22)
Z t


2
2
+
∥S(τ )∥H s−1 + ∥g(τ )∥H s−1/2 dτ .
0

establishing continuous dependence on the data.
Theorem U.11 (IBVP Well-Posedness). Let Ω ⊂ R3 be
bounded with smooth boundary, s > 5/2. Assume (H1)–
(H3), initial data (ψ0 , ψ1 ) ∈ H s (Ω) × H s−1 (Ω), source
S ∈ H s−1 (Ω), boundary data g ∈ H s ([0, T ] × ∂Ω), with
compatibility conditions up to order ⌊s − 1⌋. Then there
exists T > 0 and a unique solution
ψ ∈ C 0 ([0, T ]; H s (Ω)) ∩ C 1 ([0, T ]; H s−1 (Ω)),

(U23)

depending continuously on (ψ0 , ψ1 , S, g) in the corresponding Sobolev norms.

e.

Finite Speed of Propagation

Theorem U.12 (Finite Speed of Propagation). Assume
(H1)–(H3). Let ψ and ψ̃ be solutions of Eq. (U11) on
[0, T ] × R3 with initial data (ψ0 , ψ1 ) and (ψ̃0 , ψ̃1 ) agreeing
on BR (x0 ). There exists a characteristic speed cchar > 0
(depending only on the hyperbolicity constant λ) such that
ψ(t, x) = ψ̃(t, x)

for 0 ≤ t ≤ T, |x − x0 | ≤ R − cchar t.
(U24)

Proof. Apply the energy method to the difference w =
ψ − ψ̃, which satisfies a linearized equation. Using a cutoff
function supported inside the backward characteristic
cone and standard energy estimates yields w = 0 in the
interior. The characteristic speed cchar is determined by
eigenvalues of the principal symbol.

3.

Parabolic Extension and Long-Time Behavior

For dissipative systems or numerical relaxation, consider the parabolic extension:


∂t ψ − ∇ · µ(|∇ψ|)∇ψ = f (t, x).
(U25)

213
Let A : W01,p (Ω) → (W01,p (Ω))′ be the monotone operator A(ψ) = −∇ · a(∇ψ).
Theorem U.13 (Parabolic Well-Posedness). Under
(A1)–(A4), for ψ0 ∈ L2 (Ω) there exists a unique evolution
ψ ∈ Lp (0, T ; W 1,p (Ω)) ∩ C([0, T ]; L2 (Ω)).

(U26)

If f is time-independent and boundary operators are dissipative, solutions converge to a steady state as t → ∞.
Proof. By Crandall-Liggett theory [160], −A generates a
contraction semigroup on L2 (Ω). The result follows from
standard nonlinear semigroup theory.

4.

Stability and Continuous Dependence

Theorem U.14 (Stability Estimate). Let ψ1 , ψ2 be solutions with data (f1 , BC1 ) and (f2 , BC2 ). If a is strongly
monotone and locally Lipschitz, then


∥∇(ψ1 − ψ2 )∥Lp (Ω) ≤ C ∥f1 − f2 ∥V ′ + ∥BC1 − BC2 ∥ .
(U27)
This stability result is essential for numerical convergence and for justifying perturbative analyses around
equilibrium configurations.

5.

Open Problems

Several mathematical questions remain open:
• Global existence: Under what conditions on the
source f and initial data do solutions, classical or
weak, exist for all time, and are weak solutions
unique?
• Gradient blow-up: Partly settled, in the vacuum
equation on a non-zero uniform background gradient.
There |∇ψ| does not become unbounded: it stays
bounded, and at the first singular time it is still
continuous, with |∂ 2 ψ| → ∞ (a vertical tangent).
The continuation past that time, the singularity
structure, and the sourced and degenerate (∇ψ → 0)
cases remain open.
• Horizon regularity: The “ingoing flux only” horizon condition is physically motivated but mathematically non-standard. Full justification within
elliptic PDE theory remains open.
• Coupling to tensorial sectors: Mathematical
treatment of the full DFD system with electromagnetic and matter fields.

6.

Summary: Mathematical Status of DFD

Mathematical Well-Posedness Summary
Static (elliptic) problem:
• Existence: Browder-Minty theorem (monotone
operators)
• Uniqueness: Strict monotonicity of
µ(x) = x/(1 + x)
1,α
• Regularity: Cloc
for smooth data
• Exterior domains: solutions with |∇ψ| → 0 at
infinity exist
• Optical BC: the horizon (Neumann) condition is
open (§U 5)
Dynamic (hyperbolic) problem:
• Uniform hyperbolicity: holds for the regularised
galactic approximant µs0 (λ = 1 + 1/s0 ); the
unregularised approximant degenerates at ∇ψ = 0
• Local well-posedness: H s solutions for s > 5/2
(regularised approximant)
• IBVP: Well-posed with compatibility conditions at
corners
• Finite speed: cchar bounded by hyperbolicity
constant λ
Parabolic extension:
• Semigroup generation: Crandall-Liggett theory
applies
• Long-time behavior: Convergence to steady states
for dissipative BCs
Conclusion: DFD is mathematically as robust as
standard quasilinear wave and diffusion equations used
throughout mathematical physics. The analysis is
independent of phenomenological applications: it
establishes that the static constraint is well-posed and
that its c-speed relaxation is locally well-posed for the
regularised approximant.

214
Appendix V: Extended Phenomenology and
Numerical Methods

This appendix addresses three areas that complete the
DFD phenomenological framework: the external field effect (EFE), wide binaries, and numerical implementation
via finite element methods.

1.

The External Field Effect (EFE)

In nonlinear theories like DFD, the internal dynamics
of a subsystem depend on its external gravitational environment. This external field effect (EFE) arises from the
nonlinearity of the field equation.

a.

Physical Origin

The DFD field equation


8πG
(V1)
∇ · µ(|∇ψ|/a⋆ )∇ψ = − 2 ρ
c
is nonlinear in ∇ψ. For a subsystem (e.g., a dwarf galaxy)
embedded in an external field (e.g., a host galaxy) with
|∇ψext | ≫ a0 /c2 but |∇ψint | ≪ a0 /c2 , the host field acts
even if internal accelerations are in the deep-field regime.
This “Newtonianizes” the internal dynamics.

b.

Satellite D (kpc) gobs /a0 (observed) gN /a0 (baryonic)
Fornax
Sculptor
Draco
Crater II

138
79
76
117.5

0.071
0.124
0.129
0.083

0.0040–0.0061
0.0121–0.0187
0.0131–0.0202
0.0055–0.0084

ne
12–15
6.8–8.6
6.6–8.2
10–13

is set by the quadrature composition of App. NL; every satellite in the table lies far below the threshold of
Axiom NL.14 (ne = 6.6–15), so none is frozen.
a. Falsification criterion. If dwarf satellites uniformly show enhanced dynamics independent of their
position relative to the Milky Way, the EFE mechanism
(and hence DFD’s nonlinear structure) would be falsified.

2.

Wide Binaries

Wide stellar binaries with separations s ≳ 5000 AU
probe the low-acceleration regime where DFD deviates
from Newtonian gravity.

Quantitative Formulation

Define the dimensionless acceleration ratios:
c2 |∇ψint |
,
(V2)
xint =
2a0
c2 |∇ψext |
xext =
.
(V3)
2a0
Inside a host the subsystem’s
relative
acceleration is
p


νemb gN,s with νemb = ν
τs2 + τh2 for τh < ln 2 and
νemb = 1 for τh > ln 2 (App. NL); the functional never
forms the vector sum of the two fields, whose local reading
is excluded by the Solar-System quadrupole (§VII I).
a. Limiting cases.
• Isolated system (xext → 0): νemb = ν(τs ), standard DFD dynamics.
• Strong external field (xext ≫ 1, xext ≫ xint ):
νemb = 1, Newtonian dynamics restored.
c.

TABLE XCIX. External fields of four Milky Way satellites, in
units of the bound-matter a0 (distances of McGaugh & Wolf
2010; Crater II at 117.5 kpc). The observed field is gobs =
Vc2 /D with the Milky Way’s Vc = 190 km/s. The baryonic
field is the Newtonian field of the Milky Way’s baryons, gN =
10
2
GMb /D
p with Mb = 6.5–10 × 10 M⊙ ; it sets the host depth
τext = gN /a0 and the host-mode occupation ne = 1/(eτext −
1) used by the composition law of App. NL (computed).

Observational Signatures

The EFE predicts that satellites at smaller galactocentric radii (higher gext ) show less enhanced dynamics
than isolated dwarfs with similar internal properties. How
strongly the host field suppresses each satellite’s boost

a.

The Crossover Scale

For a binary with total mass M and separation s, the
internal acceleration is:
GM
aint = 2 .
(V4)
s
The crossover to deep-field behavior occurs when aint ∼
a0 :
r

1/2
GM
M
scross =
≈ 7040 AU ×
.
(V5)
a0
M⊙
For solar-mass binaries, scross ≈ 7040 AU at the boundmatter a0 .
b.

Predicted Velocity Anomaly

In the deep-field regime (s ≫ scross ), the orbital velocity
is enhanced:

1/2
s
vDFD = (GM a0 )1/4 = vNewton ×
. (V6)
scross

215
The velocity ratio relative to Newtonian prediction:
#1/2
"
p
1 + 1 + 4/yN
aNewton
vDFD
,
yN ≡
=
,
vNewton
2
a0
(
1
s ≪ scross ,
≈ p
s/scross s ≫ scross .
(V7)
Isolated-pair values from this formula (vDFD /vNewton =
1.42 at 104 AU and 1.84 at 2 × 104 AU for a 1 M⊙ pair)
are upper envelopes only: the Galactic external field at
the Sun (ggal = 1.79 a0 ) caps the force boost below that
envelope, at γ = gobs /gN = 1.37–1.52 at 104 AU and 1.59–
1.74 at 2 × 104 AU under the completed scale-additive
law, i.e. a velocity enhancement of 17–32%; these are not
realised at the Sun, where the environmental-confinement
axiom makes local pairs exactly Newtonian (App. NL),
and pairs beyond R ≃ 13–16 kpc, in their weaker host
field, keep a boost.

a.

Weak Form for FEM

The weak formulation (U6) translates directly to FEM
assembly:
XZ
XZ
µ(|∇ψh |)∇ψh · ∇vh dx =
f vh dx, (V8)
e

Ωe

e

Ωe

where ψh , vh are finite element approximations on mesh
elements Ωe .
b.

Newton Iteration for Nonlinearity

The nonlinear system is solved via Newton iteration.
The Jacobian matrix is:
∂i ψ ∂j ψ
Jij (∇ψ) = µ(|∇ψ|)δij + µ′ (|∇ψ|)
.
(V9)
|∇ψ|
For µ(s) = s/(1 + s):
1
.
(V10)
(1 + s)2
a. Regularization at small gradients. At |∇ψ| → 0,
the Jacobian may become ill-conditioned. A standard
remedy is regularization:
p
|∇ψ| → |∇ψ|2 + ϵ2 ,
(V11)
µ′ (s) =

c.

GAIA DR3 Constraints

Recent analyses of GAIA DR3 wide binary data show
conflicting results:
• Some analyses report enhanced relative velocities
consistent with MOND-like dynamics at s > 5000
AU [51].
• Other analyses find no significant deviation from
Newtonian predictions [52].
a. DFD interpretation. The EFE complicates wide
binary tests: binaries whose host’s baryonic Newtonian
field exceeds (ln 2)2 a0 ≃ 0.48 a0 (in the Milky Way, R ≲
13–16 kpc) are exactly Newtonian (Axiom NL.14). A
definitive test requires:
• Selection of binaries beyond R ≃ 16 kpc.
• Proper treatment of projection effects and orbital
phase.

with ϵ ∼ 10−10 in dimensionless units.

c.

The deep-field regime features steep gradients near
sources. Adaptive mesh refinement (AMR) is recommended:
• Refine where |∇ψ| changes rapidly (gradient indicator).
p
• Refine near crossover radius r⋆ = GM/a0 .
• Use logarithmic radial spacing for exterior domains.

• Statistical comparison with DFD predictions including EFE.
b. Falsification criterion. If wide binaries beyond
the frozen zone (R ≳ 16 kpc) show strictly Newtonian
dynamics at s > 104 AU, or local pairs show a boost,
DFD’s external-field sector is falsified.

Mesh Refinement Strategy

d.

a.

Boundary Conditions

Dirichlet (fixed ψ).
ψ|ΓD = ψD .

Used for outer boundaries with known asymptotic value.
b. Neumann (fixed flux).
µ(|∇ψ|)∇ψ · n̂|ΓN = gN .

3.

Finite Element Implementation

The DFD field equation is directly implementable via finite element methods (FEM). We outline the key elements
for numerical solution.

(V12)

(V13)

Used for symmetry planes or specified matter flux.
c. Robin (mixed).
µ(|∇ψ|)∇ψ · n̂ + κ(ψ − ψ∞ ) = 0.

(V14)

Used for approximate radiation conditions at finite boundaries.

216
e.

Convergence Verification

For code verification, use the analytic deep-field solution:
2p
ψ(r) = ψ0 − B ln(r/r0 ), B = 2 GM a0 .
(V15)
c
Richardson extrapolation on mesh sequences should
yield:
∥ψh − ψexact ∥L2 = O(hp+1 ),

(V16)

where p is the polynomial order of the elements.
FEM Implementation Checklist
1. Assemble weak form with µ(|∇ψ|)∇ψ flux
2. Newton iteration with analytic Jacobian
3. Regularize |∇ψ| at small values
4. Adaptive mesh refinement near crossover
5. Verify against analytic deep-field solution
6. Richardson extrapolation for convergence rate

decomposes as V ∗ ⊗V ∗ = S 2 (V ∗ )⊕Λ2 (V ∗ ) with S 2 (V ∗ ) =
1 ⊕ 5 and Λ2 (V ∗ ) = 3.
One pairing-symmetry selection rule follows:
1. Angular cancellation: the quintet 5 exchange
channel couples maximally to s-wave condensates
(isotropic gap) but vanishes for d-wave condensates
R 2π
( 0 cos 2ϕ dϕ/(2π) = 0).
√ 2
The conjectured coefficient
√
pis δ = 3 α = 92.23 ppm
(0.01σ match), with 3 = Ngen from incoherent amplitude addition of three generation channels and α2 from
the ψ–EM vertex structure. It is a conjecture outside
DFD’s claims: no derivation from the theory’s action fixes
the coefficient, so neither a match nor a miss bears on
the theory. As stated it is universal for s-wave superconductors and zero for d-wave materials, a distinction
that multi-material London-moment measurements at
≤ 20 ppm precision can test.

6.
4.

Summary

Matter Power Spectrum from ψ-Screen

Extended Phenomenology Summary
a.

Scale-Dependent ψ Perturbations

Density perturbations δρ source δψ via the linearized
field equation:
∇2 δψ = −

8πG
δρ.
c2

(V17)

In Fourier space:
8πG
δ ρ̃(k).
c2 k 2
The ψ-perturbation power spectrum is:

2
8πG
Pψ (k) =
Pρ (k).
c2 k 2
δ ψ̃(k) =

b.

(V18)

(V19)

Observational Signatures

At linear order this is the Newtonian Poisson equation,
and under the Rest-Mass Channel the growth has Qχ = 1
(App. GR); a lensing or clustering signal would need a
nonzero screen, which the current solution does not have
(Sec. XVI A).

5.

Cooper-Pair Mass Anomaly from A5 Pair Space

The Cooper-pair mass in niobium, measured by Tate et
al. (1989) via the London moment, exceeds the predicted
0.999992 × 2me by δ = 92 ± 21 ppm—a 4.4σ anomaly
unexplained for 36 years [161]. Within the A5 microsector,
each electron’s generation quantum number lives in the
fundamental V ∗ (dim V ∗ = 3). The pair tensor product

External Field Effect:
• Nonlinear µ-function causes environmental
dependence
• Subsystems in hosts above (ln 2)2 a0 are Newtonian;
satellites are not
• Testable via dwarf galaxy velocity dispersions vs.
position
Wide Binaries:
• Newtonian-acceleration crossover at scross ≃ 7040
AU for solar-mass binaries
• γ = 1 exactly near the Sun (environmental
confinement, App. NL), with the null analyses and
against Chae (2023); a boost and a step beyond
R ≃ 13–16 kpc; the isolated-pair formula is an
upper envelope
• EFE complicates interpretation; requires low-ggal
samples
Cooper-Pair Mass Anomaly (§V 5):
• Conjecture
outside DFD’s claims:
√
δ = 3 α2 = 92.23 ppm (universal for s-wave)
• Selection rule: d-wave → 0 (angular cancellation)
• Test: multi-material London-moment measurement
at ≤ 20 ppm
Numerical Methods:
• Standard FEM with Newton iteration for
nonlinearity
• Regularization needed at small |∇ψ|
• Adaptive mesh refinement near crossover scale
• Verification against analytic deep-field solution

217
Appendix W: Experimental Protocols and Sensitivity
Analyses

This appendix provides detailed, pre-registered experimental protocols for the key DFD discriminators. Each
protocol specifies the observable, prediction, systematics
budget, decision rule, and falsification criteria.

1.

• Upper station at h2 = h1 + ∆h: Second atomic
clock and auxiliary diagnostics.
• Link: Phase-stabilized optical fiber at < 10−18
level.
• Height difference: ∆h ∼ 100 m (tower, elevator
shaft, or mine).

Cavity-Atom LPI Test: Complete Protocol

The height-separated cavity–atom comparison remains
a valuable protocol, but after geometric cancellation
(Sec. XII A) it is best viewed as a demanding long-horizon
residual test. The tree-level cavity/atom response canres
cels; only a screened residual ξLPI
survives. This section
preserves the full protocol details for completeness and
for future experiments that may reach the required sensitivity.

c.

Measurement Cycle

Each measurement cycle consists of:
1. Lock cavity and lower atomic clock; record R(h1 )
for integration time τ .
2. Reconfigure for upper station measurement.
3. Record R(h2 ) for integration time τ .
4. Repeat with randomized order to decorrelate slow
drifts.

a.

Observable and Predictions

The frequency ratio at height h is:
νC (h)
R(h) ≡
.
νA (h)
a. GR prediction.


∆R
= 0.
R GR

a. Integration budget. For τ ∼ 104 s per height and
N ∼ 50 cycles:
(W1)

(W2)

10−18
∼ 1.4 × 10−19 .
σstat ∼ √
N
With systematic floor σsyst ∼ 2 × 10−19 :
q
2
2
∼ 2.5 × 10−19 .
σtot = σstat
+ σsyst

b. DFD prediction. After the constitutive-chain cancellation, the surviving signal is:


∆R
res g ∆h
= ξLPI
,
(W3)
R DFD
c2
res
where ξLPI
is the screened residual coupling that remains
once the leading geometric (tree-level) effect is removed.
At Earth’s surface, the screening analysis of Sec. XI C
res
and the BACON constraints of Sec. XII D restrict ξLPI
to be small — far below order unity.
c. Numerical estimate. For ∆h = 100 m and g = 9.8
m/s2 :

d.

Systematics Budget

Effect

Contrib.

Mitigation

Temp. gradients
Magnetic fields
Pressure/refr. index
Vibrations
Fiber link noise

< 10−19

mK stab., shielding
nT stab., shielding
Vacuum
Isolation
Phase stab.

< 10−19
< 10−20
< 10−20
< 10−19
∼ 2 × 10−19

(W4)

The DFD signal is this factor multiplied by the small
res
screened residual ξLPI
, making the target signal extremely
demanding. This is why the cavity–atom channel is
ranked below cross-species and nuclear-clock tests in the
current experimental priority ordering (Sec. XI I).

b.

(W6)

TABLE C. Systematics budget for cavity-atom residual test.

Total

g ∆h
≃ 1.1 × 10−14 .
c2

(W5)

Experimental Configuration

• Lower station at h1 : High-stability optical cavity
(ULE or Si) and reference atomic clock (Sr or Yb
lattice clock).

e.

Blinding Protocol

To prevent experimenter bias:
1. A secret offset δ at the 10−18 level is added to all
recorded R(h) values.
2. All data selection and systematic modeling performed on blinded data.
3. Analysis pipeline frozen before unblinding.
4. Offset removed only after all cuts finalized.

218
f.

b.

Pre-Registered Decision Rule

\ be the unblinded estimator with uncertainty
Let ∆R/R
σtot :
\ < 3σtot ⇒ consistent with
• Null regime: |∆R/R|
GR and with geometric cancellation. Upper bound
on the residual:
3σtot
res
.
(W7)
|ξLPI
|<
g∆h/c2

TABLE CI. Clock species and α-sensitivities. The unscreened
α
pure-α law KA = kα SA
is excluded at Earth by the same-ion
E3/E2 null unless screened (Sec. XI); the Th-229 entry is the
strong channel.
Species

α
SA

Transition

KA /kα (unscreened)

Cs
Ground HFS 2.83
Rb
Ground HFS 2.34
H
1S–2S
≈0
1
Sr
S0 –3 P0
0.06
1
Yb
S0 –3 P0
0.31
+
1
Al
S0 –3 P0
0.008
2
Hg+
S1/2 –2 D5/2 −3.2
Yb+ (E3) 2 S1/2 –2 F7/2 −5.95
Th-229
Nuclear
∼ 104

• Detection regime: | \
∆R/R| > 5σtot ⇒ a non-zero
screened residual is measured:
\
∆R/R
σtot
res
ξLPI
=
±
.
(W8)
2
g∆h/c
g∆h/c2
• Intermediate: 3–5σtot ⇒ extend campaign.
g.

Species Selection

a.

Sensitivity Reach

2.83
2.34
≈0
0.06
0.31
0.008
−3.2
−5.95
∼ 10

α
α
− SB
|:
Maximize |SA

Optimal pairs.

• Yb+ (E3)/H: ∆S ≈ 6
For the benchmark parameters above, the minimum
detectable residual is:
σtot
res
∼ 2 × 10−5 .
(W9)
ξLPI,
min ∼
g∆h/c2
This is sensitive enough to detect a residual at the level
predicted by the screened formalism if it is near the upper
end of the surviving window, but would require spacebased or long-baseline platforms to push significantly
deeper.

• Yb+ (E3)/Al+ : ∆S ≈ 6
• Cs/H: ∆S ≈ 2.8
• Th-229/Sr: ∆S ∼ 104 (nuclear clock)
c.

Analysis Protocol

1. Fit ∆AB (t) to model: A0 + A1 cos(Ωt + ϕ).
2.

2. Extract amplitude A1 and phase ϕ.

Multi-Species Clock Comparison Protocol

3. Compare phase to predicted solar ephemeris.
The full channel-resolved coupling of Eq. (291) produces
differential clock responses that can be measured without
(α)
height separation. The simplified pure-α scaling KA =
α
kα SA is only the leading same-ion term and is not the
canonical master law for the clock program.

4. If phase matches and A1 > 5σ: detection.
5. If A1 < 3σ: upper bound on |KA − KB |.
3.

a.

Observable

For clock species A and B at the same location, measure:
∆AB (t) ≡ ln
a. DFD prediction.
frequency Ω = 2π/yr:

νA (t)
νA
− ⟨ln
⟩.
νB (t)
νB

Matter-Wave Interferometry: T 3 Protocol (Not a
DFD Prediction)

Long-baseline matter-wave interferometers (MAGIS100, AION) can detect the parity-isolated T 3 phase signature (DFD’s own prediction for it is null).

(W10)
a.

Observable

Solar potential modulation at

∆Φ⊙ (t)
,
(W11)
c2
where ∆Φ⊙ /c2 ∼ 3 × 10−10 over Earth’s orbit; since
Φ = −c2 ψ/2, each per-unit-ψ coupling KA of Eq. (291)
and Table CI enters this slot with twice its magnitude.
∆AB (t) = (KA − KB ) ·

The formula under test, for interrogation time T :
ϕDFD = ϕGR + ∆ϕKC
DFD ,

(W12)

3

where the T correction is the kinematic expression of
Sec. XIII (Eq. (337)):
∆ϕKC
DFD =

2
ℏkeff
g 3
T .
m c2

(W13)

219
Status: not a DFD prediction (the matter-wave sector
is null); see the status note after Eq. (337) in Sec. XIII.
(The overall prefactor of Eq. (337) is not a phase (§XIII):
see the dimensional-bookkeeping item noted in Sec. XIII;
the parity-isolation protocol below is independent of that
prefactor.)
b.

a.

Observable signal.

26 Hz ≲ annual amplitude ≲ O(1 kHz)

b.

The T 3 term is even under keff → −keff , while the
T Newtonian term and laser-phase systematics are odd
(Sec. XIII). This allows isolation via:
1. k-reversal: Repeat with the effective wavevector
reversed.

Experimental Requirements

• Continuous comparison with optical clock (Sr or
Yb) over ≥ 1 year.
• Analysis for annual modulation at solar frequency.
c.

2. Sum: ϕ+k + ϕ−k .
3. Result: T 2 terms cancel; T 3 terms add.
Sensitivity Requirements
5.

Space Mission Protocols

Space-based tests provide access to larger potential
differences and different systematic environments.

a.

ACES (ISS)

Interpretation of Outcomes

If the parity-isolated T 3 phase is measured to be:
• Consistent with zero — not a falsification channel:
DFD and GR coincide through 1PN (§XIII).
• Non-zero at > 5σ with the T 3 time scaling, g-parity
2
and keff
/m recoil scaling of Eq. (337) ⇒ not a DFD
signal: no DFD-specific T 3 is predicted, and the
2
even-k, ℏkeff
/m T 3 term is the Newtonian gravitygradient recoil phase common to DFD and GR
(§XIII).
4.

Timeline

Nuclear clock technology is expected to reach required
precision within 5–7 years.

Status: not a DFD prediction (the matter-wave sector
is null); see the status note after Eq. (337) in Sec. XIII.
The expression of Eq. (337) has units s2 /m, so it sets no
phase-resolution target.
d.

(Sec. XI F).
(W15)

• Nuclear clock operational with systematic uncertainty < 10−16 .

Parity Isolation

2

c.

For solar potential modulation:

Prediction

The extended strong channel, with ks a matching coefficient not derived from the minimal action (Sec. XI F),
gives:

approximately 3× larger than Cs/Sr difference.

• Microwave clock comparisons with ground.
• Sensitivity to KA − KB at 10−7 level.
b.

Dedicated LPI Mission

A dedicated mission with optical clocks could achieve:

• Cavity-atom comparison in space.

The 229 Th nuclear isomer transition provides sensitivity to strong-sector couplings, with ds ∼ 1.3 (order of
magnitude larger than de ).

KTh − KSr ∼ 8 × 10−5 ,

• ∆Φ/c2 ∼ 10−10 (ISS altitude).

• Highly elliptical orbit: ∆Φ/c2 ∼ 10−9 .

Nuclear Clock Protocol: Th-229

a.

The Atomic Clock Ensemble in Space provides:

(W14)

res
• Sensitivity to screened residual ξLPI
at 10−6 level.

220
TABLE CII. DFD experimental verification timeline.
Time

Test

Prediction

Falsification

1–3 yr
1–3 yr
3–7 yr
>7 yr
>7 yr

Cross-species clocks
Nuclear clocks
Matter-wave T 3 (null prediction)
Cavity–atom
Space missions

Channel residuals
26 Hz–kHz window
As GR (1PN)
Screened residual
Enhanced prec.

All-channel nulls
No annual signal
None
Null at target
—

6.

Summary: Experimental Roadmap

Appendix X: Neutrino Mass Spectrum from DFD
Microsector

Experimental Protocol Summary
Each protocol fixes in advance:
• Observables and predictions specified before data
collection
• Decision rules fixed in advance
• Blinding protocols where applicable
• Clear falsification criteria for both GR and DFD
Key discriminators:
• Cavity–atom residual: screened non-metric
mismatch after tree-level cancellation
• Multi-species clocks: channel-resolved species
dependence governed by Eq. (291)
• Matter-wave T 3 : parity-isolated phase (null
prediction, not a discriminator)
• Nuclear clocks: Strong-sector coupling ds ∼ 1.3
Current status:
• Awaiting experimental implementation

This appendix derives a complete closed-form neutrino
sector from DFD microsector relations. Using tribimaximal (TBM) mixing geometry, a discrete S2 residual symmetry, and microsector-normalized α-power exponents,
we obtain neutrino mass ratios with zero continuous
parameters.

1.

DFD Inputs from the Microsector

DFD provides three ingredients:
1. TBM mixing geometry (Appendix G): The
“neutrinos-at-center” overlap rule gives the tribimaximal mixing matrix
p
p

p2/3 p1/3 p0
UTBM = −p 1/6 p1/3 p1/2 .
(X1)
1/6 − 1/3
1/2
2. Heavy Majorana scale (Appendix P):
MR = MP α3 ≈ 4.7 × 1012 GeV.
3. Electroweak scale (Section XVII):
√
v = MP α8 2π ≈ 246 GeV.

(X2)

(X3)

TBM fixes the eigenvectors but not the eigenvalues
(m1 , m2 , m3 ). The question is whether the residual symmetry structure can fix the mass ratios without continuous
parameters.

2.

Why S3 Invariance Cannot Split the Doublet

Let three generations carry the permutation representation of S3 . The S3 -invariant endomorphisms are spanned
by I3 and J = 11T .
The representation decomposes as 3 ∼
= 1 ⊕ 2, where
1 = span(1, 1, 1) is the singlet and 2 = {x1 + x2 + x3 = 0}
is the doublet.
On the doublet, J acts as zero (since Jx = (x1 + x2 +
x3 )1 = 0), so any S3 -equivariant operator restricted to
the doublet is proportional to the identity:
A|2 = a I2

⇒

degenerate eigenvalues.

(X4)

221
namely α:
m2
= α−2/13
m1

Any m2 /m1 ̸= 1 requires breaking S3 to a proper
subgroup; that breaking is the mechanism.

3.

TBM Selects a Canonical Residual S2

TBM naturally singles out the µ ↔ τ transposition as
residual symmetry:


1 0 0
Sµτ = 0 0 1 .
(X5)
0 1 0
Its eigenvectors in the µ–τ plane are the even and odd
parity axes:
1
1
v+ = √ (0, 1, 1),
v− = √ (0, 1, −1),
(X6)
2
2
with Sµτ v± = ±v± .
The third TBM column is exactly v+ . Thus TBM
motivates a canonical residual transposition subgroup
S2 = ⟨Sµτ ⟩.
4.

Microsector-normalized residual-S2 spurion

The rigid choice O = I3 + P− (which assigns m2 /m1 =
2) is the minimal-integer deformation of the identity
consistent with residual µ ↔ τ symmetry. Here we replace
that rigidity by a microsector-normalized coefficient that
is still knob-free: the coefficient is fixed as a discrete
channel-fraction exponent of α determined by the alreadylocked microsector integers.
a. Setup. Let P− be the rank-1 projector onto the
odd axis v− as before, and define the residual-S2 spurion
family
O(κ) := I3 + κ P− ,

(X7)

so that on parity eigenstates,
O(κ) v− = (1 + κ) v− ,

O(κ) v+ = 1 · v+ .

(X8)

The doublet ratio is then assigned as
m2
= 1 + κ,
(X9)
m1
an assignment of the spurion strength to the doublet ratio,
not a ratio of eigenvalues of O(κ) on tribimaximal mass
eigenstates: v− is not a column of UTBM .
b. No-hidden-knobs microsector normalization. In
the microsector construction, the line-bundle degree is
fixed by minimal-padding to (a, n) = (9, 5), and the CP1
Toeplitz truncation used elsewhere in the unified derivations has canonical channel count
dCP1 (k) = k + 4

⇒

dCP1 (a) = a + 4 = 13. (X10)

Residual µ ↔ τ splitting is a two-channel deformation (a
doublet), so a canonical knob-free choice is to assign the
spurion strength to the doublet channel fraction 2/13 in
the only universal dimensionless base available to DFD,

⇒

κ = α−2/13 − 1 . (X11)

c. Canonical-shift variant (Branch B). A second,
equally canonical knob-free option replaces the numerator
2 (doublet count) by the CP2 canonical shift 3 (the K −1
degree on CP2 ), while the denominator is fixed by the
CP1 channel count induced by the microsector dimension
dim(CP2 × S 3 ) = 7:
dCP1 (dim M ) = dim M + 4 = 7 + 4 = 11,
yielding the alternative
m2
= α−3/11
⇒
m1

(X12)

κ = α−3/11 − 1 . (X13)

d. Singlet-doublet hierarchy (microsector-normalized).
Replace the rigid r = α−1/3 ansatz by a microsectornormalized hierarchy built from locked integers dim M =
7 and n = 5:
m3
= r := α− dim M/(4n) = α−7/20 .
(X14)
m2

5.

Combined mass pattern (microsector-normalized)

With either choice for m2 /m1 above and the
microsector-normalized r, the mass pattern is fixed up to
one overall scale:
m1 : m2 : m3 = 1 : k : kr,
(X15)
k ∈ {α−2/13 , α−3/11 }, r = α−7/20 .

6.

Parameter-free oscillation invariant
(discriminator)

Fix the overall scale by matching ∆m221 , so that
∆m221
,
m2 = k m1 ,
m3 = r m2 . (X16)
k2 − 1
Then the dimensionless oscillation invariant becomes a
pure α-function:
m21 =

∆m232
(k 2 r2 − k 2 )
=
2
∆m21
(k 2 − 1)

7.

(k, r as above).

(X17)

Complete numerical predictions

Using α−1 = 137.035999177 and ∆m221 = 7.49 ×
10 eV2 (NuFIT 6.0), the two branches give:
Branch B matches NuFIT 6.0 to < 0.1σ.
−5

Remark X.1 (Branch selection). Branch A (k = α−2/13 ,
doublet count 2 over dCP1 (a) = 13) and Branch B
(k = α−3/11 , canonical shift 3 over dCP1 (dim M ) = 11)
are equally canonical knob-free constructions; the present

222
TABLE CIII. Neutrino mass branch predictions.
Branch
A
B

k

r

(m1 , m2 , m3 ) [meV] Σmν [meV] ∆m232 [10−3 eV2 ] ∆m232 /∆m221

α−2/13 α−7/20 (4.60, 9.80, 54.84)
α−3/11 α−7/20 (2.34, 8.96, 50.16)

NuFIT 6.0 (NO):

—

axioms do not force the choice between them. The selection of Branch B is made by the oscillation data: Table CIII shows Branch A’s ∆m232 missing NuFIT 6.0 by
≈ 24σ, while Branch B (2.437 × 10−3 eV2 ) matches to
< 0.1σ. The sector is therefore closed at zero continuous
parameters with one discrete choice anchored empirically;
downstream of that bit, both exponents, the dimensionless oscillation invariant, and the absolute-scale formula
14
m3 = 14
are α-locked with no further reference
13 πMP α
to data. The quoted goodness-of-fit (χ2 = 0.025, p = 0.99)
is that of the selected branch. The printed integers allow
four such fractions (numerator 2 or 3, denominator 11 or
13); with the absolute scale of this appendix 3/11 fits both
splittings best, and the next best (3/13) misses ∆m221
by 1.5σ (χ2 = 2.4 over both splittings; computed), so
the look-elsewhere factor is at least four. The data-free
priming scale of App. X 8 reproduces the ∆m221 -anchored
Branch-B spectrum to 0.1%, whereas the same scale under
Branch A misses the observed ∆m221 by ≈ 6σ: a second,
correlated empirical concordance conditioned on Branch
B. The companion volume’s forcing argument for Branch
B fixes its level from the branch it selects, so it cannot
force the choice, and the selection stays data-selected.
The CKM apex branch ambiguity is treated the same way
before its Euler-projection closure (Appendix AO).
With sin2 θ13 = 3α and the trimaximal weights |Ue1 |2 =
2
1
2
2
3 , |Ue2 | = 3 − 3α, |Ue3 | = 3α, Branch B gives
mβ = 9.15 meV,

mββ ∈ {0.13, 2.33, 3.25, 5.45} meV

(X18)
(the four CP-conserving points derived below), with
Σmν ≈ 61.5 meV. The tribimaximal limit Ue3 = 0 would
give mβ ≈ 5.52 meV and mββ ∈ [1.43, 4.55] meV.
a. Structural identity. For the Branch B pair (k, r) =
(α−3/11 , α−7/20 ) one has
2 2

k r =α

−(6/11+7/10)

=α

−137/110

(X19)

so the combined hierarchy exponent has numerator 137.

8.

Absolute-scale closure for Branch B from finite-d
priming

This subsection replaces the ∆m221 anchoring step with
a DFD-internal absolute-scale closure. The key input is
a forced finite-dimensional normalization factor from the
same Toeplitz truncation and determinant priming used
in the closed form for α.

69.24
61.46

2.911
2.437

38.87
32.54

—

2.438 ± 0.020

32.55

a. Bundle-degree bookkeeping (no knobs). The microsector bundle decomposition is E = O(a) ⊕ O⊕n with
minimal-padding (a, n) = (9, 5). The Toeplitz truncation on CP 1 ⊂ CP 2 carries the Spinc determinant shift
Ldet = K −1 = O(3). For a Yukawa/Dirac vertex, one
inserts the Higgs hyperplane factor: generation wavefunctions are holomorphic sections of O(1), so the Dirac
overlap lives in
O(a) ⊗ O(3) ⊗ O(1) ∼
on
CP 1 .
= O(a + 4)
(X20)
Thus the Toeplitz level is forced to be mν = a + 4 for the
neutrino Dirac sector.
Lemma (Forced finite dimension). With mν = a + 4,
the truncated holomorphic state space has dimension
dν = dim H 0 (CP 1 , O(mν )) = mν + 1 = a + 5.

(X21)

For a = 9: dν = 14 .

b. Why d/(d − 1) appears (not a fit). The primed determinant prescription removes the null channel from the
finite-dimensional spectrum. At the level of normalized
traces, passing from an unprimed average over d channels
to a primed average over d −1 nonzero channels multiplies
the normalization by d/(d − 1).
Define the neutrino finite-d priming factor:
14
dν
=
.
(X22)
Fν :=
dν − 1
13
c. DFD absolute-scale closure. The seesaw closure
gives m3 ∝ πMP α14 . The finite-d priming factor lifts this
to:
m3 = Fν πMP α14 =

14
πMP α14 .
13

(X23)

With Branch B ratios k = α−3/11 , r = α−7/20 , we get
m2 = m3 /r and m1 = m3 /(kr). Using α−1 = 137.036:

223
DFD-Closed Neutrino Predictions (Zero Continuous
Parameters)
Quantity

NuFIT 6.0

m1
m2
m
P3
mν

2.34 meV
8.96 meV
50.16 meV
61.46 meV

—
—
—
—

∆m221
∆m231

7.48×10−5 eV2
2.51×10−3 eV2

(7.49 ± 0.19)×10−5
(2.513 ± 0.020)×10−3

mββ
mβ

{0.13, 2.33, 3.25, 5.45} meV
9.15 meV

—
—

Both splittings match NuFIT 6.0 to < 0.2σ at zero
continuous parameters (Branch B).

d. What was used. Besides the stated Dirac rung
2
yD
= Fν α (Prop. AT.1), the absolute-scale closure uses
only DFD inputs already present:
1. Minimal-padding microsector integer a = 9
2. Spinc determinant shift +3
3. Higgs hyperplane factor O(1)
4. Primed-channel prescription
No continuous tuning is introduced. The 14/13 factor is
forced by (a, n) = (9, 5).
The explicit mass matrix (TBM eigenbasis)

With TBM eigenvectors and the microsectornormalized hierarchy, the mass spectrum is
m1 ,

m2 = k m1 ,

m3 = kr m1 ,

(X24)

k ∈ {α−2/13 , α−3/11 },

r = α−7/20 .

(X25)

where
In the tribimaximal limit (θ13 = 0, before the trimaximal correction sin2 θ13 = 3α used in this appendix) the
neutrino mass matrix is
Mν = m1 P1 + (km1 ) P2 + (krm1 ) P3

(X26)

in terms of the TBM projectors Pi = ci cTi .
10.

Falsification criteria

The DFD-closed Branch B (with absolute scale from
finite-d priming) gives concrete predictions summarized
below (normal ordering).
TABLE CIV. Falsification criteria for DFD neutrino sector.
Observable DFD prediction Falsification
∆m221
2
∆m
P 31
mν
mβ
mββ
Ordering

7.48×10−5 eV2
2.51×10−3 eV2
61.46 meV
9.15 meV
≤ 5.45 meV
Normal

External global-fit verification

a. NuFIT 6.0 Table 1 check (conservative Gaussian).
NuFIT 6.0 publishes best-fit values and 1σ uncertainties
for the mass-squared splittings [162]. Using the “IC24
with SK-atm” Normal Ordering line in Table 1 of their
JHEP update:

DFD

9.

11.

NuFIT > 3σ
NuFIT > 3σ
< 45 or > 80 meV
β-decay incomp.
measured mββ > 5.5 meV
Inverted confirmed

∆m221 = (7.49 ± 0.19) × 10−5 eV2 ,

(X27)

−3
∆m23ℓ = (2.513+0.021
eV2 .
−0.019 ) × 10

(X28)

Symmetrizing the second uncertainty to σ3ℓ = 0.020 ×
10−3 eV2 , the normalized pulls for the DFD Branch B
predictions are:
7.48 − 7.49
pull21 =
= −0.053 σ,
(X29)
0.19
2.51 − 2.513
pull3ℓ =
= −0.15 σ.
(X30)
0.020
The conservative uncorrelated Gaussian statistic is:
χ2Gauss = pull221 + pull23ℓ ≈ 0.025

(X31)

with 2 degrees of freedom. corresponding to a p-value
of 0.99. Branch B lands essentially on the published
global-fit best point.
b. Including realistic |Ue3 |2 (discrete 0νββ comb).
Because DFD imposes µ–τ reflection symmetry to lock
θ23 = π/4 and δCP = −π/2 (Thm. AY.3), the same symmetry is the Takagi/Autonne reality condition S Mν∗ S =
Mν (S = 2–3 reflection) on the Majorana mass matrix,
and forces both Majorana phases to the CP-conserving
values α21 , α31 ∈ {0, π} (Grimus–Lavoura). The Majorana phases are therefore not free, and the 0νββ effective mass does not scan a continuous band; with
the locked TM1 weights |Ue1 |2 = 23 , |Ue2 |2 = 13 − 3α,
|Ue3 |2 = P
3α and the Branch-B spectrum it collapses to
2
mββ =
i ηi |Uei | mi , ηi ∈ {±1}:
mββ ∈ {0.13, 2.33, 3.25, 5.45} meV

(X32)
(4 CP-conserving points, not a band).
pP
2 2
The kinematic mass is phase-free, mβ =
i |Uei | mi =
9.15 meV, and Σmν = 61.5 meV. Which of the four parities is physically realized is fixed by the (deferred) sign
structure of Mν ; the symmetry-forced, theorem-grade
content is the four-point discreteness and the phase-free
mβ . All four points lie 1–2 orders of magnitude below
current laboratory 0νββ limits, so no datum is consulted
to obtain them. (A c213 -scaled-TBM scan gives a continuous band [0.29, 5.55] meV; its endpoints are just the
all-aligned/anti-aligned CP-conserving points, so the symmetry replaces that band with the four discrete teeth
above.)
c. Reproducibility. A helper script scripts/
scripts nufit table1 gaussian eval.py reproduces
this conservative check:
python3 scripts/scripts_nufit_table1_gaussian_eval.py \
--dm21 7.48e-5 --dm3l 2.51e-3

d. Optional: profile-level ∆χ2 evaluation. The Gaussian check above is intentionally conservative (it uses only

224
the published Table 1 central values and 1σ widths). NuFIT additionally publishes 1D ∆χ2 profiles for each oscillation parameter. To evaluate the DFD prediction against
those profiles, we include scripts/scripts nufit chi2
eval.py, which (i) loads the NuFIT profile tables, (ii)
interpolates ∆χ2 (x), and (iii) reports the total χ2 for
the predicted parameter vector under the chosen ordering/data set.
We do not hard-code the profile files here (NuFIT
periodically updates file names), but the script documents
the expected plain-text format and directory layout.

12.

Summary: DFD-closed neutrino sector (zero
continuous parameters)

Theorem X.2 (Conditional neutrino-sector closure).
Conditional on the stated inputs of this sector in Ta2
ble CXXVIII (the Dirac rung yD
= Fν α, sin2 θ13 = 3α,
trimaximal mixing with µ–τ reflection) and a single dataselected discrete input, the choice of Branch B among
four equally natural fractions (the doublet-ratio bit, fixed
empirically by ∆m232 ; see the branch-selection remark of
§X 7)—the entire light-neutrino sector is fixed at zero
continuous parameters:
1. Ordering is structural (normal). The microsector exponents k = α−3/11 > 1 and r = α−7/20 > 1
force m1 < m2 < m3 ; inverted ordering is impossible.
14
2. Absolute scale is α-locked. m3 = 14
=
13 πMP α
50.16 meV (Eq. X23), with the finite-d priming factor 14/13 obtained two independent ways
1
(14/13 = 1 + Ngen
sin2 θW with sin2 θW = 3/13,
and dν /(dν − 1) with dν = a + 5 = 14), giving (m1 , m2 , m3 ) = (2.34, 8.96, 50.16) meV and
Σmν = 61.46 meV.
3. Mixing
is
parameter-free
(TM1).
cos2 θ12 cos2 θ13 = 
 2/3 exactly and sin2 θ12 =
(1 − 9α)/ 3(1 − 3α) = 0.3184 (Thm AY.3).
The resulting splittings match NuFIT 6.0 to < 0.2σ (χ2 =
0.025, p = 0.99) with no continuous fit parameter. The
single branch bit is the only empirical input.

Neutrino Sector Summary (Zero Continuous Parameters)
Derivation chain (zero continuous parameters;
one data-selected binary branch choice, see the
branch-selection remark in App. X 7):
1. TBM from “neutrinos-at-center” → µ ↔ τ residual
S2
2. Microsector integers (a, n) = (9, 5), dim M = 7 lock
channel fractions
3. k = m2 /m1 = α−3/11 (Branch B; selected over
Branch A by the oscillation data)
4. r = m3 /m2 = α−7/20 (from dim M/(4n))
5. Dirac overlap in O(a + 4) → dν = a + 5 = 14
6. Finite-d priming factor Fν = 14/13
14
7. m3 =
π MP α14 (absolute scale)
13
Arithmetic identities:
k2 r2 = α−137/110
14
π MP α14 (14 = a + 5, 13 = a + 4)
13
DFD predictions vs NuFIT 6.0:
m3 =

Observable
∆m221
∆m231

DFD

NuFIT 6.0

7.48×10−5

(7.49 ± 0.19)×10−5

Pull

−0.05σ
2.51×10−3 (2.513 ± 0.020)×10−3 −0.15σ

Combined: χ2 = 0.025 (2 dof ), p = 0.99.
Complete predictions:
• (m
P 1 , m2 , m3 ) = (2.34, 8.96, 50.16) meV
•
mν = 61.46 meV
• mβ = 9.15 meV (tribimaximal limit 5.52 meV)
• mββ ∈ {0.13, 2.33, 3.25, 5.45} meV at the four
CP-conserving points (tribimaximal limit 4.55
meV)
Status: DFD-closed at zero continuous
parameters, given the stated inputs. No continuous
empirical input; one discrete branch choice (B, among
four equally natural fractions) is selected by the
oscillation data. Every number then derives from α, MP ,
and locked microsector integers. The prediction matches
NuFIT 6.0 with χ2 = 0.025.

225
Appendix Y: Finite Yukawa Operator, Chiral Basis,
and the Af Prefactors
1.

Purpose and Scope

The charged-fermion mass formula used in the main
text,
v
mf = Af αnf √ ,
(Y1)
2
separates a localization (power-law) factor αnf from a
finite microsector prefactor Af . To make Af a derived
quantity (rather than an asserted number), one must
specify:
(i) the finite Hilbert space HF ,
(ii) the chiral states χL,f , χR,f ∈ HF for each fermion
f , and
(iii) a concrete finite Yukawa operator Yfinite acting between the chiral subspaces.
Only then does the definition
Af ≡ ⟨χR,f | Yfinite | χL,f ⟩

Finite Higgs Connector as an Explicit Matrix

Let H ∈ C2 be the weak doublet column H = (h1 , h2 )T .
Embed it into M6 (C) as the off-diagonal connector
b := h1 E4,6 + h2 E5,6 ,
b † = h∗1 E6,4 + h∗2 E6,5 . (Y9)
H
H
In block form,



03×3 03×2 03×1
Φ(H) = 02×3 02×2 H2×1  .
†
01×3 H1×2
01×1

e = iσ2 H ∗ and its
Similarly, define the conjugate Higgs H
be
embedding H by replacing (h1 , h2 ) with (h̃1 , h̃2 ) in (Y9).
After electroweak symmetry breaking in unitary gauge,
we take
 
1 0
v
H→√
⇒ h2 = √ , h1 = 0,
(Y11)
2 v
2
and analogously for e
H.
5.

Finite Hilbert Space and Normalization

Chiral Subspaces and Canonical Link-States

A minimal, explicit choice consistent with the (3, 2, 1)
connectivity is to realize chiral states as normalized link
basis elements (off-diagonal blocks). Define the following
canonical link-states:
a. Quark doublet left states (color → weak). For a ∈
{1, 2, 3},
ba,4 ,
ba,5 .
χQ (a, ↑) := E
χQ (a, ↓) := E
(Y12)
L

We work with the regular-module finite Hilbert space
HF := Md (C),

(Y3)

equipped with the normalized Hilbert–Schmidt inner product
1
⟨X, Y ⟩ := Tr(X † Y ).
(Y4)
d
Let Eab ∈ Md (C) denote matrix units, (Eab )ij = δai δbj .
Then the rescaled units
√
bab := d Eab
E
(Y5)
form an orthonormal basis:
bab , E
bcd ⟩ = δac δbd .
⟨E
3.

(Y6)

Block Decomposition for the (3, 2, 1) Microsector

To align with the (3, 2, 1) sectoral split used throughout
the manuscript, take d = 6 and order basis indices as:
{1, 2, 3} (color),

{4, 5} (weak),

{6} (singlet).
(Y7)
Every X ∈ M6 (C) is then written in (3, 2, 1) block form



X33 X32 X31
X = X23 X22 X21  ,
X13 X12 X11

(Y10)

(Y2)

become computable.
This appendix makes those objects explicit and states
the minimal additional structure required to reproduce
species-dependent Af .
2.

4.

L

b. Quark singlet right states (color → singlet). For
a ∈ {1, 2, 3},
ba,6 .
χqR (a) := E
c.

Lepton doublet left states (weak → singlet).
b
b
χL
χL
(Y14)
L (↑) := E4,6 ,
L (↓) := E5,6 .

d. Charged-lepton singlet right state (singlet → singlet).
b6,6 .
χℓR := E

(Y8)

(Y15)

Important: At this stage these are canonical basis
states of the minimal (3, 2, 1) connector model. Speciesresolution beyond multiplet type (e.g., distinguishing t
from τ at the level of Af ) requires additional finite structure; see Section Y 8.
6.

Yfinite as an Explicit Operator and Its Matrix
Elements

To make (Y2) explicit, we must specify an operator
Yfinite : HL → HR .

dim(X33 , X22 , X11 ) = (3, 2, 1).

(Y13)

(Y16)

The most concrete realization on HF = Md (C) is an
operator of multiplication type (then fully specified by a
fixed matrix). Two natural choices are:

226
a.

Right-multiplication insertion (Higgs on the right).
(R)
(Yfinite X) := X b
H,

b.

(Y17)

Left-multiplication insertion (Higgs on the left).
(L)
b † X.
(Yfinite X) := H

(Y18)

Given χL,f , χR,f ∈ HF and the inner product (Y4),
the finite matrix element is

1 
⟨χR,f | Yfinite | χL,f ⟩ = Tr χ†R,f (Yfinite χL,f ) . (Y19)
d
7.

Explicit Evaluation in the Canonical Link Basis
(R)

With the canonical link-states above and Yfinite = Yfinite
from (Y17):
a. Down-type quark (example). Take χL = χQ
L (a, ↓
q
b
b
) = Ea,5 and χR = χR (a) = Ea,6 . Using Ea,5 E5,6 = Ea,6
and Ea,5 E4,6 = 0,
(R)
b = h2 E
ba,6 .
Yfinite χL = χL H

(Y20)

Then orthonormality gives
(R)

⟨χR |Yfinite |χL ⟩ = h2 .
(Y21)
√
After EWSB (Y11), h2 = v/ 2.
(L)
b. Charged lepton (example). Taking Yfinite = Yfinite
L
ℓ
b5,6 and χR = χ = E
b6,6 .
from (Y18), let χL = χL (↓) = E
R
Then E6,5 E5,6 = E6,6 implies
(L)
b † χL = h∗ E
b
Yfinite χL = H
2 6,6 ,

(Y22)

and hence
(L)

⟨χR |Yfinite |χL ⟩ = h∗2 ,
(Y23)
√
whose magnitude again becomes v/ 2 after EWSB.

8.

Universality Wall and the Required Additional
Structure

The computations above reveal a structural fact:
Proposition Y.1 (Universality of the Minimal (3, 2, 1)
Connector Yukawa). In the canonical link-basis realization of HF = M6 (C) with Yfinite defined by the bare
Higgs connector (Y17) or (Y18), the finite matrix element
⟨χR,f |Yfinite |χL,f ⟩ depends only on the Higgs component
selected (and on gauge convention), not on the fermion
species label f beyond its multiplet type.
In particular, this minimal structure cannot generate nontrivial, species-dependent Af factors.
9.

The Ldet Twist: A 3ngen Ratio Pattern (Not a
Parameter-Free Mass)

The universality wall of Proposition Y.1 is set by the
H, which carries the trivial O(1)
leading Higgs connector b

twist on the color block. There is, however, a forced
subleading connector that the bare Higgs insertion omits,
and it carries a nontrivial twist. This subsection states
exactly what that twist buys and what it does not buy.
The grade is that of Appendix FM: the ratio pattern
is structurally suggestive at the level of a power law
in Nc , but the overall normalization and the generation
assignment remain, respectively, an imported number and
an already-open dictionary. No parameter-free light-mass
claim is made here.
a. The subleading connector and its determinantbundle charge. On the color block, quantization of the
internal geometry proceeds on CP 2 (the (3, 2, 1) sector’s
projective fiber). Its canonical bundle is KCP 2 = O(−3),
since c1 (CP 2 ) = 3H with H the hyperplane class; the
Spinc determinant (anticanonical) line is therefore
−1
Ldet = KCP
2 = O(+3),

(Y24)

a forced geometric object of degree 3 (not an invented
b of Eq. (Y9) is a
charge). The leading Higgs connector H
section of the trivial O(1)-grade connector and carries no
unit of Ldet . The subleading connector
b ⊗ (Ldet insertion)
Y1 = X H
(Y25)
carries exactly one unit of the O(3) determinant twist
on the color block; this is the only structural difference
b
between Y1 and the leading Y0 = X H.
Proposition Y.2 (Ldet pattern: 3ngen within-multiplet
spread). Suppose the within-multiplet quark/lepton mass
ratio is controlled by the number of units of the Ldet twist
carried by the dominant connector, and that one such
unit contributes a multiplicative factor equal to the color
degree Nc . Since Nc = 3 (the colour input, App. FK)
and the determinant line has degree 3 by Eq. (Y24), the
within-multiplet ratio scales as
mquark
∝ Ncngen = 3 ngen ,
ngen ∈ {+1, −1, 0},
mlepton gen
(Y26)
with the Georgi–Jarlskog exponent set ngen = (+1, −1, 0)
for
P generations (1, 2, 3). The exponent set is traceless
( g ngen = 0), so 3ngen contributes only a spread, not an
overall scale. At the level of this pattern, the degeneracy
of Proposition Y.1 is lifted: distinct generations acquire
distinct, Nc -quantized within-multiplet ratios.
Remark Y.3 (Grade of Proposition Y.2: pattern-suggestive, not uniquely forced). Two welds in Proposition Y.2
are not supplied by the core action and must be stated
explicitly. (i) The bridge “one unit of the degree-3 determinant line ⇒ a multiplicative factor of exactly Nc = 3
per generation” equates two a-priori-distinct 3’s (a bundle
degree and a color multiplicity); Berezin–Toeplitz quantization of Ldet yields a highest-weight degree shift, not
an automatic amplitude factor of 3. (ii) A free-base leastsquares fit of the observed within-multiplet spread returns
base ≈ 3.22 (about 7% from 3); the non-circular test —
does Nc = 3 predict the spread of the deviations — gives
per-generation exponents (+1.1, −1.0, −0.1) against the

227
claimed (+1, −1, 0), i.e. excellent on generation 2 but
∼10–12% off on generations 1 and 3. Hence 3ngen is a
structurally suggestive pattern (and Nc = 3 does bestdescribe the spread among small integers: RMS ≈ 11% at
Nc = 3 versus ≈ 39% at Nc = 2 and ≈ 19% at Nc = 4),
but it is not a precision identity and is not uniquely
forced.
Remark Y.4 (The QCD enhancement ηq is imported,
not derived). Promoting the pattern of Eq. (Y26) to
absolute light masses requires one overall common factor
ηq multiplying the whole down/lepton column,
mdi
= ηq 3 ngen (i) .
(Y27)
mℓi
This ηq ≈ 2.67 is a single imported factor, not a perspecies fit — that is the economy of the construction.
But its value is not derived from the DFD action: it
equals the determinant-extracted / textbook light-quark
QCD enhancement,

1/3
md ms mb
ηq = m
= 19.01/3 = 2.669,
(Y28)
m
m
e
µ
τ
the cube root of the observed down/lepton mass determinant. Because the exponent set (+1, −1, 0) is traceless,
the product of the three resulting ratios is identically
ηq3 = 19.0, the observed determinant, by construction.
Consequently the often-quoted “ms /mµ to 0.6%” is a
consistency check that re-uses data already absorbed into
ηq (three data points fix one parameter, then the same
data are checked), not a parameter-free prediction; its
siblings md /me and mb /mτ land ∼12–13% off. This is
the circularity recorded in Appendix FM, and it is binding: the Ldet mechanism does not remove the 23.4%
forced-ceiling muon error in a parameter-free way.
Remark Y.5 (The (+1, −1, 0) assignment is the same already-open dictionary). Which generation receives ngen =
+1, −1, or 0 in Eq. (Y26) is fixed by the down-type conjugation / sign rule already in this appendix — Proposition Y.15 together with Assumption Y.14 — which
the appendix itself labels “not derived from the core
DFD action.” The Ldet mechanism introduces no new
assumption to set these signs: it inherits the same open
generation↔sector dictionary. (Of the six possible orderings, only (+1, −1, 0) achieves the ∼11% RMS spread;
the other five give 122–468% errors — so the assignment
is data-selected, but it is selected by the pre-existing
dictionary, not by a fresh postulate.)

Ldet result (caveats in both directions)
Gain: the Ldet = O(3) determinant twist that the
leading Higgs connector lacks upgrades the universality
wall of Proposition Y.1 from “degenerate” to “a 3ngen
ratio pattern” — a structural reorganization, with Nc = 3
(the colour input) and the degree-3 anticanonical line a
forced geometric object. Limit: the absolute light
masses remain not parameter-free. The common
factor ηq ≈ 2.67 is imported (it is the cube root of the
observed mass determinant, hence circular as a
“prediction”), the degree-3 → ×3 amplitude bridge is a
weld rather than a derivation, and the (+1, −1, 0)
assignment is the same already-open dictionary
(Prop. Y.15, Assump. Y.14). This subsection therefore
does not claim to “remove the muon 23% failure.” For
the binding status see Appendix FM.

10.

The ηq -Free Double Ratio and the Nc3 Color
Clebsch

The box above isolates the one weld that blocks a
parameter-free result: the common down/lepton factor
ηq ≈ 2.67 is imported (it is, by construction, the cube root
of the observed determinant). This subsection extracts
the part of the Ldet mechanism that survives after the
imported factor is algebraically removed, and tests it. It
is the explicit color-/family Clebsch demanded by Theorem FM.2; the imported factor cancels, but the result
depends on the scale at which the masses are taken.
a. Construction of the eta-free observable. Write the
per-generation down/lepton mass ratio in the Proposition Y.2 form
mdi
si = (+1, −1, 0) for gen i = (1, 2, 3),
= ηq Ncsi ,
mℓi
(Y29)
and form the double ratio
D :=

(md /me ) (mb /mτ )
= ηq(1−2+1) Nc(s1 −2s2 +s3 ) = Nc3 .
(ms /mµ )2

(Y30)
The exponent of ηq is 1−2+1 = 0: the imported factor
cancels identically, regardless of its value or origin (it
is an imported number: DFD’s own running gives 4.21,
Rem. Y.6). What remains is the pure within-multiplet
spread, and the Ldet mechanism forces it to be
s1 − 2s2 + s3 = (+1) − 2(−1) + (0) = 3 =⇒ D = Nc3 = 27.

(Y31)
b. Forcedness of the exponent set. The set
{+1, −1, 0} is not chosen; it is the traceless equivariant weight system of the anticanonical line
−1
Ldet = KCP
= O(3) restricted to the three fixed
2
points of the (Z3 ) generation action on CP 2 . The three
fixed points carry Z3 charges q = (0, 1, 2); the degree-3
line contributes normalized weight 3q/Nc = q, which
after the mandatory traceless centering (the fixed-point
weights of any line bundle on a compact toric variety sum
to a multiple of the canonical class and are measured

228
relative to their mean) becomes
q − q̄ = (0, 1, 2) − 1 = (−1, 0, +1),

(Y32)

i.e. the multiset {+1, −1, 0}. The base is Nc = 3, the
colour input. Hence D = Nc3 rests on a forced weight set
and a stated input, and D does not contain ηq .
c. Numerical confrontation. Because ηq is a running
quantity, it is common to the three generations only
when all six masses are taken at one renormalization
scale. There the data give D = 33.1 ± 0.8 anywhere between 2 GeV and the top
√ mass, and 29.3 ± 0.7 at the
matching scale Λtop = α MP , 8% above 27. PDG’s
mixed reporting convention gives 27.7, but in that convention D moves from 21 to 31 as the light-quark reference
scale moves from 1 to 3 GeV. The same weights predict
(md /me )/(ms /mµ ) = 9 at any scale, against 10.35 ± 0.15
observed, so (+1, −1, 0) is a pattern, not a precision identity.
Remark Y.6 (What this does and does not buy — the
two residual welds). (1) The overall scale ηq is an
imported number, not a forced connector coefficient. The Georgi–Jarlskog column (3, 13 , 1) has geometric mean unity, so ηq ≈ 2.67 would have to be the
accumulated colored-vs-colorless renormalization-group
enhancement,
and it is not. Started at the matching scale
√
Λtop = α MP and run with DFD’s own printed couplings (αs (MZ ) = 0.1187, sin2 θW = 3/13; full StandardModel running, then four-loop QCD and one-loop QED
to the scales at which the masses are quoted), the column gives ηq = 4.21, 58% above 2.67: it overshoots
md /me by 38%, ms /mµ by 62% and mb /mτ by 75%,
and five variants give 4.02–4.21 (computed). Only a
start near 2.6 × 106 GeV, where nothing is printed, or
αs (MZ ) = 0.076 would give 2.67. The near-coincidence
ηq ≈ 8/3 = 2CF is therefore not earned by DFD’s running, and ηq is carried as one imported input: the absolute
light masses remain not parameter-free, exactly as Appendix FM states. (2) The ordering of the exponent
set is data-selected among three cyclic choices.
The unordered set {+1, −1, 0} is forced; assigning +1 to
generation 1 and −1 to generation 2 (vs. the five other
orderings) is fixed by the same down-type conjugation
rule (Prop. Y.15) and lands at ∼11% RMS, whereas the
five competitors give 122–468%. So the ordering is a
discrete sign/label, not a continuous fit, but it is still
the already-open dictionary. Net: the Ldet weights keep
the “pattern-suggestive” grade of Remark Y.3: the double
ratio D = Nc3 is a pattern, not a precision identity, and
the two named welds (ηq running, ordering) stay open.
Consequence: To make Af computable and speciesdependent (and thereby to “re-earn” any table of numerical Af values), one must introduce at least one of the
following:
d. (i) Species projectors/embeddings in the finite space.
Define explicit finite projectors or partial isometries
ΠL,f , ΠR,f ∈ Md (C),

(Y33)

and replace the bare insertion by a species-resolved
Yukawa map, e.g.,
(f )
b
Yfinite (X) := ΠR,f X ΠL,f H
(Y34)
(f )
b † ΠR,f X ΠL,f .
or Yfinite (X) := H
Then
(f )

Af = ⟨χR,f |Yfinite |χL,f ⟩

(Y35)

becomes an explicit, computable function of (ΠL,f , ΠR,f )
and the chosen finite basis states.
e. (ii) Enlarged finite Hilbert space carrying full SM
representation content. Replace the minimal (3, 2, 1) connector space by a finite space large enough to encode
distinct chiral multiplets and flavor structure as orthogonal finite states, with a correspondingly nontrivial finite
Dirac/Yukawa operator DF (block matrix) whose entries
are determined by the microsector rules.

11.

A5 Species Projectors: Breaking the
Universality Wall

The minimal (3, 2, 1) connector produces Yukawa matrix elements that do not distinguish fermion species beyond multiplet type (Proposition Y.1). This section provides an explicit construction of species projectors compatible with the microsector identification kmax = |A5 | = 60.
The manuscript uses kmax = 60 = |A5 | (order of the
alternating group). Under the identification of Lemma H.5
(its derivation is open; status box below), take the channel
Hilbert space to be the group algebra C[A5 ], with species
projectors from A5 structure (not from (Z3 )2 , which has
order 9 and is not a subgroup of A5 ).
Resolution: The alternating group A5 has 5 conjugacy
classes, including two distinct classes of 5-cycles (5A and
5B), providing a natural discrete species label without
additional structure.

a.

Channel Space as Group Algebra

The channel Hilbert space is the group algebra
Hch := C[A5 ], {|g⟩ : g ∈ A5 },
dim Hch = |A5 | = 60.

(Y36)

For x ∈ A5 , define the right-regular unitary action
R(x) |g⟩ := |gx⟩,

(Y37)

so R(x) is a 60 × 60 permutation matrix in the {|g⟩} basis.

b.

Generators and Universal Connector

Fix the standard generators of A5 :
a = (123),

b = (12345),

S = {a, a−1 , b, b−1 }.
(Y38)

229
Define the channel connector as the Cayley adjacency
operator
Xch :=

X

R(s)

(Y39)

s∈S

This is an explicit sparse 60 × 60 matrix (each row has
|S| = 4 nonzero entries).
c.

Higgs Kernel from Derived εH

Let ℓ(g) be the word length of g in the Cayley graph
(A5 , S). With the derived Higgs width εH = Ngen /kmax =
3/60 = 0.05 (Theorem H.4), define the diagonal kernel
X ℓ(g)
b ch :=
H
εH |g⟩⟨g|
(Y40)
g∈A5

This is fully determined by (A5 , S, εH ) with no free
parameters.
d.

The alternating group A5 has exactly 5 conjugacy
classes:
Class Representative Size Element Order
e (identity)
(12)(34)
(123)
(12345)
(12354)

1
15
20
12
12

1
2
3
5
5

For each class C ⊂ A5 , define the class projector
X
PC :=
|g⟩⟨g|
(Y41)
g∈C

These are explicit, mutually commuting, diagonal idempotents on Hch .

Define the minimum class-to-class hop distance:
ℓ(x−1 y).

This is the mechanism that breaks the universality
wall: pure Cayley geometry combined with derived εH
generates species-dependent hierarchy.

Species-Resolved Prefactors

a. Canonical species assignment. Given the channel Hilbert space Hch = C[A5 ], the class projectors of
Eq. (Y41), and the generation hierarchy projectors, the
species assignment rule defines a canonical map f 7→ |ψf ⟩
from fermion species to channel states:
(Y43)

(Y42)

(Y44)

No additional phenomenological species-label assignment
is required: gauge quantum numbers determine the class
projector, generation determines the hierarchy projector,
and their ordered product on the seed state yields a unique
channel state.
In the explicit SM embedding, this takes the form:
Let PLgauge (f ), PRgauge (f ) denote the standard SM gauge
projectors on the internal factor HSM . Define the full
species projectors:
ΠL,f := PLgauge (f ) ⊗ PCL (f ) ,

ΠR,f := PRgauge (f ) ⊗ PCR (f ) .

(Y45)
The species prefactor is then the finite matrix element
b ch |ψL,f ⟩
Af = ⟨ψR,f | ΠR,f Xch ΠL,f H

g.

Cayley Geometry and Hierarchy Mechanism

min

Proposition Y.7 (Hierarchy from Cayley Geometry).
The two 5-cycle classes 5A and 5B differ by one hop
from identity. For any Yukawa functional weighted by
ℓ(g)
εH , this produces an automatic discrete suppression scale
of order εH between the two 5-cycle sectors, up to path
multiplicities and edge-count factors.

Af = ⟨ψf |Y|ψf ⟩.

• 5B contains b2 and b3

x∈C, y∈D

∆(1A, 3A) 1
a = (123) ∈ S
∆(1A, 5A) 1
b = (12345) ∈ S
∆(1A, 5B) 2
b2 ∈
/S
∆(1A, 2A) 3 Double transpositions

where Pq(f ) is fixed by gauge quantum numbers, P(g(f ))
by generation, and |ψ0 ⟩ is the universal channel seed state.
The Yukawa prefactor is then uniquely determined as the
expectation value of the channel Yukawa operator Y:

• 5A contains b and b4 = b−1

∆(C, D) :=

Comment

|ψf ⟩ = Pq(f ) P(g(f )) |ψ0 ⟩,

The 5-cycles split into two distinct conjugacy classes
5A and 5B of equal size. This provides a natural ± label
(related to quadratic residues mod 5) that can distinguish
species without additional structure.
With the generator b = (12345):

e.

Class Pair ∆

f.

Species Projectors from Conjugacy Classes

1A
2A
3A
5A
5B

For the generating set S = {a, a−1 , b, b−1 }:

(Y46)

Class-Amplitude Formula

P
For class-superposition states |C⟩ := |C|−1/2 g∈C |g⟩,
the channel-only overlap reduces to an explicit weighted

230
1. Generation = multiplicity-3 in V ⊗ V ∗ factorization

edge count:
X ℓ(h)
1
A(CR , CL ) = p
εH · #{s ∈ S : hs ∈ CR }.
|CR ||CL | h∈CL
(Y47)
This is purely determined by (A5 , S, εH ) with no mass
data input.

2. Canonical generation projectors Mr with rank 3
3. Down-type selection via mod-3 conjugation automorphism

a.
h.

Regular Module Factorization

Proposed Species Assignment Rule

A minimal assignment principle compatible with the
structure:
1. Element order rule: The odd spinc label kf ∈
{1, 3, 5} selects the element order sector (identity /
3-cycles / 5-cycles)
2. 5-cycle split rule: Weak isospin sign (up vs down
component) selects between 5A and 5B for kf = 5
3. Gauge sector: Lepton vs quark distinction regauge
mains in the gauge projector factor PL/R
(f )

Definition Y.8 (Regular module, left/right actions). Let
C[A5 ] be the group algebra (regular A5 -module). Define
left- and right-regular actions
(L(g)f )(x) = f (g −1 x),

(R(g)f )(x) = f (xg), (Y48)

so L(g) and R(h) commute for all g, h ∈ A5 .
Definition Y.9 (Z3 × Z3 phases and Fourier projectors).
Fix any element a ∈ A5 of order 3 (a 3-cycle) and set
U := L(a),

V := R(a),

ω := e2πi/3 .

(Y49)

Define the phase projectors
2

Pr(L) :=

This rule can be tested by computing Af and comparing
to observed masses.

1 X −rm m
ω
U ,
3 m=0

2

Ps(R) :=

1 X −sn n
ω
V ,
3 n=0

r, s ∈ {0, 1, 2}.
(Y50)
The joint projector is

12.

Complete Status Summary

Mass Sector Status (Complete Assessment)
What is derived:
• The exponent structure αnf from CP 2
localization/overlap construction
• The Higgs-width parameter
εH = Ngen /kmax = 3/60 (Theorem H.4)
Universality wall (Proposition Y.1): The minimal
(3, 2, 1) connector with bare Higgs insertion cannot
distinguish species within a multiplet.
Resolution via A5 conjugacy classes:
• Channel space Hch = C[A5 ] (consistent with
kmax = 60)
• Species projectors from 5 conjugacy classes (sizes 1,
15, 20, 12, 12)
• Built-in hierarchy from 5A vs 5B 5-cycle split (hop
distance difference)
b ch using derived εH
• Explicit Higgs kernel H
• Connector Xch as Cayley adjacency (explicit
60 × 60 sparse matrix)
Complete derivation: See Section Y 13 for the full
generation projector construction and down-type
selection rule.

Pr,s := Pr(L) Ps(R) =

2
1 X −(rm+sn) m n
ω
U V
9 m,n=0

Remark Y.10 (Independence of the choice of a). All 3cycles in A5 form a single conjugacy class. Replacing a
by a′ := gag −1 conjugates U, V by unitary permutation
matrices, permuting the labels (r, s) without changing
any invariant. All physical statements are label-invariant.

b.

Phase Factorization on Isotypic Blocks

Proposition Y.11 (Phase factorization). Let Π be either
Π3 or Π3′ , the projector onto a 9-dimensional isotypic
block. Under the canonical decomposition
M
C[A5 ] ∼
Vλ ⊗ V ∗ ,
(Y52)
=
λ

λ

the Π-block is V ⊗ V ∗ with
U = ρ(a) ⊗ 1,

V = 1 ⊗ ρ(a)∗ .

Complete Derivation: Generation Projectors
and Down-Type Selection

This section gives the construction of the species projector mechanism. The key results are:

(Y53)

The joint Fourier projector factorizes:
Π Pr,s Π = Π (Pr(L) ⊗ Ps(R) ) Π

13.

(Y51)

(Y54)

Here r labels a left-factor Z3 phase and s labels a rightfactor Z3 phase.

231
c.

Canonical Generation Projectors

One-Hop Bin Scan (Bins Chosen per Fermion)

Proposition Y.12 (Generation projectors). Fix Π ∈
{Π3 , Π3′ }. Define
Mr := Π Pr(L) Π,

r ∈ {0, 1, 2}

b 3′ )| with
Using the trace formula |yf | = |Tr(PR XPL HΠ
derived εH = 3/60:

(Y55)

Fermion

Then {M0 , M1 , M2 } are orthogonal projectors:
Mr2 = Mr ,

Mr Mr′ = 0 (r ̸= r′ ),

2
X

t
c
τ
b

Mr = Π. (Y56)

mf /mt

L-bin R-bin Computed Err

1.000
(0, 0) (0, 0)
1.000
0%
7.34×10−3 (2, 0) (1, 0) 7.28×10−3 0.8%
1.03×10−2 (0, 0) (2, 0) 9.23×10−3 10%
2.42×10−2 (0, 2) (1, 2) 1.83×10−2 24%

r=0

Each Mr has rank 3 (fixing the left eigenspace leaves the
3D right factor).
Physical interpretation: The three generations are
the three irreducible phase sectors under the left Z3 action inside the multiplicity space. This is a canonical
construction, not a phenomenological ansatz.
a. Status of the construction. The statements proved
in this appendix are canonicality statements given the
species–class dictionary and the Higgs-conjugation rule.
They do not by themselves derive the species–class dictionary from the core DFD action. The mathematical gain
is that, once the dictionary is fixed, no further arbitrariness remains in the generation projectors, phase-sector
decomposition, or finite Yukawa operator evaluation.
d.

Four heavy fermion masses are fitted within 25% using
discrete bin labels (r, s) ∈ Z3 × Z3 and derived εH . No
continuous parameters fitted.

f.

The diagonal bins (L = R) exhibit the expected εH
power hierarchy:

s 7→ −s ≡ s + 2

(mod 3)

(0, 0)
(1, 2), (2, 1)
(0, 1), (0, 2)
(1, 0), (2, 0)
(1, 1), (2, 2)

Proposition Y.15 (Down-bin shift under the Higgs-conjugation assumption). For shared QL with left label sL ,
(u)
(d)
(u)
if up-type selects sR , then down-type selects sR ≡ −sR
(u)
(mod 3). With the successful up-type choice ∆s = 2:
⇒

(d)

sR = 2

ε0H
ε0.1
H
ε1.0
H
ε1.1
H
ε1.6
H

The suppression factor εH = 0.05 is verified numerically.

g.

Light Fermion Limitation

The one-hop kernel achieves minimum ratio ∼ 3.6 ×
10−3 (≈ ε1.9
H ), insufficient for light fermions requiring
|y|/|ymax | ∼ 10−4 to 10−6 .
Resolution: Light fermion masses require the genera(L)
tion projectors Mr = ΠPr Π combined with walk-sum
kernels.

(Y58)

Under Assumption Y.14 this gives the map (1, 0) 7→ (1, 2),
which the bin scan below does not confirm.
e.

1.000
0.759
0.050
0.036
0.009

(Y57)

Assumption Y.14 (Higgs-conjugation dictionary). Uptype Yukawa couplings implement the conjugation κ on
the right-phase sector (finite analogue of H̃ ∝ H ∗ ); downtype use identity.

∆s(d) ≡ 1

|y|/|ymax | Approximate power

Bin

Down-Type Selection via Conjugation

Definition Y.13 (Right-phase conjugation). Complex
conjugation on the right factor sends eigenvalue ω s to
ω s = ω −s , inducing:

Diagonal Bin Structure

h.

Generation Projector Results

Using generation-2 projector M2 with one-hop kernel,
with bins chosen per fermion, yields heavy fermion predictions:

Numerical Bin Scan

Fermion mf /mt (obs) L-bin R-bin Computed Error
The conjugation rule (1, 0) 7→ (1, 2) follows from Assumption Y.14 that required numerical verification. Full
bin scanning (below) reveals the correct assignments differ
from the simple conjugation prediction.
The one-hop kernel computation reveals the correct bin
assignments.

t
b
τ

1.0000
0.0242
0.0103

(2, 1) (2, 1)
(0, 1) (2, 1)
(1, 0) (2, 0)

1.0000
0.0241
0.0096

0.0%
0.4%
6.6%

Using generation-1 projector M1 with walk-sum kernel:

232
Fermion mf /mt (obs) L-bin R-bin Computed Error
5.4 × 10−4
6.1 × 10−4

s
µ

14.

(2, 2) (1, 2) 4.7 × 10−4 12.9%
(2, 2) (1, 2) 4.7 × 10−4 23.4%

Bin–Overlap Lemma and the Structural
Scale

√

2

Pr(L) :=

1 X −rm
ω
L(a)m ,
3 m=0

Ps(R) :=

1 X −sn
ω
R(a)n ,
3 n=0

r ∈ {0, 1, 2},

(Y64)

s ∈ {0, 1, 2}.

(Y65)

2

20

This section provides the exact computation of the Z3 ×
Z3 bin overlaps that determine the rational multipliers in
the Af prefactors.
a.

Fix an order-3 element a ∈ A5 and ω = e2πi/3 . Define
the Z3 projectors

Normalized Class-State Matrix Elements

Let G = A5 and let S = {a, a−1 , b, b−1 } with a = (123)
and b = (12345). Define the Cayley operator (rightregular action)
X
T =
Rs ,
(Rs )g,h = δg,hs .
(Y59)

Let C3 ⊂ A5 denote P
the order-3 conjugacy class (so |C3 | =
20), and let PC3 := g∈C3 |g⟩⟨g|.
We define the Z3 × Z3 bin-overlap weights:


r(C3 ; r, s) := Tr PC3 Pr(L) Ps(R)
(Y66)
Lemma Y.16 (Exact bin-overlap evaluation). For the
regular representation of A5 one has the closed form:

1
r(C3 ; r, s) =
20 + 2ω −(r+2s) + 2ω −(2r+s)
9
(
(Y67)
8/3, r = s,
=
2,
r ̸= s.

s∈S

For each conjugacy class C ⊂ G, define the normalized
class state
1 X
|C⟩ = p
|g⟩.
(Y60)
|C| g∈C
Then the induced operator on the class subspace has
matrix elements
N (Ci ← Cj )
⟨Ci |T |Cj ⟩ = p
,
(Y61)
|Ci ||Cj |
where N (Ci ← Cj ) is the total number of Cayley edges
from elements of Cj into Ci .
In particular, for the unique order-3 class C3 of size
|C3 | = 20 in A5 and the identity class {e}, only the two
order-3 generators {a, a−1 } contribute, giving
⟨C3 |T |{e}⟩ = p

2

2
1
= √ = √ ≈ 0.4472
20
5
|C3 |

(Y62)

This exhibits structurally (i.e., without fitting)
p how√the
conjugacy-class normalization produces a |C3 | = 20
scale in any overlap built from class-localized states and
Cayley-graph operators.

b.

Let G = A5 act on HF := ℓ2 (G) by the left and right
regular actions
R(h)|g⟩ := |gh⟩.

The complete bin-overlap matrix is:
8
3




W = r(C3 ; r, s) r,s=0,1,2 =  2

2 2
8
3




2

(Y69)

8
3

2 2

4
The diagonal/off-diagonal ratio is 8/3
2 = 3.

c.

Species Projector Closure

Definition Y.17 (Complete species projector). For a
fermion species f with LH generation index i ∈ {0, 1, 2}
and RH generation index j ∈ {0, 1, 2}:

Bin–Overlap Lemma for the Order-3 Class

L(h)|g⟩ := |hg⟩,



Proof sketch (counting). Using Tr PC3 L(a)m R(a)n =
#{g ∈ C3 : am gan = g}, we reduce the problem to
counting fixed points in C3 under the map g 7→ am gan .
A direct computation in A5 gives
Nm,n := #{g ∈ C3 : am gan = g}


20, (m, n) = (0, 0),
(Y68)
= 2, (m, n) ∈ {(1, 2), (2, 1)},

0, else.
P2
Substituting into r(C3 ; r, s) = 19 m,n=0 ω −rm−sn Nm,n
yields the closed form above.

(Y63)

(f )

(L)

ΠL,f = PC Pi

(f )
Pgauge
,

(f )

(R)

ΠR,f = PC Pj

(f )
Pgauge

(Y70)
where:
(f )
• PC : class projector (quarks → C3 , leptons → {e})
(L)
(R)
• Pi , Pj : Z3 generation projectors (left/right)
(f )

• Pgauge :
gauge quantum
(color/isospin/hypercharge)

number

selector

233
Definition Y.18 ((r, s) → species map). The bin index
(r, s) encodes the Yukawa matrix entry:
Yij ←→ bin (r = i, s = j)

(Y71)

where i is the LH generation index and j is the RH
generation index.

d.

Af Prefactor Structure

Proposition Y.19 (Microsector Af formula). The
Yukawa prefactor for quark species f in generation g has
the overlap structure:
p
Yf = gY εH |C3 | · r(C3 ; g−1, g−1) · Gg · Rg,t (Y72)
where:p
√
• |C3 | = 20: structural class normalization
• r(C3 ; g−1, g−1) = 8/3: computed diagonal bin
weight
• Gg : generation suppression factor
• Rg,t : up/down type factor (Gg and Rg,t are not
computed in this volume, so the formula does not
yet fix Af )
• gY εH : single global normalization (one convention)
The mass prefactor convention absorbs gY εH · (8/3)
into the normalization, giving:
p
(Y73)
Af = |C3 | × Gg × Rg,t
Closure Status: What Is Derived vs. Convention
Derived (no fitting):
• |C3 | = 20 (A5 group
√
√ theory)
• ⟨C3 |T |{e}⟩ = 2/ 20 = 1/ 5 (Cayley matrix
element)
• r(C3 ; i, i) = 8/3, r(C3 ; i, j) = 2 for i ̸= j
(fixed-point counting)
• Closed form:
r(C3 ; r, s) = 19 (20 + 2ω −(r+2s) + 2ω −(2r+s) )
One global convention:
• gY (the global Yukawa normalization) is fixed once
from mτ /mµ (Appendix H)
• εH = 3/60 is closed at the value level: both
premises of Theorem H.4 (uniform Higgs connector;
equal 3-channel generation coupling) are forced by
A5 representation theory once Hch ∼
= C[A5 ] — the
identification this appendix’s own kernel
ℓ(g)
b ch = P
H
g∈A5 εH |g⟩⟨g| already uses (Lemma H.5,
Appendix H). The residual, one level deeper:
deriving Hch ∼
= C[A5 ] itself from the CP 2 × S 3
harmonic spectrum remains open

Final Status
Discrete bin fit (sub-10%): Heavy fermions t, b, τ via
generation-2 projector
Discrete bin fit (∼20%): Middle fermions s, µ via
generation-1 walk-sum
Mechanism: εH = 3/60 suppression, rank-3 orthogonal
generation projectors, √
discrete bin assignments
Structural closure: 20 normalization and {8/3, 2}
bin weights proven from A5 fixed-point counting
Open: Light fermions (u, d, e), c quark inter-generation
normalization, and derivation of the channel
identification Hch ∼
= C[A5 ] from the CP 2 × S 3 harmonic
spectrum (given that identification, εH = 3/60 is forced —
Lemma H.5, Appendix H)

Appendix Z: Complete Parameter Derivation

This appendix presents the microsector derivation chain
for Standard Model parameters from the topology of the
internal manifold X = CP 2 × S 3 . Combined with the
results of Appendices K–Y, this demonstrates that within
the stated microsector framework, the Standard Model
parameters follow with two continuous inputs, the fitted Higgs prefactor and the Higgs quartic (the corollary
to Theorem Z.4), once kmax = 60 is fixed as in Theorem Z.4. Beyond them, the absolute charged-fermion
masses other than the top and the charm rest on seven
data-selected prefactors, deployed by four discrete labels
that are data-selected rather than derived (App. K 5,
caveat box; App. FM). The individual derivations below
should be read in the context of the claim taxonomy in
Sec. I C.

1.

The Weinberg Angle

Theorem Z.1 (Weinberg Angle from Partition). Let
X = CP 2 × S 3 with gauge partition (3, 2, 1) corresponding
to SU(3)c × SU(2)L × U(1)Y . Then:
sin2 θW =

3
= 0.230769 . . .
13

(Z1)

Proof. Write the gauge action in trace form over the internal blocks,
X Z
Sgauge ∝
κr Tr(Frµν Fr µν ) ,
r

with stiffness scaling κr = nr κ0 for the partition
(n3 , n2 , n1 ) = (3, 2, 1). To identify the physical couplings
one must convert the trace-normalized terms to canonical Yang–Mills normalization. 
With the standard SU(2)
generator convention Tr T a T b = 21 δ ab ,
Tr(F2µν F2 µν ) = 12 F2a µν F2aµν ⇒ g −2 ∝ κ2 · 12 .
For U(1)Y the trace weight is fixed by the SM hypercharge

234
spectrum (per generation),
Tr(F1µν F1 µν ) = Tr Y F1µν F1 µν ,
X


Tr Y 2 =
d3 d2 Y 2 = 10
3 ,


2

1 gen

using QL : (3, 2, 16 ), uR : (3, 1, 23 ), dR : (3, 1, − 31 ), LL :
(1, 2, − 12 ), eR : (1, 1, −1). Hence g ′−2 ∝ κ1 · 10
3 , with the
hypercharge weight taken over a generation and the SU(2)
weight over one doublet, a stated choice. Taking the ratio
and using κ2 /κ1 = 2 gives
κ2 ( 12 )
3
g ′2
=
=
,
g2
10
)
κ1 ( 10
3

(App. AO). (iii) The CP phase η̄ scales with dim2 = 49.
(iv) The amplitude A scales with Ngen × h0 (7) = 108. (v)
All parameters are suppressed by α.
c. Status. The magnitude integers (λ, A, η̄) =
(31, 108, 49)α are line-bundle cohomology counts selected
before any apex rule. The apex selection ρ̄ = 43
2 α is
theorem-grade inside the strengthened DFD–SD branch
via the Euler-projection postulate (App. AO); consistent
with the no-forcing anti-theorem of App. AO, this postulate is a structural strengthening, not a consequence
of the pre-strengthened axioms. The mapping from each
cohomology sum to its Wolfenstein parameter remains
the open ingredient at full theorem grade.

3
g ′2
=
.
2
′2
g +g
13

sin2 θW =

TABLE CV. CKM parameter pattern verification.
Parameter

a. Experimental comparison. sin2 θW (MS, MZ ) =
0.23129 ± 0.00004 [2]. Agreement: 0.23%. Here 3/13 is
the bare angle; its radiative relation to the MS value at
MZ is not computed.
Corollary Z.2. The ratio α1 /α2 = 1/2 holds exactly
for the bare couplings, with α1 = (5/3) g ′2 /4π; the hypercharge coupling itself gives g ′2 /g 2 = 3/10.
2.

Pattern

λ

31α = 0.2262 0.22501 ± 0.00068

A

108α = 0.788

0.826+0.016
−0.015

4.6%

ρ̄

43
α = 0.157
2

0.1591 ± 0.0094

1.4%

η̄

49α = 0.358

0.3523+0.0073
−0.0071

1.5%

ρ̄ = 43
2 α,

η̄ = 49α

(Z2)

31 = h0 (2) + h0 (3) + h0 (4) = 6 + 10 + 15,

(Z3)

A = 108α,

where the integers arise as:
43
D
2 = 2

0.5%

Per-channel (no mean quoted); A is the soft spot
(|Vcb | incl./excl. tension). See App. AO.

3.

2

Agreement

The CKM Matrix

The CKM matrix shows a pattern when expressed
in terms of α and line bundle cohomology integers. The
numerical agreement is close, but a complete selection rule
identifying which cohomologies govern each parameter
remains an open problem.
a. CKM Pattern from Line Bundle Cohomology. Let
h0 (k) := dim H 0 (CP 2 , O(k)) = (k + 1)(k + 2)/2. The
Wolfenstein parameters match the pattern:
λ = 31α,

PDG 2024

− χ(CP 2 ) = 49
2 − 3,
2

3

2

2

(Z4)
2

49 = [dim(CP × S )] = 7 = D ,

(Z5)

108 = Ngen × h0 (7) = 3 × 36.

(Z6)

The CP-even apex ρ̄ = 43
2 α = 21.5α (Path A) is the Euler-

projected value: the minimizer of the reduced geometric

2
action Iρ (r) = 12 r − D2 /2 + χ(CP 2 ) r, r = ρ̄/α, derived in full in Appendix AO (Euler-projection postulate).
The raw assignment ρ̄ = 19α = [h0 (1) + h0 (2) + h0 (3)]α
does not give the apex.
b. Interpretation. (i) The Cabibbo angle λ controls
1 ↔ 2 mixing. The bundles O(2), O(3), O(4) give sections
6, 10, 15. Sum: 31. (ii) The apex coordinate ρ̄ = 43
2 α
is the Euler-projection of the ideal A5 covering-space
apex D2 /2 after the Euler localization cost χ(CP 2 ) = 3

The Higgs Sector

Theorem Z.3 (Higgs from Dimension 8). The number
8 = dim(CP 2 ×√S 3 ) + 1 determines the exponent of v,
whose prefactor 2π is fitted, and the conjectured quartic:
√
v = MP · α8 · 2π = 246.09 GeV,
(Z7)
1
λH =
(conjectured),
(Z8)
8
v
mtree
= 123.05 GeV.
(Z9)
H =
2
√
Proof. In v = MP α8 2π the exponent 8 = dim X + 1
is derived (the dimension √
count of CP 2 × S 3 plus the
Higgs radial mode). The 2π prefactor is fitted: it is
the
√ best simple constant near the best-fit value 2.508
( 2π = 2.507, a 0.054% gap), and no printed rule fixes
it. Three routes fail: a Gaussian zero-mode measure on
every rung spoils the χ relic, the winding-period rule that
fixes χ’s period caps v below 10 GeV, and the spectralaction normalization that fixes α and G gives v of order
the Planck mass (computed). Integer winding fixes a
normalization only on a compact direction, while the
vacuum value lies on the radial one, and DFD’s own permode determinant ledger (Lemma O.5) cancels such 2π
factors; the prefactor is a property of the Higgs potential’s
loop-corrected minimum, which this volume does not
compute (App. H). The 17-order α8 hierarchy is derived;
the O(1) prefactor is fitted.

235
For the quartic coupling: the conjecture λH = 1/d =
1/8 arises from the expectation that dimensional reduction
on a Kähler manifold of total dimension d = 8 (= dim X +
1 for the radial mode) yields λH = 1/d. A complete
derivation from the microsector action remains to be
established, and the scale at which the value
√ holds is not
derived. Assuming λH = 1/8: mH = v 2λH = v/2 at
tree level.
a. Radiative corrections. Imposed at the top-quark
mass, where the Standard-Model quartic is matched,
λH = 1/8 gives mH = 124.63 ± 0.15 GeV under twoloop Standard-Model running (computed): the loop
shift from the tree value v/2 = 123.05 GeV is +1.3%,
and the result lies 0.40% (2.7σ) below the measured
125.13 ± 0.11 GeV (PDG 2026). The running quartic
equals 1/8 at 182 ± 4 GeV; the matching scale is not
derived.

on 17 stated inputs: 15 discrete (the strong scale, seven
charged-fermion prefactors, three CKM counts, the Euler
projection, the neutrino branch bit, the Dirac-neutrino
Yukawa rung and sin2 θ13 = 3α) and 2 continuous (the
fitted Higgs prefactor and the Higgs quartic, conjectured
to be 1/8 at a matching scale that is not derived), with
three structural postulates (the conjugation-odd CP offset,
the C[A5 ] channel space, and trimaximal mixing with µ–τ
reflection). DFD needs far fewer continuous numbers than
the Standard Model, but about as many stated items.

6.

Integer Catalog

TABLE CVI. Master integer catalog.
Int. Geometric Origin

Physical Application

dim(S 3 ), N

4.

The PMNS Correction

a. Reactor Angle (Conjecture). The PMNS angle θ13
receives a geometric correction to tribimaximal:
√
sin θ13 = 3α = 0.148
(Z10)
√
where the factor 3 arises from Ngen = 3 or dim(S 3 ) = 3.
b. Status. sin2 θ13 = 3α is a stated conjectural input:
it fits (−0.1σ against NuFIT 6.0), and no mechanism for
the µ ↔ τ breaking that generates it from the TBM base
is printed. With trimaximal mixing and µ–τ reflection
imposed it fixes sin2 θ12 = 0.3184, which JUNO’s first
result, 0.3092±0.0087, places 1.1σ away and which JUNO
at its target precision tests at about 6σ (computed).

5.

Master Theorem

Theorem Z.4 (Complete Parameter Determination).
Given the internal manifold X = CP 2 × S 3 , the level
kmax = 60 (fixed by the generation count, the derived
q1 = 3 and the five matter types of a family), the unit MP
and the stated inputs of Tables CXXVII and CXXVIII,
the parameters not listed as inputs follow; in the fermion,
Higgs and neutrino sectors these are the top and charm
Yukawas, θ̄ = 0 at tree level, the three neutrino masses,
θ12 , θ23 and δCP .
Proof. Follows from App. K 1 (α), Theorem Z.1 (sin2 θW ),
App. AO (CKM), Theorem Z.3 (Higgs), App. K 5
(masses), Theorem L.3 (θ̄), Theorem X.2 (neutrinos),
§Z 4 (θ13 ), Theorem O.7 and Eq. (479) (H0loc at C = 1,
Postulate ZS.11; H0early with the measured C), with the
inputs stated there.
Corollary Z.5. Of the Standard Model’s 26 free continuous numbers in the fermion, Higgs and neutrino sectors, DFD derives 9, pins the two Majorana phases to
one of four CP-conserving points, and rests the other 15

3
Generations, εH = 3/60
gen
4 dim(CP 2 )
Gauge structure
7 dim(CP 2 × S 3 )
η̄ = 49α
8 dim +1
exponent of v; λH (conjectured); ka
13 3 + 10 (EW)
sin2 θW = 3/13
19 h0 (1)+h0 (2)+h0 (3) ΛQCD (not the ρ̄ apex)
31 h0 (2)+h0 (3)+h0 (4) λ = 31α
49 72
η̄ = 49α
60 kmax = χ(CP 2 , E) α−1 , εH
64 kmax + 4
Hilbert space dim.
108 3 × 36
A = 108α
137 Closed form
α−1

7.

Strong Coupling Constant

The strong coupling αs (MZ ) follows from the selected
QCD scale and its selected scheme-matching constant
(App. FK).
Theorem Z.6 (QCD Scale). The QCD confinement scale
is selected on the α-tower (App. FK):
19/2
ΛDFD
= 61.20 MeV.
QCD = MP · α

(Z11)

a. Proper-time
to
MS
matching. The
spectral/proper-time regulator produces the one-loop
effective action
 2 
Z
b0
d4 p
ΛPT
2
W1-loop ⊃
log
Fµν
.
4
2
(2π)
p2
The MS scheme defines the renormalization scale by µ̄2 :=
4πe−γE µ2 , so
 2
 2
µ̄
µ
log 2 = log 2 + log(4π) − γE .
p
p
Matching
log-arguments
gives
the scheme conversion
√
ΛMS = 4π e−γE /2 ΛPT .
The DFD definition absorbs the Euler constant:
ΛDFD := e−γE /2 ΛPT .

236
Lemma Z.7 (Scheme Matching). The proper-time to MS
conversion is:
ΛMS =

√

8.

Summary

Summary: Standard Model Parameters from Topology
4π ΛDFD

(Z12)
Results (derived, selected and input rows marked):

b. Numerical evaluation. Using MP = 1.220890 ×
1019 GeV (CODATA 2022) and α−1 = 137.036:
ΛDFD = MP × α19/2 = 61.20 MeV,
√
(5)
ΛMS = 4π × 61.20 MeV = 216.95 MeV.

Value

Agreement

α−1

137.036

21 ppm (integers)

(Z13)
(Z14)

Running to MZ = 91.1876 GeV with 4-loop QCD
(nf = 5, fixed coefficients):
αs (MZ ) = 0.1187

Parameter

θ̄

(Z15)

c. Experimental comparison. PDG 2024: αs (MZ ) =
0.1180 ± 0.0009. Agreement: 0.8σ (0.6%).
d. Trace weight sanity check. For completeness, we
verify that nonabelian trace weights cannot provide a
“10/3 miracle” for αs . Per SM generation, using the fundamental index Ifund (SU(N )) = 1/2:
SU(3): QL (2 weak components in 3) → 2 × 12 = 1;
uR , dR each → 12 . Total: TrF (T32 ) = 2.
SU(2): QL (3 colors of doublet) → 3 × 12 = 32 ; LL → 12 .
Total: TrF (T22 ) = 2.
Hence A2 = A3 = 2 from SM fermion content alone—no
nontrivial ratio emerges. The hypercharge trace Tr(Y 2 ) =
10/3 is special because it sums over different Y values; the
nonabelian traces are representation-independent. This
is why αs must be derived via ΛQCD + RG, not trace
normalization.

v
Ngen
sin2 θW
αs (MZ )
εH

Status

Derived
(identification
with 1/e2 open,
§X I)
0
≲ 10−16 (bound
Derived
(θYM = 0 fixed by
10−10 )
the spectral action
given that nothing
is added to it for
the gauge fields)
√
MP α8 2π
0.05%
Exponent
derived†
3
exact
Input†
3/13
0.23%
Derived
0.1187
0.8σ
Selected†
3/60 = 0.05
exact
Derived

Conditional (require full Af computation):
• Light fermion masses: exponent structure derived; prefactors
need overlap computation
• CKM matrix elements: integer×α pattern observed; selection
rule pending
Species construction (Section Y 13):
(L)
• Generation projectors Mr = ΠPr Π with rank 3 (canonical,
not fitted)
• Down-type selection: s 7→ −s (mod 3) under
Assumption Y.14; the bin scan selects other bins
• t/b/τ within 7% via gen-2 projector (bins chosen per
fermion)
Conjectures (need proofs):
Parameter
λH
sin θ13

Conjecture

Agreement

1/8
√
3α

1.7% (tree); −0.40% in mH at mt
0.1%

Key rigorous results:
• α−1 = 137.036 from Chern-Simons quantization
(Appendix K 1; bracket coefficient open; the identification of
the hypercharge-trace coefficient with 1/e2 is not derived)
• sin2 θW = 3/13 from the stated trace normalization +
partition (Theorem Z.1)
√
• αs (MZ ) = 0.1187 from ΛQCD = MP α19/2 + 4π matching
(selected scale; Theorem Z.6)
• θ̄ = 0 (tree level; loop-induced ≲ 10−16 ) from the
det-orthogonal tree-level protection together with the
η-invariant argument, with θYM = 0 fixed by the spectral
action given that nothing is added to it for the gauge fields
(Appendix √
L)
√
• v = MP α8 2π (exponent 8= dim X+1 derived; 2π
prefactor fitted, App. AY)
• Ngen = 3: discrete input (numerically χtop (CP 2 ); not an
index-theorem output — App. F generation-count remark)
• εH = 3/60 from channel counting (Theorem H.4)
• Generation = left Z3 phase sectors in V ⊗ V ∗
(Proposition Y.12)
Conclusion: Six fundamental parameters plus the generation
structure follow from topology under the stated inputs;
Ngen = 3 is a discrete input and the v prefactor is fitted († ;
App. F/AY status remarks).

237
Appendix AA: The Padé Identity: GR’s Isotropic
Lapse and the Exponential

This appendix establishes a precise mathematical
identity between general relativity’s isotropic-coordinate
Schwarzschild lapse-squared scalar and the exponential
exp(2u), the value of DFD’s lapse-squared eψ at ψ = 2u.
The result is purely structural: GR’s Schwarzschild form
is the squared [1, 1] Padé approximant of the exponential, with matching argument. Agreement through second
order in the gravitational potential reproduces the postNewtonian parameter β = 1 in both theories (consistent
with all current gravitational-redshift observations). Beyond that order DFD’s exterior is not the exponential:
with the scalar-longitudinal piece σ of the parent strain
kept, it is the Schwarzschild metric in DFD’s coordinates
(App. WS), so the pole of the rational form at u = 2 is a
horizon of DFD’s black holes as well. The Padé identity
operates on the lapse-squared scalar; the spatial-curvature
PPN parameter γ is established separately from DFD’s
physical metric (Sec. IV).

1.

giving the lapse-squared scalar

2
1 + u/2
LGR (u) =
.
1 − u/2

Theorem AA.1 (Padé Identity). The lapse-squared
scalar of isotropic-coordinate Schwarzschild is mathematically identical to the square of the [1, 1] Padé approximant
of eu . Writing Pm,m (u) for the diagonal [m, m] Padé
approximant of exp(u) (so Pm,m (u) ≈ eu ):

2
(AA5)
LGR (u) = P1,1 (u) ,
where P1,1 (u) = (1 + u/2)/(1 − u/2) is the unique rational
function with degree-1 numerator and denominator whose
Taylor expansion matches that of exp(u) through O(u2 ).
Proof. The [1, 1] Padé approximant of exp(u) about u = 0
is constructed by requiring
a0 + a1 u
u2
=1+u+
+ O(u3 ).
(AA6)
1 + b1 u
2
Cross-multiplying and matching coefficients at orders
u0 , u1 , u2 gives the unique solution a0 = 1, a1 = 1/2,
b1 = −1/2:

Statement of the Padé Identity

P1,1 (u) =

Define the lapse-squared scalar
L(u) ≡

c2
,
|gtt |

(AA1)

a clock-rate / redshift observable, with u ≡ GM/(ρc2 )
the dimensionless gravitational potential, where ρ denotes the isotropic radial coordinate in GR and the flatspace radial coordinate in DFD. DFD’s matter-coupling
(physical) metric has gtt = −c2 e−ψ (Sec. IV); on the
spherically symmetric exterior in the µ → 1 regime,
ψ(r) = 2GM/(c2 r) = 2u through second order in u,
giving
ψ

LDFD (u) = e = exp(2u)

(AA2)

at that order (Sec. VI A, Eq. (147)); with the parent
strain’s σ kept the full exterior is the Schwarzschild metric in DFD’s coordinates, ψ = 2u − 43 u3 + · · · (App. WS).
Independently, by Postulate P1 the optical refractive
index governing light propagation through the optical
metric ds̃2 = −c2 dt2 /n2 + dx2 is nDFD (u) = eψ , equal to
exp(2u) at the same order. The lapse-squared LDFD and
the refractive index nDFD are distinct physical quantities
arising from different metrics in DFD’s two-metric structure, but they coincide as functions of u on the exterior
solution. The Padé identity below operates on L (the
clock-rate scalar).
The corresponding Schwarzschild metric in isotropic
coordinates takes the form
ds2GR,iso = −

(AA4)



1 − u/2
1 + u/2

2



4
c2 dt2 + (1 + u/2) dρ2 + ρ2 dΩ2 ,

(AA3)

1 + u/2
.
1 − u/2

(AA7)

Squaring and comparing with (AA4) gives (AA5).

2.

Order-by-Order Agreement

Taylor expanding the exponential and GR’s form about
u = 0:
4 5
u + O(u6 ),
exp(2u) = 1 + 2u + 2u2 + 43 u3 + 23 u4 + 15
(AA8)

LGR (u) = 1 + 2u + 2u2 + 23 u3 + u4 + 58 u5 + O(u6 ).
(AA9)
The series agree identically through O(u2 ), which is the
defining property of the [1, 1] Padé approximant. They
first diverge at O(u3 ), with coefficient difference 3/2 −
4/3 = 1/6. DFD’s own lapse-squared agrees with exp(2u)
through O(u2 ) in DFD’s radius; beyond that order it is
Schwarzschild’s (App. WS).
TABLE CVII. Taylor coefficients of exp(2u) and LGR through
O(u5 ).
Order n [un ] exp(2u) [un ] LGR Difference
0
1
2
3
4
5

1
2
2
4/3
2/3
4/15

1
2
2
3/2
1
5/8

0
0
0
−1/6
−1/3
−43/120

238
a. PPN consequence. The O(u) and O(u2 ) coefficients of L = c2 /|gtt | are set by the PPN expansion of
gtt , from which the post-Newtonian parameter β = 1
in both theories is read off, consistent with β − 1 =
(−4.5 ± 5.6) × 10−5 from Hofmann–Müller [121] lunar
laser ranging. The PPN parameter γ is determined by
the spatial metric gij and is not accessible from the lapsesquared sector alone; for DFD, γ = 1 follows from the
physical metric gij = e+ψ δij at first post-Newtonian order and is established in Sec. IV independently of the
present identity, consistent with γ − 1 = (2.1 ± 2.3) × 10−5
from Cassini [35]. The two formulations are therefore observationally indistinguishable at all currently measured
post-Newtonian orders, with β = 1 following directly from
the Padé identity and γ = 1 following separately from
DFD’s physical-metric structure.

3.

The Padé Pole is GR’s Horizon

The rational approximant (AA4) develops a double
pole at u = 2, where 1 − u/2 = 0. In isotropic coordinates
this corresponds to ρ = GM/(2c2 ). The transformation
to standard Schwarzschild coordinates,

u 2
r =ρ 1+
,
(AA10)
2
maps u = 2 (equivalently ρ = GM/(2c2 )) to r = 2GM/c2 ,
precisely the Schwarzschild event horizon. The pole is
structurally a feature of the rational form.
By contrast, the exponential function exp(2u) is entire:
it has no pole anywhere in the finite complex plane. At
u = 2 it takes the finite value exp(4) ≈ 54.598. In
DFD’s variables the horizon of a static exterior is reached
as ψ → ∞: the lapse e−ψ/2 has no zero at finite ψ,
and with the parent strain’s σ kept DFD’s exterior is
Schwarzschild’s (App. WS), so the pole is a horizon of
DFD’s black holes as well.

4.

Distinction from Yilmaz-Type Exponential
Metrics

The exponential refractive-index form of DFD should
not be conflated with the Yilmaz–Rosen–Papapetrou exponential metric [163, 164],


ds2Yilmaz = −e−2m/r dt2 + e+2m/r dr2 + r2 dΩ2 ,
(AA11)
which arises as a solution of Einstein–Klein–Gordon equations with a massless antiscalar field. When analyzed as a
4-metric within GR, (AA11) was shown by Boonserm et
al. [165] to represent a traversable wormhole with exotic
matter at the throat.

Theorem AA.2 (No wormhole in the DFD flat-R3 formulation). On the fundamental flat-R3 spatial manifold
of DFD, there exists
p no wormhole throat. The areal radius
function R(r) ≡ A(r)/4π of the arena metric on any
constant-t slice satisfies R(r) = r and dR/dr = 1 for all
r > 0, admitting no minimum; that of the physical metric
of a static exterior increases monotonically outside the
horizon.
Proof. DFD postulates a flat Euclidean spatial metric
dl2 = δij dxi dxj = dr2 + r2 dΩ2 . The proper area of
a 2-sphere at coordinate radius r is A(r) = 4πr2 , giving R(r) = r. Differentiating, dR/dr = 1 identically
on r > 0. No critical points exist, hence no minimum, hence no throat. With the parent strain’s σ
kept the physical metric of a static exterior has lapse
e−ψ/2 = (1 − 2GM/c2 R)1/2 and det gij = e3ψ (App. WS),
so its areal radius R obeys r2 dr = R2 (1 − 2GM/c2 R) dR,
that is r3 = R3 − 3(GM/c2 )R2 + const, and dR/dr =
r2 /[R(R − 2GM/c2 )] > 0 outside the horizon: no minimum there either. By contrast, the Yilmaz metric (AA11)
has spatial metric dl2 = e2m/r (dr2 + r2 dΩ2 ), giving
R(r) = r em/r with dR/dr = em/r (1 − m/r), which vanishes at r = m and is a minimum (second derivative
e/m > 0). The Yilmaz throat is located at r = m with
Rmin = em ≈ 2.718 m.
Theorem AA.3 (No exotic matter in the DFD fundamental formulation). DFD’s field equation (21) is sourced by
ordinary matter density ρ ≥ 0. The ψ-field energy density
uψ = (a20 /8πG) W (|∇ψ|2 /a2⋆ ) is nonnegative for all configurations satisfying the monotonicity condition (A4) of
Sec. III A. The physical metric obeys the Einstein–Hilbert
equations with this ordinary source (Proposition WS.5);
no exotic matter is invoked, and the null energy condition
holds.
Proof. By construction, Eq. (21) is an elliptic PDE for
ψ on R3 sourced by ρ. Under premise G1 of Proposition WS.5 the Einstein–Hilbert term is the dynamics of the physical metric, with the parent strain’s σ
kept, and its source is the matter stress tensor, which
satisfies the null energy condition for ordinary matter.
The isotropic exponential exterior would need, in addition, the stress of a massless scalar with negative kinetic energy, Yilmaz’s antiscalar: its Einstein tensor is
Ga b = |∇U |2 diag(1, −1, 1, 1) with U = GM/(c2 r), which
no ordinary source supplies (computed). The nonnegativity of uψ = (a20 /8πG) W (s) follows from W≥ 0, which
√
√
holds for W (X) = X − 2 X + 2 ln 1 + X , the primitive of the derived µ(x) = x/(1+x) (see Appendix N).
a. Ontological distinction. DFD and Yilmaz share
the exponential refractive-index form but differ fundamentally:

239
DFD

Yilmaz

Fundamental geometry
Flat R3
Curved 4D Lorentzian
3
Field content
Parent strain on R (ψ and its traceless part) Metric gµν + antiscalar φ
Field equation
Einstein–Hilbert with ordinary ρ ≥ 0
Einstein–Klein–Gordon
3
Spatial topology
R
Wormhole (throat at r = m)
Matter requirement
Ordinary ρ ≥ 0
Exotic (NEC-violating)
Horizon
GR’s event horizon
None (throat instead)

The Boonserm–Visser wormhole/exotic-matter critique
applies to (AA11) but not to DFD, because DFD does
not treat (AA11) as a fundamental object. The optical metric used in DFD for null-geodesic tracking,
ds̃2 = −c2 dt2 /n2 + (eh )ij dxi dxj , is flat in its spatial part
only through first post-Newtonian order; the physical
metric solves Einstein’s equations with ordinary matter
where µ → 1 (App. WS). The Padé identity itself is a
statement in the lapse-squared scalar L = c2 /|gtt |, computed in DFD from the matter-coupling physical metric
(Sec. IV); on the spherically symmetric exterior this coincides numerically with the optical refractive index, but the
two are distinct physical quantities arising from DFD’s
two-metric structure.

5.

The Padé Hierarchy

The [1, 1] Padé is the first nontrivial rational approximant of the exponential. Higher-order approximants
match the exponential to increasing Taylor order: the
squared [m, m] diagonal Padé approximant satisfies

2
Pm,m (exp(u)) = exp(2u) + O(u2m+1 ).
(AA12)
The full exponential is the m → ∞ limit: the unique
entire function agreeing with every order. Any finite-m
approximant has at least one pole in the complex plane,
which lies on the positive real axis, and so gives a finiteradius singularity, only for odd m.
This hierarchy provides a natural interpretation of extended theories:
• A hypothetical gravity theory with n(u) =
[P2,2 (exp(u))]2 would agree with the exponential
through O(u4 ), i.e. two orders beyond standard
PPN, indistinguishable at second post-Newtonian
precision before any deviation would appear.
• Each additional Padé order matches two additional
orders in u.
• The exponential is singled out not by any specific
post-Newtonian test but by the simplicity and analyticity of its underlying functional equation —
Cauchy’s multiplicative composition law (Sec. II) —
which admits no pole structure at all.

6.

Framing Summary

Padé Identity: Structural Framing
Core identity (Theorem AA.1):
LGR (u) = [P1,1 (exp(u))]2 ,

LDFD (u) = exp(2u) + O(u3 ),

where L(u) ≡ c2 /|gtt | is the lapse-squared scalar
(clock-rate / redshift observable). GR’s isotropic
lapse-squared is the m = 1 slot of the Padé
hierarchy of the exponential.
Weak-field (observational) consequence: Agreement
through O(u2 ) by construction of the Padé approximant.
PPN parameter β = 1 in both theories follows directly
from the lapse identity, matching LLR at ≲ 10−4 . PPN
parameter γ = 1 for DFD follows separately from the
physical-metric structure gij = e+ψ δij at first
post-Newtonian order (Sec. IV), matching Cassini at
≲ 10−5 .
Strong-field consequence: First divergence of the two
functions at O(u3 ); pole at u = 2, the Schwarzschild
horizon r = 2GM/c2 . With the parent strain’s σ kept
DFD’s exterior is the Schwarzschild metric in DFD’s
coordinates (App. WS), so the pole is a horizon of DFD’s
black holes too, and DFD’s photon sphere and shadow
are Schwarzschild’s.
Firewall from Yilmaz: DFD’s fundamental arena is
flat R3 ; its spatial topology is globally trivial; its matter
source is ordinary ρ ≥ 0, and its exterior is
Schwarzschild’s, not Yilmaz’s. Theorems AA.2 and AA.3
establish that the Boonserm–Visser
wormhole/exotic-matter critique of (AA11) does not
apply to DFD.

240
Appendix AB: Uniqueness of the Internal Manifold
CP 2 × S 3

a global frame on Mg is what makes that sector globally well defined. Only the consequence
χtop (Mg ) = 0 is used in the proof below.

The preceding appendices have used the internal manifold X = CP 2 × S 3 for the Standard Model gauge structure, the generation count, the fine-structure constant,
the CKM pattern, the Strong-CP vanishing, and the G–
H0 invariant. A natural question is whether this manifold
is an ansatz or whether it is uniquely forced by first principles. This appendix closes that gap: we show that
X = CP 2 × S 3 is the unique compact internal manifold
satisfying a finite list of axioms motivated by the optical
structure of DFD, the observed Standard Model gauge
group, and the observed generation count.

(V5) Three generations. χtop (Mc ) = 3, supplying
the generation count by identification: the observed family number is an input, and no index on
the internal space supplies it (App. F, Rem. F.21).
The side-condition dimC Mc ≥ 2 is not assumed:
it follows from (V5)+(V10), since the only Fano
curve is CP 1 with χtop = 2 (Corollary AB.3, Step
1).

1.

Vacuum Axioms

We seek a compact Riemannian manifold X —the internal fiber of the vacuum on which the refractive field ψ
lives— satisfying:
(V1) Compactness and smoothness. X is a compact, connected, smooth, orientable Riemannian
manifold without boundary.
(V2) Minimality. dim X is minimal among all X
satisfying the remaining axioms. The value
dim X = 7 is not assumed: it is derived in Corollary AB.3, together with the split 7 = 4 + 3. An
isometric (3, 2, 1) partition cannot justify it (see
Lemma AB.13).
(V3) Not an axiom. Every closed orientable 4-manifold
and every closed 3-manifold admits a Spinc structure, so this condition selects nothing within
the class considered. Spinc -ness of the winner
— needed for the Dirac operator and the Spinc
HRR index of Appendix K 4 — is recovered as
Corollary AB.12.
(V4) Nondegenerate product structure. X =
Mc × Mg where Mc is a compact Kähler manifold
and Mg is a compact manifold with dim Mg ≥ 1.
The complex factor supplies the chiral (holomorphic) Yukawa structure; the frame factor
supplies the topological protection mechanisms.
(V4) does not require “a compact Lie group or
group quotient”: that clause is simultaneously
too strong (the group structure is never used)
and fatally too weak (the quotient branch admits Mg = S 2 = SU(2)/U(1), which is simply connected with π3 (S 2 ) = Z and would give
dim X = 6, defeating minimality). (V4′ ) is used
instead.
(V4′ ) Global transgression frame. Mg is parallelizable. The Chern–Simons transgression that
defines the χ sector is framing-dependent, so

(V6) Internal mode partition (3, 2, 1) — a dictionary entry, not a selector. The rank-6 internal
mode bundle Hint = C3 ⊕ C2 ⊕ C of Sec. XVII B
carries the block partition (n3 , n2 , n1 ) = (3, 2, 1)
of complex ranks, whose U (6) stabiliser is U (3) ×
U (2) × U (1), that is SU(3) × SU(2) with two
abelian factors modulo the overall phase, together
with a singlet sector. Under the electroweak identification the single hypercharge is the abelian
factor of the isotropy split C3 = ℓ⊥
p ⊕ ℓp of
CP 2 , whose stabiliser is S(U (2) × U (1)) = U (2)
(App. F, Rem. F.2). This is a condition on an
associated bundle over spacetime, not on T X:
by Lemma AB.13 the tangent bundle of any
CP 2 × N 3 admits no splitting into subbundles of
real ranks (3, 2, 1, 1), and none is required. (V6)
is not used in the proof of Theorem AB.1; it is
recorded here only to fix the dictionary. No isometric (3, 2, 1, 1) splitting of T X is asserted.
(V7) Simply connected frame factor. π1 (Mg ) = 1.
π3 (Mg ) = Z is then a theorem, not an axiom: see
Corollary AB.12. The winding number of Theorem F.26 is therefore derived from the internal
geometry rather than posited. Simple connectedness is also the strictly stronger of the two bits:
assuming π3 = Z instead leaves Mg ranging over
the infinite family {S 3 /Γ}∪{S 2 ×S 1 , RP 3 #RP 3 },
i.e. no uniqueness at all.
(V8) Not an axiom: it is a theorem. Since dim TCP =
dim X +1 and dim X = 7 is odd (Corollary AB.3),
dim TCP = 8 is automatically even, so η(DTCP ) =
0 by the spectral-symmetry theorem of Appendix L. Recorded as Corollary AB.12.
(V10) Positive internal curvature. Mc admits a
Kähler metric of positive Ricci curvature. This is
the stiffness hypothesis that Lemma AB.4 needs;
without it that lemma is false, with two counterexample families (see the remark following it).
By Kobayashi’s theorem it also forces Mc to be
Fano and simply connected.
a. Remark on the axiom count. The list above has
seven substantive statements — (V1), (V2), (V4), (V4′ ),
(V5), (V7), (V10) — and it is satisfiable, with the explicit witness verified axiom-by-axiom in Lemma AB.11.

241
An axiom asserting a splitting of T X into subbundles
of real ranks (3, 2, 1, 1) would make the list unsatisfiable:
Lemma AB.13 shows that such a splitting is topologically
impossible on CP 2 × N 3 for every closed 3-manifold N ,
and an unsatisfiable axiom set makes a uniqueness theorem vacuously true, which is a worse failure mode than losing uniqueness. Four properties are theorems rather than
assumptions: dim X = 7, Spinc , π3 (Mg ) = Z, and even
mapping-torus dimension; and kmax = 60 is a computed
property of the winner. kmax = 60 is not claimed to follow
from (V2), (V4) and (V6): that implication would need
the tangent-bundle reading of (V6), which (V6) above
does not make. kmax = 60 is established for the winner
by Hirzebruch–Riemann–Roch (Corollary AB.12(iv)), not
by an axiom.
2.

Uniqueness Theorem

Theorem AB.1 (Uniqueness of CP 2 ×S 3 ). Let X satisfy
axioms (V1), (V2), (V4), (V4′ ), (V5), (V7), (V10). Then
Mc is biholomorphic to CP 2 , Mg is diffeomorphic to S 3 ,
and up to diffeomorphism
2

3

X = CP × S .

(AB1)

The axiom set is satisfiable — the manifold (AB1) is an
explicit model for it (Lemma AB.11) — so the statement
is not vacuous.
Remark AB.2 (What is and is not fixed). The axioms
fix the smooth structure of X and the complex structure
of Mc ; they do not fix the metric. The Fubini–Study
scale and the S 3 radius remain as two continuous moduli, pinned downstream by the kmax = 60 / α57 closure
(Appendix O). “Up to diffeomorphism” is exact; “up to
isometry” would overstate it.
The proof proceeds by reducing each axiom to a structural constraint and showing that the intersection of these
constraints admits a unique solution.
3.
a.

Reduction of the Axioms

Dimension, product structure, and the split 7 = 4 + 3

Corollary AB.3 (Internal dimension — dim X = 7 is
derived). Under (V1), (V4), (V4′ ), (V5), (V7), (V10)
the dimensions obey dimR Mc ≥ 4 and dim Mg ≥ 3; both
bounds are attained simultaneously. Hence by minimality
(V2),
dim X = 4 + 3 = 7,

curve has deg K < 0, hence genus 0, hence is CP 1 with
χtop = 2. Both contradict (V5). So dimC Mc ≥ 2.
Step 2: dim Mg ≥ 3. Mg is closed, connected (X is),
orientable (X and Mc are), of dimension ≥ 1 by (V4),
simply connected by (V7), and parallelizable by (V4′ )
— hence it admits a nowhere-vanishing vector field and
χtop (Mg ) = 0 by Poincaré–Hopf. In dimension 1 the
only closed connected orientable manifold is S 1 , excluded
by (V7). In dimension 2 the closed connected orientable
manifolds are S 2 and the genus-g surfaces; the latter
have π1 ̸= 1, and S 2 has χtop = 2 ̸= 0, excluded by
(V4′ ). (Dimension 0 is excluded by the nondegeneracy
clause of (V4); note Poincaré–Hopf does not reach it, since
χ(pt) = 1.)
Step 3: attainment. CP 2 × S 3 satisfies every axiom
(Lemma AB.11) with dim = 7. Hence the minimum in
(V2) is exactly 7, and since dim Mc + dim Mg = 7 with
dim Mc ≥ 4, dim Mg ≥ 3, the split is forced to 4 + 3.

b.

(4)

Identification of Mc

Lemma AB.4 (Kähler surface with χ = 3 and positive
Ricci). A compact Kähler surface admitting a metric of
positive Ricci curvature and having χtop = 3 is biholomorphic to CP 2 .
Proof. By Kobayashi’s theorem the surface is Fano, i.e.
del Pezzo. Noether’s formula χtop = 12χ(O) − K 2 with
χ(O) = 1 gives χtop = 3 for CP 2 (K 2 = 9), 4 for CP 1 ×
CP 1 (K 2 = 8), and 3 + k for dPk = Blk CP 2 (K 2 = 9 − k,
1 ≤ k ≤ 8), i.e. 4, . . . , 11. Exactly one entry equals 3.
Remark AB.5 (Why the positivity hypothesis is not optional). Without (V10) the lemma is false, and a Spinc
hypothesis does not save it (every compact complex surface is Spinc via c1 ). Two counterexample families have
χtop = 3: (i) the fake projective planes, ball quotients
B2 /Γ with pg = q = 0, c21 = 9, c2 = 3 and K ample —
100 of them, in 50 conjugate pairs; and (ii) the infinite
family Bl4g−1 (Cg × CP 1 ) for a genus-g curve Cg , which
has χtop = 4 − 4g + k = 3 for every g ≥ 1. Both have infinite π1 , so Kobayashi’s simple-connectedness conclusion
removes both at once.

c.

(3)

Identification of Mg

Lemma AB.6 (Frame factor). A closed, connected, orientable, simply connected 3-manifold is diffeomorphic to
S 3 ; consequently π3 (Mg ) = Z.

and the split 7 = 4 + 3 is forced.

Proof. This is the Poincaré conjecture, proved by Perelman (2003). The homotopy statement then follows from
π3 (S 3 ) = Z (Hopf degree).

Proof. Step 1: dimR Mc ≥ 4. By (V10) and Kobayashi’s
theorem, a compact Kähler manifold carrying a metric of
positive Ricci curvature is Fano and simply connected. A
0-dimensional Fano manifold is a point, χtop = 1; a Fano

Remark AB.7 (The lens-quotient menu is removed by
proof, not by selection). A 12-member discrete menu

242
L(p, 1) = S 3 /Zp , p | 60, priced in the trials-factor audit of App. CE, fails twice over. First, it is an undercount: the quotient branch of a group-quotient clause in
(V4) would also admit every spherical space form S 3 /Γ
with Γ < SU(2) finite (binary dihedral, tetrahedral, octahedral, icosahedral — not only cyclic), together with
S 1 × S 2 = (U(1) × SU(2))/U(1), which is a compact
homogeneous 3-manifold with π3 = Z and is not a lens
space at all. Second, the entire menu is removed by proof :
every member has π1 ̸= 1 and is excluded by (V7), and
Lemma AB.6 then leaves S 3 as the unique survivor. The
×12 price is therefore not owed in the App. CE trials
audit (which keeps it only as a hostile over-count), and
the one discrete bit that remains — simple connectedness
of Mg , axiom (V7) — is the only price on this coordinate:
π3 (Mg ) = Z is derived, not assumed.
A group-quotient clause in (V4) would leave a second
loophole. S 2 = SU(2)/U(1) is a compact group quotient,
is simply connected, and has π3 (S 2 ) = Z by the Hopf
fibration; so X ′ = CP 2 × S 2 would satisfy (V1), (V4)
with that clause, (V5) and (V7) at dim X ′ = 6, and no
minimality argument could produce dim X = 7. Axiom
(V4′ ) removes S 2 — and only S 2 — via χtop (S 2 ) = 2 ̸= 0.
Proposition AB.8 (Exhaustive invariants of the
12-member lens menu, and a charge-quantization interlock). For all twelve members L(p, 1), p | 60, of the menu
of Remark AB.7: the real Betti numbers of CP 2 × L(p, 1)
are identically (1, 0, 1, 1, 1, 1, 0, 1), independent of p; the
2
torsion is H1 = H3 = H5 = Zp with Htors
= Zp ; and
0
Isom = U (1) × SU(2) for p ≥ 3 but SO(3) × SO(3) for
p = 2. No twist restores a full doublet for any p ≥ 3.
Consequently every p ≥ 2 predicts discrete charge hair
on the Zp torsion classes, whereas DFD’s p = 1 predicts
none.
Falsifier AB.9 (Charge quantization). Observation of discrete Zp charge hair — a quantized charge label with no
continuous gauge-field source — selects p ≥ 2 and reopens
the lens menu, falsifying axiom (V7), which removes the
menu in Remark AB.7. DFD predicts no such hair.
Remark AB.10 (Trials-factor: the ×12 improvement is declined). It is tempting to argue that Proposition AB.8 collapses the 12-member menu and improves the App. CE αladder margin from ×159.8 to ×1918. That improvement
is declined: χ(CP 2 , E) = 60 is a Hirzebruch–Riemann–
Roch statement on the CP 2 factor alone and is untouched
by an S 3 quotient, so the menu is not reduced by this
route. The margin stays at ×159.8.
d.

Consistency with remaining axioms

Lemma AB.11 (Satisfiability — the axiom set has a
model). The manifold X = CP 2 ×S 3 , carrying the Fubini–
Study metric on the first factor and the round metric on
the second, satisfies (V1), (V2), (V4), (V4′ ), (V5), (V7),
(V10) simultaneously. In particular Theorem AB.1 is not
vacuous.

Proof. (V1) A product of two closed orientable manifolds
is closed and orientable, and both factors are connected.
(V4) CP 2 is compact Kähler; the Fubini–Study
metric
arises from the Kähler potential K = log 1 + |z|2 on an
affine chart, with giȷ̄ = ∂i ∂¯j K and det g = (1 + |z|2 )−3 .
S 3 is compact of dimension 3 ≥ 1.
(V4′ ) S 3 ⊂ H is parallelizable: q 7→ (qi, qj, qk) is a
global orthonormal frame of T S 3 , since right multiplication by a unit imaginary quaternion is an isometry of R4
carrying q ⊥ to itself. Hence χtop (S 3 ) = 0.
(V10) det g = (1 + |z|2 )−3 gives Riciȷ̄ =
−∂i ∂¯j log det g = 3 giȷ̄ , so the Fubini–Study metric is
Kähler–Einstein with positive Einstein constant; the induced real metric satisfies RicR = 3 gR > 0. (In the
normalisation with holomorphic sectional curvature 4 this
reads Ric = 6g; positivity is scale-free.) The round S 3 of
radius 1 has Ric = 2g > 0.
(V5) By Chern–Weil applied to the curvature Θ =
¯ −1 ∂g) of the same Fubini–Study metric, tr Θ = −3g
∂(g
R
(so c1 = 3H), CP 2 c21 = 9, and
Z
χtop (CP 2 ) =
c2
CP 2
Z
 2 

1
i
=
(tr Θ)2 − tr(Θ ∧ Θ) = 3.
2 2π
CP 2

3

(V7) π1 (S ) = 1.
(V2) By Corollary AB.3 no admissible X has dimension
below 7, and this one has dimension 4 + 3 = 7; so it
realises the minimum.
Corollary AB.12 (Properties of the winner: Spinc , π3 ,
the even mapping torus and kmax ). For X = CP 2 × S 3 :
(i) X is Spinc : c1 (CP 2 ) = 3H supplies the determinant
line O(3), and S 3 is parallelizable hence spin; the product carries the product Spinc structure. (ii) π3 (Mg ) =
π3 (S 3 ) = Z — the winding invariant of Theorem F.26,
derived (Lemma AB.6). (iii) dim TCP = dim X + 1 = 8 is
even, so η(DTCP ) = 0 by Theorem L.1. The CP involution
is complex conjugation [z0 : z1 : z2 ] 7→ [z̄0 : z̄1 : z̄2 ] on
CP 2 times the identity on S 3 . (iv) χ(CP 2 , O(9)⊕O⊕5 ) =
55 + 5 = 60 = kmax by Hirzebruch–Riemann–Roch, and
57 = kmax − Ngen .
e.

Uniqueness assembly

Proof of Theorem AB.1. Corollary AB.3 forces dim Mc =
4 and dim Mg = 3. Lemma AB.4 then gives Mc ∼
= CP 2
3
∼
and Lemma AB.6 gives Mg = S . Lemma AB.11 exhibits
that manifold as a model of the full axiom set, so the
solution set is non-empty and the theorem is not vacuous.
Uniqueness up to diffeomorphism follows.

4.

Tangent-Bundle Obstruction Lemma

Lemma AB.13 (No (3, 2, 1, 1) splitting of the internal
tangent bundle). Let N be any closed 3-manifold. Then

243
T (CP 2 × N ) admits no splitting into subbundles of real
ranks 3, 2, 1, 1.

T CP 2 ⊕ C = O(1)⊕3 , giving

Proof. Restrict a putative splitting to the slice CP 2 ×{pt};
restriction preserves ranks, so it suffices to obstruct a
(3, 2, 1, 1) splitting of T CP 2 ⊕ ε3 . Write H ∗ (CP 2 ; Z/2) =
Z/2[a]/(a3 ), deg a = 2. Since H 1 = H 3 = 0, every real
line subbundle is trivial and a rank-2 or rank-3 subbundle
has total class 1 + ϵa with ϵ ∈ Z/2. The total class is
unchanged by adding trivial summands: w(T CP 2 ⊕ ε3 ) =
(1 + a)3 = 1 + a + a2 (mod 2). Whitney then requires

real ranks (2, 2, 2, 1) with abelian structure group U(1)3 ×
O(1) — not SU(3) × SU(2) × U(1). Independent grouptheoretic corroboration: the minimal faithful real representation dimensions are 6, 4, 2 for SU(3), SU(2), U(1),
against slots of dimension 3, 2, 1; and dim Isom(S 3 ) =
dim O(4) = 6 < 8 = dim SU(3), so no sub-object of the S 3
factor can carry colour under any partition. The (3, 2, 1)
of (V6) is the block structure of the complex mode bundle
Hint = C3 ⊕ C2 ⊕ C (Sec. XVII B), an associated bundle
over spacetime, not a splitting of T X.

(1 + ϵa)(1 + ηa) = 1 + (ϵ + η)a + ϵη a2 = 1 + a + a2 ,
i.e. ϵ + η = 1 and ϵη = 1 simultaneously; the second forces
ϵ = η = 1, whence ϵ + η = 0 ̸= 1. Contradiction.
Remark AB.14 (What T X does split into, and what
(V6) is about). The obstruction is sharp: an exhaustive
check over all rank partitions of 7 finds (3, 3, 1), (3, 2, 1, 1),
(3, 1, 1, 1, 1), (2, 2, 1, 1, 1), (2, 15 ) and (17 ) obstructed while
the neighbours (4, 3), (3, 2, 2) and (2, 2, 2, 1) are allowed.
The genuine decomposition comes from the Euler sequence

Integer

Origin

T X = π ∗ O(1)⊕3
R ⊕ εR ,

5.

Integer Cohomology and Standard Model
Parameters

Once X = CP 2 × S 3 is established, the integers appearing throughout the Standard Model parameter derivations
acquire a uniform topological interpretation:

Role

3
dim S 3 = χtop (CP 2 )
generations, CKM integer ×α3
7
dim X
internal dimension, gauge partition
8
dim TCP
Strong-CP vanishing
13
3 + 10
Weinberg angle denominator (3/13)
19
h0 (O(1)) + h0 (O(2)) + h0 (O(3)) ΛQCD exponent 19/2 (selected); baryon-octet intercept
31
line-bundle section count
CKM λ integer
2
2
49 = 7
(dim X)
CKM η̄ integer
57 = kmax − Ngen Toeplitz rank minus generations GℏH02 /c5 exponent
60
χ(CP 2 , E) = |A5 |
Toeplitz truncation rank
0
108 = 3 · 36
Ngen · h
CKM A integer
137
spectral weighted sum
α−1

These are not free parameters: they are fixed by the
unique manifold (AB1) together with the stated inputs
and identifications of App. FK.

6.

Falsifiability

Theorem AB.1 supports a sharp falsification criterion:
• Discovery of a fourth generation (violating (V5))
would falsify the identification in (V5).
• A confirmed deviation of sin2 θW from 3/13 after
radiative corrections, or of α−1 from 137.036 beyond systematic uncertainty, would challenge the
mode-bundle block structure of (V6) or the Toeplitz

closure of Corollary AB.12(iv) — not any property
of T X.
• A robust detection of |θ̄| > 10−15 in neutron or
atomic EDM experiments would falsify the treelevel θ̄ = 0 and its loop bound (Theorem L.3).
• Any compact manifold other than CP 2 × S 3 satisfying (V1), (V2), (V4), (V4′ ), (V5), (V7), (V10)
would falsify Theorem AB.1 by counterexample.
Conversely, exhibiting that no manifold at all satisfies the list would render the theorem vacuous; that
is why Lemma AB.11 exhibits an explicit model.
a. Status. The uniqueness theorem is theorem-grade
conditional on the seven axioms (V1), (V2), (V4), (V4′ ),

244
(V5), (V7), (V10); uniqueness among all compact 7manifolds without these axioms would need an exhaustive
classification and is not claimed. The axioms are satisfiable — an explicit model is exhibited in Lemma AB.11 —
and each is used. The list is closed, and four properties
are theorems here: dim X = 7, Spinc , π3 (Mg ) = Z, and
even mapping-torus dimension. The single discrete bit
that remains on the Mg coordinate is simple connectedness (V7); it is the only price on that coordinate, and
neither a π3 = Z axiom nor the ×12 lens menu is needed
(App. CE keeps the menu only as a hostile over-count).

7.

Summary

Uniqueness of the Internal Manifold
Statement (Theorem AB.1): The unique compact
internal manifold satisfying DFD’s seven vacuum axioms
— compactness, minimality of dimension, nondegenerate
Kähler-times-framed product structure, positive internal
Ricci curvature, χtop (Mc ) = 3, and simple connectedness
of the frame factor — is X = CP 2 × S 3 , up to
diffeomorphism. The axiom set is satisfiable: that
manifold is an explicit model of it.
Derived, not assumed: dim X = 7 and the split
7 = 4 + 3; π3 (Mg ) = Z; the Spinc structure; the even
mapping-torus dimension behind Strong-CP; and
kmax = 60, through the conditional determinant-line
count of App. F given the generation count, the flux
quantum and the five multiplet types.
Implication: The integers
{3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137} appearing
throughout the Standard Model and cosmology are not
free: they are fixed by this unique manifold together with
the stated inputs and identifications of App. FK.
Falsifier: Any exhibited compact manifold
X ′ ̸= CP 2 × S 3 satisfying the same seven axioms; or any
observation of a fourth generation (violating (V5)).
Consequence: Combined with the α57 dimensionless
constraint (Appendix O), DFD’s scale sector has one
measured continuous coefficient — the clock
normalization C of the early clock in (H0 tP )2 = C α57 ,
equivalently λ = C −1/2 of Postulate ZS.11 (C = 1 on
bound-matter clocks), proven underivable from the
printed corpus by the eight-route no-go (Rem. AP.6); the
continuous calibrations of the cosmological prediction set
are graded in App. FK (Rem. FK.8). The internal
geometry is fixed up to the discrete selections booked in
Remark AB.7 and priced in the trials-factor audit
(Sec. CE 2).

Appendix AC: Linear Growth Closure from the
EFE-Screened DFD Operator
1.

AC.1

Purpose

This appendix derives the first-order linear response
operator of DFD’s spatial sector. The temporal dust
branch (Appendix Q) proves that the scalar completion
has w → 0 and c2s → 0 for homogeneous deviations
only. Here we prove the spatial perturbation operator
and evaluate it at the external field of the Hubble-EFE
hypothesis.
Cross-reference: cosmological scope of the HubbleEFE Hypothesis
The linear response operator derived below is correct
mathematics for any background gradient |∇ψ̄|. The
numerical 1.02–1.17 G envelope below depends on the
named Hubble-EFE Hypothesis aext (z) = cH(z).
Appendix AE establishes that the FRW background of
the DFD action gives |∇ψ̄| = 0, so the Hubble-EFE
Hypothesis is not what comes out of the action for
cosmological perturbations. On galactic and cluster
scales the external field is set by local matter, and the
response operator is unchanged.

2.

AC.2

Linearization of the Nonlinear Spatial
Operator

The quasi-static DFD field equation is
8πG
|∇ψ|
∇ · [µ(x)∇ψ] = − 2 ρ,
x=
.
c
a⋆
Let
ψ = ψ̄ + φ,

(AC1)

ρ = ρ̄(1 + δ),

(AC2)

and define the background quantities
ḡi
|ḡ|
.
ḡi = ∂i ψ̄,
,
ĝi =
x̄ =
a⋆
|ḡ|
The flux is

(AC3)

Fi (∇ψ) = µ(x) ∂i ψ.

(AC4)

Lemma AC.1 (First variation of the flux). The first
variation of Fi around the background is
δFi = Mij ∂j φ,

(AC5)

Mij = µ0 δij + µln,0 ĝi ĝj ,

(AC6)

with

where
µ0 = µ(x̄),

µln,0 =

dµ
= x̄ µ′ (x̄).
d ln x x̄

(AC7)

Proof. Direct chain-rule computation. Fi = µ(x)∂i ψ.
Vary: δFi = µ(x)∂i φ + µ′ (x) δx ∂i ψ̄, with δx =
(ĝj /a⋆ )∂j φ. Substituting and rearranging gives δFi =
µ0 δij ∂j φ + (x̄µ′ (x̄))ĝi ĝj ∂j φ, which is the stated form.

245
Remark AC.2 (Notation). The response coefficient appearing in the linear operator is µln,0 = x̄µ′ (x̄) = x̄/(1 + x̄)2
for µ(x) = x/(1 + x), i.e. dµ/d ln x; it is not dµ/dx =
1/(1 + x̄)2 . This appendix uses µln,0 throughout.

Therefore the external-field argument is

In Fourier space, the linearized field equation becomes
8πG
ki Mij kj φk = 2 ρ̄ δk .
(AC8)
c
Matter acceleration is aφ = (c2 /2)∇φ. The growth equation is therefore

This is independent of redshift if the same epochconsistency rule that fixes a0 (z) is used.
At x̄ = 5.85,

δ̈k + 2H δ̇k = 4πGeff (x̄, θ) ρ̄ δk ,

(AC9)

where
Geff (x̄, θ) =

G
,
µ0 + µln,0 cos2 θ

(AC10)

and cos θ = k̂ · ĝ.

x̄EFE =

1
aext (z)
= √ ≃ 5.85.
a0 (z)
2 α

µ0 = 0.854,

µln,0 = 0.125.

(AC17)

(AC18)

Thus the directional envelope is
G
G
G⊥ =
= 1.171 G,
G∥ =
= 1.021 G.
µ0
µ0 + µln,0
(AC19)
The angular mean is
√
G arctan q
1
⟨Geff ⟩ =
,
q=
= 0.146, (AC20)
√
µ0
q
1 + x̄
giving

3.

AC.3

For µ(x) = x/(1 + x) (Theorem N.8), one has
x̄
x̄
,
µln,0 =
.
(AC11)
µ0 =
1 + x̄
(1 + x̄)2
Hence

−1
x̄
x̄
2
(AC12)
GDFD
=
G
+
cos
θ
,
eff
1 + x̄ (1 + x̄)2
or equivalently
G
1
GDFD
=
,
eff
µ0 1 + q cos2 θ

4.

AC.4

⟨Geff ⟩ ≃ 1.12 G.

Specialization to the DFD Interpolation
Function

q=

µln,0
1
.
=
µ0
1 + x̄
(AC13)

Cosmological External-Field Screening

The cosmological external field supplied by the Hubble
flow is
aext (z) ∼ cH(z).

(AC14)

This is the Hubble-EFE hypothesis, named explicitly so
it can be tested independently.
Hubble-EFE Hypothesis (named for testability)
The cosmological external acceleration relevant to the
EFE screening of linear cosmological perturbations is
aext (z) = cH(z).

(AC15)

This hypothesis is named explicitly so that it can be
tested independently. The 1.02–1.17G growth envelope
follows from this hypothesis
√ combined with the
epoch-consistent a0 (z) = 2 α cH(z) relation.

The epoch-consistent DFD crossover scale is
√
a0 (z) = 2 α cH(z).

(AC16)

5.

AC.5

(AC21)

First Numerical Growth Estimates

We integrate the standard linear growth equation


d ln H
3
D′′ + 2 +
D′ − Q Ωm (a) D = 0, (AC22)
d ln a
2
with Q = Geff /G, for Ωm,0 = 0.30 and the three Q values
from the EFE-screened envelope.
TABLE CVIII. Growth rates under the Hubble-EFE hypothesis, which is not DFD’s cosmological growth law (App. GR).
These are theorem-to-observable estimates; production-level
survey-pipeline likelihoods require the standard nuisance layer.
z

fΛCDM

f∥

favg

f⊥

0.38
0.61
1.50

0.7049
0.7837
0.9268

0.7139
0.7936
0.9384

0.7529
0.8368
0.9892

0.7738
0.8599
1.0164

Using survey baseline βtheory values to infer effective
biases beff (z) and propagating through the directional
envelope, the predicted RSD parameters lie in a narrow
band of width ∼ 8% around ΛCDM. These numbers
are first-pass theorem-to-observable estimates and should
not be presented as final BOSS/eBOSS likelihood values;
the full survey confrontation requires the nuisance layer
(galaxy bias, FoG, AP, mock calibration) already used in
ΛCDM analyses.
6.

AC.6

Interpretation

The DFD dust branch supplies the pressureless temporal sector of homogeneous deviations (Appendix Q),
which does not cluster:
w → 0,

c2s → 0.

(AC23)

246
Linear cosmological growth is carried by the cold χ relic
with Q = 1 (App. GR).

Appendix AD: Regular-Module Forcing from Finite
Real-Bimodule Consistency
1.

7.

AC.7

AD.1

Purpose

Falsifier

Linear Growth Closure — Falsifier
Under the Hubble-EFE hypothesis, the first-order growth
closure is falsified if quasi-linear RSD or weak-lensing
growth data require an effective coupling outside the
EFE-screened envelope
1.02 ≲ Geff /G ≲ 1.17

(AC24)

after the standard nuisance layer of galaxy bias,
Finger-of-God damping, Alcock–Paczynski mapping, and
fiducial-cosmology dictionary translation is applied.
A second, more distinctive falsifier is absence of the
predicted directional dependence in f σ8 (z, n̂) where the
reconstructed ψ-screen gradient is nonzero.
A third falsifier is rejection of the Hubble-EFE
Hypothesis: if aext (z) ̸= cH(z) at the level needed to
displace x̄EFE outside its DFD-predicted range, the
closure must be revisited.

The microsector closed form for α−1 = 137.036 contains
a forced binary fork: the finite spectral trace may be carried either by the regular module HF = A = Md (C) or by
a d-dimensional fermion-representation module HF ≃ Cd .
The numerical comparison (Table XXVI) selects the regular module by 43 ppm. This appendix proves that the
regular module is also the minimal finite Hilbert space
compatible with the DFD microsector’s real-bimodule
structure, making the lock a structural theorem conditional on the standard-form real-bimodule conditions and
minimality, which are adopted as the spectral-triple consistency requirement, not derived from the field equation.
2.

AD.2

The Finite Algebra and Its Role

Let
A = Md (C).

8.

AC.8

Summary

Linear Growth Closure — Theorem-Grade Status
Closed first-order:
• Linear response operator Mij = µ0 δij + µln,0 ĝi ĝj
derived (Lemma AC.1)
• Effective coupling Geff (x̄, θ) = G/[µ0 + µln,0 cos2 θ]
• Under the Hubble-EFE
hypothesis, EFE screening
√
fixes x̄EFE = 1/(2 α) ≃ 5.85
• Quasi-Newtonian envelope: 1.02 G ≤ Geff ≤ 1.17 G
• Angular mean: ⟨Geff ⟩ ≃ 1.12 G
Still program-grade: Production survey-pipeline
likelihood (BOSS/eBOSS/DESI full-shape analysis with
the standard nuisance layer) requires CAMB-equivalent
modification with the DFD growth equation. This is a
numerical implementation task, not a theoretical hole.

(AD1)

The DFD microsector is not the observed fermionrepresentation space. It is the finite carrier of the vacuum
gauge-frame algebra whose spectral trace normalizes the
emergent gauge kinetic term. Gauge variations are represented by inner derivations,
ada (X) = [a, X],
3.

AD.3

a ∈ A, X ∈ HF .

(AD2)

Minimal Faithful Real-Bimodule Theorem

Theorem AD.1 (Minimal faithful real-bimodule). Let
HF be a finite-dimensional Hilbert space carrying a faithful left action πL : A → B(HF ) and a faithful right
(opposite) action πR : Aop → B(HF ) satisfying
1. Order-zero condition: [πL (a), πR (b)] = 0 for all
a, b ∈ A.
2. Mutual commutant closure (finite-factor standard form):
πL (A)′ = πR (Aop ),

πR (Aop )′ = πL (A).

(AD3)

Then the minimal such Hilbert space is
HF ≃ Cd ⊗ Cd ≃ Md (C),

(AD4)

dim HF = d2 .

(AD5)

and therefore

Proof. Every finite-dimensional faithful representation of
the simple algebra Md (C) is a multiple of the defining
representation. Hence
HF ≃ Cd ⊗ Cm ,

πL (a) = a ⊗ Im .

(AD6)

247
Its commutant is
πL (A)′ = Id ⊗ Mm (C).

(AD7)

op

For πR (A ) to be faithful and equal to the full commutant, the multiplicity space must contain a full copy of
Md (C)op . This requires
m ≥ d.

(AD8)

Minimality gives m = d. Therefore
HF ≃ Cd ⊗ Cd ≃ End(Cd ) ≃ Md (C).

(AD9)

Under this identification, A acts by left multiplication
and Aop acts by right multiplication. The adjoint gauge
action is
ada (X) = aX − Xa = [a, X],

AD.4

Corollary AD.2 (Branch B is structurally invalid). The
fermion-representation branch takes
(AD11)

The commutant of the defining representation of Md (C)
is only
πL (Md )′ = C.

(AD12)

Therefore HF ≃ C cannot carry a faithful right Md (C)
action satisfying the standard-form real-bimodule condition. Branch B is therefore not a valid DFD vacuum
microsector trace carrier.
Remark AD.3 (Scope of the exclusion). Corollary AD.2
excludes HF ≃ Cd only as the DFD vacuum gauge-frame
trace carrier. It does not imply that Cd is invalid for
ordinary fermion representations elsewhere in the framework. Physical fermion fields transform in the defining
representation; the vacuum trace carrier is a separate
structure responsible for gauge-frame normalization.
5.

AD.5

Finite-d Consequence

For the DFD Toeplitz cutoff kmax = 60 (Lemma F.11),
d = kmax + 4 = 64.

(AD13)

dim HF = d2 = 4096.

(AD14)

Hence

Corollary AD.4 (Forced trace conversion factor). The
democratic trace over HF = Md (C) and conversion to
canonical su(d) generator normalization gives
(A)

εadj =

d2
d2 − 1

=

4096
.
4095

AD.6

Status

Branch A (HF = Md (C), BOOST = 4096/4095) is forced
by the minimal faithful real-bimodule theorem
(Theorem AD.1). Branch B (HF ≃ Cd , drop
= 4095/4096) fails the bimodule consistency requirement
(Corollary AD.2); its 43 ppm miss of α−1 is a redundant
consistency check rather than the basis for selection.
The α closed form is structurally locked at the
module level:
• Bimodule structure: HF = Md (C) (forced minimal
faithful real-bimodule)
• Dimension: d2 = 4096
(A)

op

d

6.

Microsector Lock — Theorem-Grade Status

Branch B Exclusion

HF ≃ Cd .

Proof. The regular module Md (C) decomposes as
Md (C) = C · 1 ⊕ su(d) as a representation of the inner
automorphism group, where the singlet corresponds to
the identity matrix and su(d) is the (d2 − 1)-dimensional
traceless adjoint. The democratic UV trace runs over all
d2 basis elements; the gauge kinetic term in canonical normalization runs only over the su(d) factor. The conversion
is the inverse ratio of dimensions, d2 /(d2 − 1).

(AD10)

which is exactly the DFD regular-module microsector.
4.

The alternative factor (d2 − 1)/d2 = 4095/4096 belongs to
a module that fails the DFD finite real-bimodule condition.

(AD15)

This is the regular-module factor used in the successful
branch of the closed form for the fine-structure constant.

• Trace conversion: εadj = 4096/4095
(Corollary AD.4)
• Gauge action: inner derivations ada (X) = [a, X]
• α−1 = 137.03599985 (21 ppm from the integers;
the bracket coefficient is open; the identification of
the hypercharge-trace coefficient with 1/e2 is not
derived)
Theorem dependencies: order-zero + faithful
left/right actions + mutual commutant closure +
minimality. These constitute a legitimate spectral-triple
consistency requirement for the DFD vacuum trace
carrier.
Falsifier: If a finite Hilbert space HF ≃ Cd can be
shown to satisfy the faithful real-bimodule conditions
(faithful left and right actions of Md (C) and Md (C)op
with mutual commutant closure), then Theorem AD.1 is
wrong. No such construction exists in the current
literature; the commutant of the defining representation
is provably C by Schur’s lemma.

248
Appendix AE: Optical-Metric σ8 Closure
1.

AE.1

Statement and Scope

In a curvature-based gravity theory such as GR, the
lensing-inferred density contrast on R8 scales is the same
object as the physical mass density contrast. This is not
the case in DFD. Because DFD’s gravity is refractive
(n = eψ ) and not curvature-based, the relation between
physical baryon clustering and observed weak-lensing convergence is mediated by the deep-MOND optical response,
not by a linear Newtonian-Poisson inversion.
This appendix derives the closure condition for the
spherical top-hat σ8 statistic and computes the baryon
contrast that, amplified optically, would produce the
Planck 2018 amplitude σ8,obs = 0.811.
The closure is an optical, apparent-amplitude construction in which the baryons are the only clustering matter;
it is not DFD’s physical growth carrier. The physical
carrier is the cold χ relic with Qχ = 1 under the adopted
Rest-Mass Channel, which also confines the baryons’ polarization to bound low-acceleration systems, and the σ8
and S8 of record are its values on the measured-clock
background, σ8 ≃ 0.8063 and S8 ≃ 0.822 (App. GR).
The numbers this appendix derives from the closure (the

baryon contrast, QDFD , the peculiar velocity, the bias and
the RSD β) and its falsifiers belong to that construction.
Theorem AE.1 does not depend on it; Corollary AE.2
describes the bare single-W action, which the Rest-Mass
Channel supersedes on linear scales (App. GR).
Relationship to Appendix AC
The linear response operator Mij = µ0 δij + µln,0 ĝi ĝj of
Appendix AC is correct mathematics for any background
gradient |∇ψ̄|. Appendix AC then adopts the
Hubble-EFE Hypothesis aext (z) = cH(z) (named
explicitly there as testable) to evaluate the operator at
x̄EFE ≃ 5.85, giving the 1.02–1.17 G envelope.
The present appendix establishes (Theorem AE.1) that
the FRW background of the DFD action gives |∇ψ̄| = 0.
The Hubble-EFE Hypothesis is therefore not recovered
from the action for cosmological perturbations. For
cosmological scales, the response operator is evaluated at
x̄ = 0 (deep MOND, no EFE), and in the baryon-only
construction the apparent σ8 is dominated by the optical
lensing inflation rather than the matter-acceleration
enhancement.

2.

AE.2

First-Principles Derivation: aFRW
=0
ext

Theorem AE.1 (FRW-derived cosmological external field). On a homogeneous and isotropic FRW background of the
DFD action
Z
o
n a2 h



 i c2
0
Sψ = dt d3 x
W |∇ψ|2 /a2∗ + K ac0 |ψ̇ − ψ̇0 | − ψ(ρ − ρ̄) ,
8πG
2
the spatial gradient of the background scalar field vanishes:
|∇ψ̄| = 0,

aFRW
≡
ext

c2
|∇ψ̄| = 0.
2

(AE1)

Proof. The proof has three steps.
1. Matter source vanishes by homogeneity. The matter coupling is ψ(ρ − ρ̄). On a strictly homogeneous FRW
background, ρ = ρ̄ everywhere, so the source term is identically zero.
2. Spatial kinetic vanishes by isotropy. A homogeneous and isotropic background has ψ̄ = ψ̄(t) only, hence
∇ψ̄ = 0 pointwise. The argument of W is then zero, and using the Lagrangian normalization W (0) = 0, the
spatial kinetic vanishes.
3. Temporal kinetic vanishes by reference invariance. The temporal deviation invariant is ∆ = (c/a0 )|ψ̇ − ψ̇0 |
where ψ̇0 is the screen-background flow. On the FRW background treated as the reference, ψ̇ = ψ̇0 identically, so
∆ = 0 and K(∆) = 0 by the normalization K(0) = 0.
The Euler–Lagrange equation for ψ̄(t) is therefore identically satisfied for any ψ̄(t): the FRW background is a flat
direction in the action. In particular, |∇ψ̄| = 0 identically, and the acceleration form aFRW
= (c2 /2)|∇ψ̄| = 0.
ext

Corollary AE.2 (Cosmological perturbations are in deep
MOND). For perturbations on the FRW background with
no external matter to source a gradient, the response operator Mij = µ0 δij + µln,0 ĝi ĝj of Appendix AC is evaluated

at x̄ = 0, placing cosmological perturbations in the raw
deep-MOND regime µ(x) → x as x → 0.

249
3.

AE.3

Setup: spherical top-hat statistic

The observable σ8 is defined as the rms density contrast
filtered with a spherical top-hat window of comoving
radius R8 = 8 h−1 Mpc:
(
3/(4πR3 ) |x| ≤ R
2
2
σ8 ≡ ⟨δW ⟩,
WR (x) =
. (AE2)
0
|x| > R
For a spherical baryon overdensity of contrast δb filling a
top-hat of radius R8 , the Newtonian gravitational acceleration at the edge is
4πG
gN (R8 ) =
ρ̄b,0 δb R8 = 12 Ωb H02 R8 δb ,
(AE3)
3
using ρ̄b,0 = Ωb ρcrit = Ωb (3H02 /8πG).
Throughout this appendix, δb is the rms baryon density
contrast at R8 (the spherical top-hat filtered rms).
4.

AE.4

In a curvature-based theory, weak-lensing convergence
on R8 scales is proportional to the integrated source
Ωm δ m :

−→

Ωb RDFD [δb ] δb .

(AE6)

In the spherical top-hat convention, the response operator is a scalar function Q(δb ): the angle-averaged
matter response of an isolated spherical overdensity in
deep MOND with aFRW
= 0 (Theorem AE.1 and Corolext
lary AE.2). The closure condition for matched lensing
observation is
Ωb δb Q(δb ) = Ωm σ8,obs .

5.

AE.5

Closure number in the spherical top-hat
convention

√
Combining Eqs. (AE3) and a0 = 2 α c H0 :
Ωb H 0 R 8
gN (R8 )
= A8 δb ,
A8 ≡ √
.
a0
4 α c

(AE12)

A8 = 3.81546 × 10−4 .
Solving Eq.
[Eq. (AE11)]:

(AE7)
Ωb

p

in

the

(AE13)

deep-MOND

limit

δb /A8 = Ωm σ8,obs ,

(AE14)

(Ωm σ8,obs )2 A8
.
Ω2b

(AE15)

δb =

With Ωm = 0.315 and σ8,obs = 0.811 (Ωm σ8 = 0.2555):
1
δb,R8 = 1.0432 × 10−2 ,
QDFD = √
= 501.24.
A8 δ b
(AE16)
The exact-µ result from Eq. (AE10) differs by a fractional
√
correction ∼ q/2 ∼ 10−3 :
δb,R8 = 1.0411 × 10−2 ,

7.

AE.7

QDFD = 502.24.

(AE17)

Physical interpretation

(AE7)

The DFD response Q(δb )

For an isolated spherical overdensity in the no-EFE
FRW background, Eq. (AE5) integrates by Gauss’s theorem on a sphere to
 |∇ψ|  c2
µ
· |∇ψ| = gN (r),
(AE8)
a∗
2
which in matter-acceleration form a = (c2 /2)|∇ψ| becomes
µ(a/a0 ) a = gN (r).

AE.6

(AE4)

In DFD, the convergence is set by the same line-of-sight
integral but with δm replaced by the refractive-source
field RDFD [δb ] δb , where RDFD is the nonlinear response
operator obtained by solving


8πG
∇· µ(|∇ψ|/a∗ ) ∇ψ = − 2 (ρb − ρ̄b ),
(AE5)
c
with µ(x) = x/(1 + x) and a∗ = 2a0 /c2 . Schematically,
Ωm δ m

6.

Since H0 R8 /c = 8/2997.92 = 2.6685 × 10−3 (independent
of h, since R8 ∝ h−1 cancels), and using the current-clock
Ωb = ωb /h2 = 0.048856, α−1 = 137.036:

The closure equation

κGR ∝ Ωm δm .

With µ(x) = x/(1 + x) and y ≡ a/a0 , q ≡ gN /a0 ,
Eq. (AE9) reads
p
y2
q + q 2 + 4q
y
= q,
y=
,
Q(δb ) = .
1+y
2
q
(AE10)
In the deep-MOND limit q ≪ 1, this reduces to
1
1
.
(AE11)
Q(δb ) ≃ √ = p
q
gN /a0

(AE9)

In the baryon-only construction, the Planck 2018 amplitude σ8 ≃ 0.81 would correspond to a baryon density
contrast of only ∼ 1.0% at R8 , amplified optically by a
factor ∼ 500 via the deep-MOND lensing response.
Apportionment with χ-matter (two channels, different
observables). This appendix quantifies the R8 -scale optical
(ψ) lensing response, where µ(x) → 0 amplifies a small
baryon contrast. The derived χ-matter field (Ωχ h2 ≃
0.12; App. AV) carries the cosmological cold clustering
(CMB peak heights, large-scale matter power) through
the rest-mass channel (Qχ = 1; App. GR), while the
optical response is sourced by the baryons. The Rest-Mass
Channel confines that response to bound low-acceleration
systems, and the net induced mass of each bound system
is zero, so on the R8 scale the optical response adds

250
at most 0.2% of the mean matter density and raises
the lensing power by at most 7% (computed; App. GR,
Remark GR.12). The lensing statistic at R8 therefore
follows the matter power that χ and the baryons carry;
the closure number of Eq. (AE17) belongs to the baryononly construction. How the two channels compose inside
galaxies, where χ halos and the baryons’ phantom overlap,
is a separate and open question (App. AV).
The construction is consistent with three independent
corpus results:
1. The corpus particle-mesh simulation in Sec. XVI
reports 5.4× overshoot of δrms relative to ΛCDM, attributed to “cosmological perturbations in the deepMOND regime” with x ∼ 4 × 10−4 . The present
derivation identifies the same regime gN /a0 = q ≃
4 × 10−6 at the closure point.
2. The
√ deep-MOND deflection formula α̂deep =
2π GM a0 /c2 (Sec. VII) gives the same closure
when used to match GR’s deflection from the LCDM
mass at R8 , up to spherical-vs-Fourier convention
differences absorbed by the spherical σ8 window.
3. The first-principles result aFRW
= 0 (Theoext
rem AE.1): the FRW background of the DFD action
gives |∇ψ̄| = 0. The Appendix AC envelope 1.02–
1.17 G applies under the named Hubble-EFE Hypothesis rather than as the actual DFD prediction
for the cosmological regime.

8.

AE.8

Required CMB-to-z = 0 amplification

The baryon-only construction requires a dynamical
amplification of δb from recombination (δinit ∼ 10−5 ) to
z = 0 (δb,R8 ≃ 0.010) of
δb,R8
∼ 1.0 × 103 .
(AE18)
δinit
ΛCDM requires amplification ∼ 105 via cold dark matter
from CMB to z = 0. In DFD that physical clustering is
carried by the derived χ-matter (Ωχ h2 ≃ 0.12, ΛCDMlike growth; App. AV), so the same ∼ 105 is achieved by
χ. The optical factor Q ≃ 500 computed here belongs
to the baryon-only construction. In DFD’s cosmology
the baryons polarize the medium only inside bound lowacceleration systems (App. GR), so this factor does not
act on the R8 scale and nothing is stacked on the χ
clustering: χ and the baryons carry the matter power and
its lensing, and the polarization adds the lensing of each
galaxy inside its bound radius.

9.

AE.9

Relation to galaxy clustering

δb ≃ 0.010, the implied galaxy bias factor relative to
baryons is
bgalaxy ≡ δgalaxy /δb ∼ 1/0.0104 ∼ 96.

(AE19)

This is much larger than typical ΛCDM galaxy bias factors
b ∼ 1.5–2.
Within the construction this is not unreasonable:
galaxy formation is preferentially seeded by peaks in the
effective refractive source RDFD [δb ]δb rather than in the
bare baryon contrast δb . The ratio RDFD /1 ∼ Q ∼ 500
provides the natural rescaling between bare baryon clustering and the gravitationally-active source that controls
galaxy formation. Galaxies trace the refractive-source
peaks at order-unity contrast even when δb itself is small.
A rigorous derivation of galaxy bias in DFD’s framework
is identified as a program-level item; see Sec. AE 12.
10.

AE.9b

RSD compatibility: deep-MOND
velocity divergence

A potential concern is that the bias multiplier F8 =
σ8 /δb ≃ 77.9 would naively imply βRSD = f /b smaller by
the same factor, in conflict with measured BOSS/eBOSS
values β ∼ 0.27–0.40 at LRG bias. We show here that this
is a unit artifact: the appropriate RSD observable is the
velocity divergence θ = ∇ · v, not the radial velocity amplitude vr , and the proper deep-MOND velocity-density
relation gives βDFD within the observed range.
a. Velocity from DFD’s exponential refractive structure. DFD’s matter equation of motion is a = (c2 /2)∇ψ
directly, not a curvature-induced geodesic. For a spherical baryon overdensity of contrast δb at radius R, the
deep-MOND DFD field equation integrated by Gauss’s
theorem gives |∇ψ|2 /a∗ = (2G/c2 )δM/R2 . Combining
with a = (c2 /2)|∇ψ| and a∗ = 2a0 /c2 :
√
G a0 δM
.
(AE20)
aDFD =
R
Substituting δM = (4π/3)R3 ρb δb with ρb =
Ωb (3H 2 /8πG) gives gN (R) = (1/2)Ωb H 2 Rδb , and the
same factor of 1/2 that defines the spherical-top-hat σ8
closure [Eq. (AE3)] enters here. The deep-MOND identity
√
aDFD = a0 gN (corpus Appendix B; consistent with the
asymptotic-velocity formula Vc4 = GMbar a0 ) gives
q
aDFD = H 12 a0 R Ωb δb ,
(AE21)
√
sub-linear in δb (a δb scaling), reflecting the deep-MOND
amplification when matter is sparse.
The peculiar velocity from quasi-static Euler integration
a/H is
q
vpec = 12 a0 R Ωb δb .
(AE22)
At the closure point δb = 1.04 × 10−2 on R8 :

A consistency concern is that observed galaxy clustering
on R8 scales gives rms galaxy contrast of order unity. If

vpec ≃ 102 km/s,

(AE23)

about 0.9 of the LCDM linear top-hat value f H0 R8 σ8 /3 ≃
108 km/s at the same scale. Equivalently, in terms of the

251
response factor Q of Eq. (AE17):
vpec
= 12 Ωb Q δb = 0.128,
(AE24)
H0 R8
self-consistent with the same 1/2 factor used in the spherical top-hat σ8 normalization.
b. The RSD observable is velocity divergence. Kaiser
RSD measures redshift-space anisotropy controlled by the
velocity divergence θ = ∇ · v, not vr itself. For a spherical
top-hat radial flow vr (r) = (vR /R)r, exactly:
1 d
3vR
θ = 2 (r2 vr ) =
.
(AE25)
r dr
R
The factor of three is geometric and unavoidable for a
spherical top-hat profile. The implied RSD β parameter
is

precision-level comparison must wait for the full kernel
calculation.
e. Falsifier sharpening. If the full Pθg /Pgg calculation gives βDFD outside the observed BOSS/eBOSS range
at significance > 3σ for at least two redshift bins, the
optical σ8 closure is falsified. Conversely, the sphericaltop-hat estimate predicts βDFD ≃ 0.22–0.25 at LRG bias,
≃ 0.31 at ELG bias, and ≃ 0.18 at QSO bias — for every
tracer 0.94 of LCDM’s β = f /b at f ≃ 0.5 (0.27, 0.33 and
0.19 at b = 1.88, 1.50 and 2.70).

11.

AE.10

Falsifiable predictions

Optical σ8 Closure — Falsifiers
θ/(aH)
3 vpec
sph
βDFD
≃
=
.
δgal
H0 R8 δgal

(AE26)

Equation (AE17) is testable through three independent
channels:

c. Numerical evaluation. With H0 R8 = 800 km/s
(independent of h since R8 = 8h−1 Mpc cancels), vpec =
102 km/s, and an LRG-like sample bGR = 1.88 giving
δgal = bGR σ8,obs = 1.525:

1. Direct baryon-density probes. Lyman-α forest
absorption at z ∼ 2–5 measures the IGM baryon
density contrast directly. The construction predicts
the bare baryon δb on R8 -comoving scales at the
percent level, much smaller than the LCDM matter
δm ∼ σ8 (z) at the same scale. A measured δb at
R8 -comoving consistent with LCDM δm falsifies
the construction.

3 × 102/800
= 0.251.
(AE27)
1.525
This sits just below the BOSS/eBOSS-scale measurement
range β ∼ 0.27–0.40 at LRG bias, but is consistent with
it for ELG-like and lower-bias samples.
A representative table for several survey tracers:
sph
βDFD
≃

Tracer

sph
bGR δgal = bGR σ8 βDFD

Matter-equivalent 1.00
BGS / low-bias
1.20
ELG-like
1.50
DESI/BOSS LRG 1.88
LRG high-bias
2.10
QSO-like
2.70

0.811
0.973
1.216
1.525
1.703
2.190

0.472
0.394
0.315
0.251
0.225
0.175

d. Status. This is a first-order spherical-top-hat estimate. A definitive RSD prediction requires computing Pθg (k, z) and Pgg (k, z) from the DFD Euler equation with the proper mode-by-mode geometry, projecting
through survey kernels matching BOSS/eBOSS/DESI
windows, and including nonlinear corrections (Finger-ofGod, Alcock–Paczynski, fiducial-cosmology dictionary).
This is identified as an additional Open Theorem Obligation in Sec. AE 12. The first-order estimate above is a
sph
near-miss sanity check : βDFD
≃ 0.25 for LRG-like tracers
sits just below the observed range β ∼ 0.27–0.40, while
sph
βDFD
≃ 0.32 for ELG-like tracers is comfortably inside it.
The optical σ8 closure is therefore not in obvious orderof-magnitude conflict with existing RSD data, but the

Z ∞
κDFD (ℓ) =

dχ
0

2. kSZ amplitude. The kinetic Sunyaev–Zel’dovich
signal probes peculiar velocities of hot baryons.
From the deep-MOND
√ velocity-density relation
Eq. (AE22), vpec ∝ δb with the spherical top-hat
normalization, giving vpec ≃ 102 km/s on R8 scales
at the closure point. This is about 0.9 of the
LCDM linear-theory prediction at the same scale.
Current ACT/Planck and DESI×ACT 13σ kSZ
detections are consistent with both amplitudes
within current uncertainties; high-precision
amplitude measurements at R8 -equivalent scales
can discriminate.
3. Galaxy bias coherent across surveys. If the
galaxy-bias explanation in Sec. AE 9 is correct, the
bias b ∼ 96 relative to baryons (not relative to
Ωm δm ) should be internally consistent across
galaxy clustering, weak lensing, and CMB lensing
cross-correlations. Inconsistency at the 10% level
would falsify the construction.

12.

AE.11

Open Theorem Obligation: full
convergence kernel

Equation (AE17) is established in the spherical top-hat
convention that defines σ8 as a single statistic. The full
weak-lensing convergence kernel in DFD,



3H02 χ g(χ)
ℓ + 1/2
Ω
R
k
=
,
z;
δ
δb (k, z),
b
DFD
b
2c2 a(χ)
χ

(AE28)

252
with the corresponding power spectrum
Cℓκκ,DFD =

Z ∞
0



2

dχ 3H02 χg(χ)
ℓ + 1/2
Ωb PRδb ,Rδb
,z ,
χ2 2c2 a(χ)
χ

remains program-level. The Open Theorem Obligations
are:
1. Compute PRδb ,Rδb (k, z) from a fiducial baryon
transfer function, propagated through the nonlinear
RDFD response operator.
2. Project Eq. (AE29) through survey kernels matching KiDS-Legacy / DES-Y6 / HSC-Y3 weak-lensing
tomography.
p
3. Verify S8DFD = σ8,DFD Ωm /0.3 matches observation S8 ≃ 0.78–0.83 at the precision of current measurements.
4. Derive the galaxy bias relation bgalaxy (δb , z, RDFD )
from the refractive-source peak statistics, and verify
cross-consistency between galaxy clustering, weak
lensing, and galaxy-galaxy lensing. Resolved at the
linear-bias level on R8 scales: see Appendix AF
(Theorem AF.1). The bias multiplier F8 = 77.9 is
derived from the closure with no free parameters;
mass-dependent halo bias b(M ) and assembly bias
remain program-grade.
5. Compute Pθg (k, z) and Pgg (k, z) from the DFD
Euler equation in the deep-MOND regime
and project through survey kernels matching
BOSS/eBOSS/DESI-Y3 redshift-space distortion
analyses. Verify that the predicted β(z, b) is consph
sistent with the spherical-top-hat estimate βDFD
≃
0.22–0.25 at LRG bias and ≃ 0.31 at ELG bias
(Sec. AE 10), and reconciles with the observed
β ∼ 0.27–0.40 range.
The spherical top-hat result in Eq. (AE17) provides a
single-statistic anchor against which the full kernel computation must be consistent.

13.

AE.12

(AE29)

Summary

Optical σ8 Closure (baryon-only construction) —
Theorem-Grade Status
Closed at theorem grade:
• aFRW
= 0 from the DFD action (Theorem AE.1).
ext
• Closure equation Ωb δb Q(δb ) = Ωm σ8,obs in the
spherical top-hat convention.
• Spherical-top-hat closure number:
δb,R8 = 1.0411 × 10−2 , QDFD = 502.24, with no
free parameters.
• Deep-MOND
velocity-density relation
p
vpec = (1/2) a0 R Ωb δb from a = (c2 /2)∇ψ and
the field-equation Gauss law, vpec ≃ 102 km/s at
the closure point on R8 .
• First-order RSD sanity check (a near miss):
sph
βDFD
≃ 0.25 at LRG bias (just below observed
0.27–0.40) and ≃ 0.32 at ELG bias (inside observed
range), Sec. AE 10.
Still program-grade:
• Full Limber-projected convergence-kernel Cℓκκ and
survey-window S8 matching.
• Full Pθg /Pgg RSD prediction across
BOSS/eBOSS/DESI tomographic bins.
• Mass-dependent halo bias b(M ) and assembly bias
in DFD (linear bias closed in Appendix AF).
• Direct Lyman-α and kSZ confrontation at the
predicted δb ∼ 0.010.
Relationship to Appendix AC: the linear response
operator structure of Appendix AC is unchanged. For
cosmological perturbations the correct value is aFRW
=0
ext
(Theorem AE.1), placing the same response operator in
the deep-MOND limit. The two appendices are
complementary, not in conflict.

253
1. (Cosmological linear regime.) On the homogeneous FRW background, aFRW
= 0 identically
ext
(Theorem AE.1), so x̄ = 0 and the ψ-sector is in
the raw deep-MOND limit. For a linear top-hat perturbation on R8 , x = gN /a0 (z) ≪ 1 at every epoch
(≲ 7.3 × 10−3 for δ ≤ 1 at z ≤ 2; 1.3 × 10−6 at
recombination; Lemma GR.3). Linear scales are
deep-MOND, never x̄ = 5.85.

Appendix GR: Adjudication of the Cosmological
Growth/Velocity Regime

Status GR.1 (Measured-clock values). All H0 = 72.09,
Ωm ≃ 0.274, σ8 ≃ 0.820, and S8 ≃ 0.784 calculations
below belong to the C = 1 background and are not current
outputs. There σ8 = 0.820 is the carried normalization,
the value that the amplitude As = 2.08 × 10−9 (the
value of 32πα5 , a calibration; App. FK) gives on that
background; on the measured clock the same amplitude
and χ carrier give the σ8 = 0.806 of record (GR.4). A
dependency-aware high-accuracy normal-hierarchy rerun
on the current frozen inputs gives R8 = 11.848 Mpc,
Ωm ≃ 0.312, σ8 ≃ 0.8063, and S8 ≃ 0.822; KiDS-Legacy
is consistent at 0.45–0.74σ depending on the primary/joint
anchor, and current CMB lensing gives Aϕϕ = 0.9887
(−0.80σ). The structural result that Qχ = 1 is conditional
on the adopted Rest-Mass Channel postulate holds on
either background. The alternative 0.790/0.755 branch is
not a result.

1.

GR.1

Purpose and the contested scalar

Appendices AC (AC) and AE (AE) share the identical
linear response operator Mij = √
µ0 δij + µln,0 ĝi ĝj and the
identical crossover law a0 (z) = 2 α cH(z). They differ in
exactly one input: the cosmological external acceleration
on the FRW background,
AC: aFRW
= cH(z) ⇒ x̄ = 2√1 α ≃ 5.85,
ext
AE: aFRW
= 0 ⇒ x̄ = 0.
ext

(GR1)

This appendix adjudicates which value governs the cosmological linear growth observables (σ8 , S8 , f σ8 , kSZ)
and which governs collapsed galactic/cluster dynamics,
and it resolves the ambiguity that left the directional f σ8
signature “regime-contested.”
Summary of this adjudication
The conflict is a scope partition with one mis-scoped
clause. AE’s aFRW
= 0 is a theorem from the action
ext
(Theorem AE.1); AC’s aext = cH is an admitted
hypothesis the action does not produce. Neither disputed
Geff is the carrier of cosmological linear growth: that role
is played by the independently-derived χ-matter field
(Ωχ h2 ≃ 0.12, GR-like growth). Hence Geff is not the
single derived growth law for cosmology
(Proposition GR.4). The resolution is a scoping
clarification; no DFD result changes.

2.

GR.2

The regime-partition theorem

Theorem GR.2 (Growth-regime partition). Let x ≡
g/a0 (z) with g the local Newtonian acceleration
sourced
√
by the matter in question and a0 (z) = 2 α cH(z). Then
the AQUAL operator Mij of Appendix AC partitions by
scale as follows:

2. (Collapsed/cluster regime.) Inside a collapsed
halo a real local mass sources a genuine |∇ψ̄| ̸=
0; there x ranges from O(0.1) (clusters) to ≳ 1
(galactic interiors).
The transition between regimes is the single derived condition
√
g
(GR2)
x=
= 1 ⇐⇒ g = 2 α cH(z).
a0 (z)
Proof. Part (1): Theorem AE.1 establishes |∇ψ̄| = 0 on
FRW (homogeneity kills the matter source, isotropy kills
the spatial kinetic, reference invariance kills the temporal
kinetic), so no cH-scale external field exists to place a
mode at x̄ = 5.85. The residual x for a physical top-hat is
bounded by Lemma GR.3. Part (2): for a bound
pmass M
at radius r, gN = GM/r2 exceeds a0 once r < GM/a0 ,
the standard MOND radius; for M ∼ 1012 M⊙ this is ∼ 35
kpc, so galactic interiors satisfy x ≳ 1. The crossover
x = 1 is Eq. (GR2) by definition of a0 .
Lemma GR.3 (Linear R8 modes are deep-MOND at
every epoch). For a spherical top-hat of contrast δ on
comoving radius R8 (proper radius aR8 ), the edge acceleration is gN = 12 Ωm (a)H(a)2 aR8 δ and
x=

Ωm (a)H(a) aR8 δ
gN
√
=
.
a0 (z)
4 αc

(GR3)

Numerically, at the measured Ωm ≃ 0.312: x(δ=1) =
2.4 × 10−3 at z=0, 5.5 × 10−3 at z=1, 7.3 × 10−3 at z=2,
and 1.3 × 10−6 at recombination (δ∼10−5 ). All ≪ 1:
deep-MOND on all linear scales.

3.

GR.3

The AQUAL Geff is not the cosmological
growth carrier

Proposition GR.4 (Growth carrier). The cosmological
linear growth of δm is carried by the derived χ-matter field
(Appendix AV, Ωχ h2 ≃ 0.12), which gravitates through the
unscreened rest channel (Remark GR.9), has Geff = G,
and grows ΛCDM-like (Q ≡ Geff /G = 1). Neither AC’s
1.12 G nor AE’s raw deep-MOND self-field is the growth
law:
1. AC’s constant ⟨Geff ⟩ = 1.118 G, if applied from
z ∼ 1000, compounds to a D-ratio 1.54, i.e. σ8 →
1.24 — a +54% over-growth, excluded (computed
with µ = x/(1 + x); on the derived exponential
response of App. BS the same ambient argument

254
x̄ = 5.85 gives +24% on the SOURCE branch and
−1.0% on the TOTAL branch (App. AC’s angularmean construction, the same quantity as the +54%),
so on that branch the exclusion of an ambient boost
as the growth law rests on Theorem AE.1, x̄ = 0
on FRW, not on over-growth). AC’s quoted “+1–
9%” is the instantaneous f σ8 , not the integrated σ8
(Remark GR.5).
2. AE’s raw deep-MOND self-field response on δm runs
away (the corpus 5.4× δrms overshoot); App AE.7
correctly reinterprets its Q ≃ 500 as a lensing amplification of a small baryon contrast, not a growth
boost.
Verification. Both branches are integrated numerically
in growth regime adjudication.py. The constantQ=1.118 ODE from z=1000 yields D(1)/D(aini ) a factor
1.54 above the Q=1 solution; the deep-MOND self-field
branch diverges. The Q=1 (χ) solution reproduces the
corpus f (0), σ8 , S8 self-consistently (GR.4).
Remark GR.5 (AC’s instantaneous-vs-integrated conflation). AC’s 1.02–1.17 G is a statement about the instantaneous growth-rate enhancement f = d ln D/d ln a at
low z. Integrated over the full history a constant 1.118 G
compounds; the appendix’s envelope therefore must not
be read as an integrated-σ8 claim. This independently
disfavors applying the Hubble-EFE envelope to cosmology.
Scope (a proven obstruction — resolved by the
adopted Rest-Mass Channel postulate)
The carrier identification (Proposition GR.4: χ has
Q ≡ Geff /G = 1 on linear scales) requires χ-matter to sit
at x ≡ g/a0 ≫ 1 (Newtonian, AQUAL screen off ). But
Lemma GR.3 establishes that on the same linear R8
scale the top-hat edge acceleration gives
x ∼ 2 × 10−3 ≪ 1 (deep-MOND) — which is exactly the
regime needed to keep the AQUAL screen off the baryons.
These two scoping statements are mutually
exclusive on one linear scale; the adopted Rest-Mass
Channel reconciles them by confining the polarization to
bound systems (below).
Status: a proven obstruction in the linear regime. A
local-source argument — inside a collapsing halo χ
self-gravitates Newtonianly where a real source gives
g ≫ a0 — does hold post-collapse, but it switches on only
at δ ≳ 500 (nonlinear) and therefore cannot reach the
linear R8 regime that sets σ8 /f σ8 and the third-peak
transfer function. In that linear regime Q = 1 is provably
not available from the bare single-W action (“structured
⇔ gradient ⇔ screened”; the Rest-Mass Channel box
below): every clustering mode carries ∇ϕ ̸= 0 and is
screened into the deep-MOND branch. Closing Q = 1 on
linear scales therefore requires the Rest-Mass Channel
(the µ-law confined to bound systems, which leaves the
µ=1 rest-mass term, the Einstein–Hilbert term’s own
static scalar energy, unscreened) — the minimal fix
(matter moves under an energy that conserves
momentum, and χ falls like the baryons wherever it fills

the µ-law’s demand, §GR.3a), proven absent from the
bare action, which DFD adopts as a postulate; given it,
Qχ = 1 follows and the clustering gate is closed.
Why it decides the S8 story. If Q = 1 holds (the asserted
resolution), one σ8 = 0.806 sets both the galaxy-lensing
S8 = 0.822 and the CMB-lensing amplitude
Aϕϕ = 0.9887, and the sharper S8 = 0.755 is not
available. If instead the AQUAL screen leaked into linear
growth, Q would run with redshift (Q ∼ 20 at z = 0 to
∼ 12 at z = 2), the growth factor D(z) would not be
flat/ΛCDM-like, the σ8 -vs-S8 lock would break — at the
cost of a z-dependent growth signature testable against
f σ8 (z). The single-W action cannot settle this (the
obstruction below); the Rest-Mass Channel, which DFD
adopts as a postulate and which confines the polarization
to bound systems, settles it with Q = 1 on linear scales,
as stated next.

Adopted postulate: the Rest-Mass Channel (twochannel gravity — closes Qχ = 1)
Postulate (adopted). The gravitational response of ψ
splits by the type of energy that sources it,
ψ = ψrest + ψgrad , with
ρrest ,
∇2 ψrest = − 8πG
c2

i
h 
|∇ψgrad |
∇ψgrad = − 8πG
ρgrad/pol .
∇· µ
a⋆
c2

(GR4)
(GR5)

Rest-mass energy sources an unscreened Poisson channel;
only gradient/polarization (edge) energy sources the
AQUAL screen. Cosmological χ perturbations are
rest-mass dust, ρχ = mχ nχ , so they feed ψrest and cluster
CDM-like: Qχ = 1 . The MOND/RAR phenomenology
stays confined to the gradient/polarization sector of
bound low-acceleration systems, a bound subsystem
counting as low-acceleration only while the Bose mode of
its host√is typically occupied (App. NL, Axiom NL.14), so
a0 = 2 α cH0loc is untouched.
Status: an adopted action completion, from which
Qχ = 1 is then a theorem (Thm GR.10, §GR.3a).
This split is not
R present in the bare single-W action:
varying Sψ = [(a20 /8πG)W (|∇ψ|2 /a2⋆ ) − (c2 /2)ψ(ρ − ρ̄)]
gives the single screened equation
∇· [µ∇ψ] = −(8πG/c2 )(ρ − ρ̄) with µ = W ′ acting on
the one field ψ upstream of the metric — so a massive
slow χ, reading only g00 = −c2 e−ψ , still reads a screened
ψ. The weak-response kinetic

√
√ 
W (X) = X − 2 X + 2 ln 1 + X , X = |∇ψ|2 /a2⋆ , the
primitive of µ(x) = W ′ (x2 ) = x/(1 + x), has W ′ (0) = 0,
i.e. µ(0) = 0 exactly, so no latent unscreened piece hides
in W , and adding a floor µ → c1 + x destroys v 4 = GM a0
(the RAR result). A term-by-term audit finds no second
ρ-sourced µ=1 elliptic operator already in the action, and
′
none is needed: W = X + Wnl exactly, with Wnl
= µ − 1,
and where the µ-law does not act the X term,
(c4 /32πG)|∇ψ|2 in the action, is the unscreened operator;
it is the Einstein–Hilbert term’s own static scalar energy
on DFD’s metric (App. WS, Proposition WS.6). The
proven obstruction below establishes that Qχ = 1 cannot
be a theorem of the single-W action. DFD therefore
adopts the Rest-Mass Channel as a postulate: the

255
µ-law acts only inside bound systems — the
minimal choice: the split is by energy type, not species
(cf. gravitational polarization, Blanchet; Verlinde); matter
moves under one energy that conserves momentum
(§GR.3a), and χ falls like the baryons wherever it fills the
µ-law’s demand. This raises DFD’s gravitational-sector
axiom count by one. Given the postulate, Qχ = 1 for
rest-mass dust follows, the dark-sector linear clustering
(third peak, lensing, f σ8 ) is consistent, and the
cluster-vs-galaxy signature below becomes a prediction
(its cluster-side amplitude follows from the derived
abundance under ΛCDM-like assembly, at estimate grade;
see below). The cost is the added axiom; the gain is a
closed clustering gate and a falsifiable signature.
(Deriving the postulate from a deeper principle would
remove the axiom; the obstruction shows that is not
possible within the single-W action.)
Proven structural obstruction (why no rearrangement can
supply Qχ = 1). Three candidate derivations were
computed explicitly and all collapse to one tautology of
the single-W action. (a) The polarization identity
∇· [µ∇ψ] = ∇2 ψ + ∇· [(µ − 1)∇ψ] is algebraically true,
but for χ’s linear R8 mode (x ∼ 2 × 10−3 , µ − 1 ≃ −0.95)
the “polarization
p current” carries essentially the entire
field: ρeff /ρ ∝ a0 /gN , the full deep-MOND boost
(Q ≈ 20), not Q = 1 — it rewrites the same screened
equation, it is not a two-sector split. (b) The coherent
condensate is dust on these scales: c2s /c2 ∼ 5 × 10−63 , its
Jeans scale sits ∼ 1014 above kLSS (and ∼ 1015 even after
1/4
the deep-MOND Geff boost), and ψ sees only the
time-averaged ρχ ∝ a−3 . (c) The local-source (g ≫ a0 )
resolution only switches on at δ ≳ 500 (nonlinear
collapse), never in the linear regime that sets σ8 /f σ8 .
The common root: the unscreened flat direction is
reserved for the spatially-homogeneous mean,
while any clustering perturbation carries ∇ϕ ̸= 0 —
and ∇ϕ is exactly what W screens. “structured ⇔
gradient ⇔ screened” is a theorem of the single-W action.
Hence Qχ = 1 is provably not a rearrangement of the
single-W action: it requires confining the µ-law to bound
systems, which leaves the X term, the Einstein–Hilbert
term’s own static scalar energy, unscreened for clustering
modes — a postulate about where the µ-law acts, not a
new field.
Theorem GR.6 (Trace-blindness obstruction,
strengthening GR.3a). Let the metric be un-truncated
from the transverse-traceless projection hTijT to the full
symmetric tracelessR hij , with the printed universal
coupling Sint = + 12 hij T ij . Its scalar sector is the single
mode Dij E = (∂i ∂j − 31 δij ∇2 )E, whose source is exactly
S[E] = ∂i ∂j T ij − 13 ∇2 T k k .
ij

i j

(GR6)

ij

For a perfect fluid T = ρv v + p δ the pressure terms
cancel identically and the rest-mass term is absent,
leaving S[E] = ∂i ∂j (ρv i v j ) − 13 ∇2 (ρv 2 ) = O(ρv 2 /c2 ),
which vanishes exactly for cold dust at rest. Hence no
un-truncation of a traceless tensor sector can
supply a ρ-sourced µ=1 elliptic operator: ρc2 is a
pure-trace source and a traceless tensor is blind to it.
Proof. Direct computation, verified symbolically: with
T ij = λ2 ρv i v j + p δ ij the source above evaluates to 0

identically at λ R= 0, and ρ enters only at O(λ2 ).
Independently, dΩ (ni nj − 13 δij ) = 0 exactly, and the
geodesic acceleration on g = diag(−c2 e−ψ , eψ (δ + λh))
with h an arbitrary symmetric traceless tensor is
2
ai = c2 e−2ψ (δij − λhij )∂j ψ, which is identically zero
when ∇ψ = 0.
Remark GR.7 (What this adds, and the self-defeat
corollary). GR.3a’s obstruction is scoped to “the
single-W action,” and the corpus’s own hij field
(§formalism) makes DFD a two-potential theory.
Theorem GR.6 covers that case and is stronger: it holds
for any minimally-coupled metric theory with any
number of scalar and tensor slots, for dust sources. The
conclusion — that Qχ = 1 requires a postulate —
therefore holds for the two-potential theory as well.
Self-defeat corollary. The very property that makes
the un-truncation force η = 1 (trace-blindness: coupling
purely to anisotropic stress) is precisely the property that
makes it useless for Qχ . The η-repair and Qχ = 1 are
mutually exclusive uses of one field. To obtain a
ρ-sourced µ=1 operator one must un-truncate the trace
slot — and g00 = −c2 e−ψ already has that slot filled by
ψ, whose Einstein–Hilbert static energy is that operator,
with the same coefficient (App. WS); the Rest-Mass
Channel adds no field of its own, only the confinement of
the µ-law to bound systems.
Remark GR.8 (Kinetic function and TT action
normalization). (i) The µ-function argument
convention and the kinetic primitive. The
Rest-Mass Channel box writes W (|∇ψ|2 /a2⋆ ), i.e. a
squared argument, and the formalism identifies
µ(x) = W ′ (x2 ). The weak-response kinetic function is
the primitive of the simple interpolation,
√ 
√
W (X) = X − 2 X + 2 ln 1 + X with
√
√
W ′ (X) = X/(1 + X), i.e. µ(x) = x/(1 + x) → x and
4
v = GM a0 ; W (0) = W ′ (0) = 0, W ′′ (X) > 0 for X > 0,
and W (x2 ) = 23 x3 + O(x4 ). The argument stays squared.
(ii) The hRij action prefactor and interaction
sign.
R
Varying A [ḣ2 /c2 − (∇h)2 ] against Sint = + 21 hT gives
□h = T /(4A) with □ = c−2 ∂t2 − ∇2 , so the field equation
□hij = (16πG/c4 )Tij printed in §formalism and
§gravitational waves requires A = c4 /(64πG), the
linearized Einstein–Hilbert value. A prefactor
R
A = c4 /(32πG), a coupling Sint = − 12 hT or a wave
equation of the opposite sign would be inconsistent with
it.
Why this is the right axiom (minimality and
motivation). (i) The split is by energy type, not species,
by analogy with gravitational polarization (Blanchet;
Verlinde), although DFD’s polarization carries no inertia
of its own; under the adopted energy action and reaction
hold and χ falls like the baryons wherever it fills the
µ-law’s demand, so any species dependence is confined to
χ below the demand, a dark-only effect (§GR.3a). (ii) A
genuine two-channel structure already operates in DFD:
the homogeneous mean ψ̄ gravitates unscreened into the
Friedmann equation (App. AU, ρdep /ρP = (3/8π)α57 ; the
mean-subtraction (ρ − ρ̄) rides every species’ mean
unscreened into H(t)), so the postulate extends an
existing unscreened channel from the mean to the

256
rest-mass perturbation rather than inventing one
wholesale. (iii) χ’s misalignment condensate is produced
at rest (zero momentum, coherent, near-homogeneous),
the natural occupant of the rest-mass channel. These
make the Rest-Mass Channel the minimal and most
strongly-motivated axiom that closes Qχ = 1; a future
derivation from a deeper principle would demote it from
axiom to theorem (the single-W obstruction shows that
derivation is not available within the present action).
Remark GR.9 (Which matter feeds which channel). The
rule that only gradient/polarization energy sources the
AQUAL screen fixes which matter feeds which channel,
and that assignment follows from the two-vertex picture
of App. N (Remark N.9). The optical medium is
polarized through
photon–fermion vertices, each
√
contributing α, so matter with charged constituents
polarizes it: neutral atoms and ionized gas alike. χ has
no charged constituents and no photon coupling: its
photon coefficient is E = 0 (App. AV, Theorem AV.35).
It therefore does not polarize the optical medium: its
crossover acceleration is zero, χ is always in the
unscreened regime, gravitates through the rest channel
only, and Qχ = 1 holds with no photon-coupling
correction. Electrically neutral cosmic neutrinos behave
the same way. This is a conditional derivation: it takes
the two-vertex picture of App. N, a heuristic count, as the
mechanism of the polarization. What stays postulated is
the confinement of the µ-law to bound systems; the rest
channel’s Poisson term is the Einstein–Hilbert term’s own
static scalar energy (App. WS), which no traceless tensor
could supply (Theorem GR.6).
Consistency. Third-peak: consistent (it was computed
with Q = 1 cold χ). Galactic RAR: holds only if χ’s pull
does not add to the µ-law inside discs. Under the
additive composition of the two channels, at the 0.03 dex
tolerance of the relation (a +3–4% rise in Vrot ), SPARC
bounds χ to ≲ 20% of Mbar inside 2.2Rd for the median
disc and ≈ 4% for the best-measured discs, and to about
half these if χ’s mass also feeds the µ-law (computed),
while the same physics makes χ form ΛCDM-like halos,
which SPARC excludes (App. AV). Under the floor
composition of App. AV (an adopted input, in energy
form) the requirement is instead that χ’s enclosed mass
stay below (ν − 1)Mb at each radius: below Mb where
gN > a0 /2, and a median 3–5 times Mb inside the last
measured radius (computed). Under the adopted energy
χ assembles as in ΛCDM (estimate), and the requirement
is met when the inner halos are cored and, in discs above
about 1011 M⊙ of baryons, lighter than abundance
matching gives (App. AV). A barometric construction
(collisionless χ in isothermal Jeans equilibrium in DFD’s
own field, with a universal dispersion ≳ 337 km/s) meets
the additive bound (worst bright spiral ≈ 2–4%), but
DFD’s structure formation supplies no route to it
(computed). PPN: the rest-mass Poisson channel is the
clean GR limit, so γ = β = 1 and α1 = α2 = 0
(Thm IV.1) are preserved; any residual χ-sector fifth
force is dark-only and lab-unconstrained (under the
adopted energy χ departs from the baryons’ fall only
where it falls short of the demand; App. AV). Net: with
the Rest-Mass Channel adopted as a DFD postulate the
Qχ = 1 clustering gate is closed. The factor-∼44

χ-abundance gate is likewise resolved : the finite-SU (2)60
Chern–Simons modular measure gives QCS = 0.073 and,
with the exact radiation-era bridge and the flux-period
identification (Lemma AV.6), Ωχ h2 = 0.1193 (−0.6σ;
App. AV), the continuum π 2 /3 overshoot being an
artifact of mixed units (radians2 vs turns2 ) and a
continuum measure — see the state-preparation theorem
and Bricks 1–3 of App. AV. The abundance is not the
open item of the dark sector (its standing conditionality
is the no-boundary preparation premise, the flux-period
identification, the stated multiplicity 158, two stated
normalization conventions and the asserted premise P4′
stated there); how much χ galaxies hold follows, at
estimate grade, from the adopted energy (ΛCDM-like
halos), and the open item is whether their inner parts are
cored as the rotation curves require (App. AV); χ’s
photon coupling is zero.
Falsifiable signature (galaxies versus clusters — a
prediction of the adopted two-channel gravity).
The two-channel structure predicts that the dark mass
seen beyond the µ-law differs between galaxies and
clusters, which separates DFD from both ΛCDM and
pure MOND. Clusters (rest-mass channel) need χ. If χ
adds its Newtonian pull to the baryons’ µ-law field, as
printed, the sixteen clusters of App. I need 4.5 ± 0.6 Mb
inside r500 , 0.84 ± 0.12 of the cosmic mix Ωχ /Ωb ≈ 5.4,
with hydrostatic bias b = 0.21 (computed). Under the
floor composition of App. AV clusters are Newtonian plus
χ and carry the ΛCDM budget: 1.28 ± 0.16 of the cosmic
mix inside r500 at b = 0 and 1.51 ± 0.14 at b = 0.21, the
usual baryon depletion (computed). The amplitude is
what the derived abundance gives under ΛCDM-like
assembly: 1.21 ± 0.05 of the cosmic mix predicted inside
r500 (estimate; App. I). Galaxies (baryon-dominated,
gradient channel) show little dark mass beyond the µ-law:
under the additive composition χ must stay ≲ 20% of
Mbar inside 2.2Rd for the median disc; under the floor
composition χ thinner than the baryons’ phantom is
invisible, and only χ in excess of it lifts the relation,
one-sidedly. Pure MOND under-predicts cluster
dynamical mass by a factor ∼2 (Sanders 2003;
Angus–Famaey–Buote 2008; Pointecouteau–Silk 2005),
historically patched with ∼2 eV sterile neutrinos; an
acceleration-independent χ component can place residual
mass even in cluster cores where g ≳ a0 (high
acceleration), where pure MOND cannot generate it. The
cold, non-thermal nature of χ is essential here: its
de Broglie/Jeans scale (∼10−13 Mpc for the 32 eV relic,
which is produced at rest) is ∼1013 × below cluster scales,
so it can cluster as ordinary CDM; a thermal χ of the
same mass would free-stream over ∼20–30 Mpc and fail
clusters (estimate). Falsified if: (a) disc galaxies hold
more χ than the composition tolerates (under the floor
composition, massive or concentrated χ halos must lift
the relation above the µ-law, and measured halos of that
kind without the lift falsify that reading); (b) clusters
need no rest-mass component beyond the µ-law, which is
decided: they need one (App. I), so this clause does not
fire; or (c) χ is shown thermal/hot (this last cuts against
DFD). The fingerprint is the galaxy-vs-cluster
asymmetry: little dark mass beyond the µ-law in
baryon-dominated disks, rising to the cosmic mix in rich

257
clusters.

4.

GR.3a The Rest-Mass Channel Theorem (the
corrected DFD gravitational action)

Scope. What follows is an action completion, not a
derivation from the single-operator action. The single-µ
form ∇· [µ∇ψ] = −(8πG/c2 )ρtot provably cannot yield
Qχ = 1 (§GR.3): it applies the MOND/polarization
constitutive law to all energy density, including cold χ
rest mass. The action below separates conserved rest
mass from optical-polarization response; from it, Qχ = 1
follows as a theorem.
a. Motivation. The DFD optical response has two
physically distinct sources of gravitational acceleration:
(i) conserved rest-energy density, carried by massive
particles and cold coherent matter; (ii) optical gradient/polarization energy, generated by strain, boundary,
and low-acceleration medium response. The MOND/RAR
constitutive law applies to the second channel, not to conserved rest mass. Accordingly the nonrelativistic response
is written in two channels, ψ = ψrest + ψpol , with ψrest the
unscreened response to rest mass and ψpol the nonlinear
optical-polarization response.
b. Action.
Z


h 4
|∇ψpol |2
a20
c
|∇ψrest |2 + 8πG
F
Sψ = dt d3 x 32πG
2
a⋆
(GR7)
i
2
2
− c2 ρrest ψrest − c2 ρpol ψpol ,
with µ(y) ≡ F ′ (y 2 ), y = |∇ψpol |/a⋆ . Matter moves under
the energy W of App. AV, Eq. (AV28), a functional of
the rest channel’s Newtonian fields of the baryons and
of χ that gives the baryons the floor composition exactly
in spherical symmetry; the field equations below state
the channel split. An action in which all matter coupled
to the total ψ would make all rest mass a source of ψpol
(derived), and the rule that every species moves in the
total ψ breaks action and reaction: for a 1010 M⊙ clump
100 kpc from a 5 × 1010 M⊙ galaxy it gives a drift of
45 km/s per Gyr (computed). Under W momentum and
angular momentum are conserved exactly, and energy
at fixed a0 ; the same pair does not drift, and with the
galaxy’s own χ halo in place it attracts at the Newtonian
value where that halo fills the demand at the clump and
up to about 4% more weakly beyond it (computed). The
price is χ’s equivalence principle where χ falls short of
the µ-law’s demand (App. AV).
Constitutive reading of ρpol (essential). Here ρpol is the
induced optical-polarization (strain) charge, ρpol = −∇· P
with P the polarization sourced by the baryonic rest mass
— not an independent matter density and not a duplicate
of ρrest . It is constitutively slaved to the baryonic source
and self-extinguishes, ρpol → 0 as µ → 1 (g ≫ a0 ), so only
the rest channel survives in the solar-system/strong-field
limit (Geff = G, no doubling). In the deep-MOND limit

the induced charge reproduces ρpol ≈ ρbaryon , closing
p the
observed RAR gtot = ν(gN /a0 ) gN with ν → a0 /gN
for gN ≪ a0 . This induced reading (an analogue of
Blanchet’s gravitational polarization, though here the
polarization carries no inertia) is the unique internally
consistent one: a second independent copy of the rest
mass would give Geff = 2G at g ≫ a0 and break the
PPN normalization. Equivalently, the channel split is by
energy type (conserved rest mass vs. optical strain), not
by particle species, and under the energy W the baryons
and χ fall alike wherever χ fills the µ-law’s demand.
Theorem GR.10 (Rest-Mass Channel). Variation of the
two-channel action (GR7) gives an unscreened Poisson
channel for conserved rest mass and the screened AQUAL
channel for polarization energy,
∇2 ψrest = − 8πG
c2 ρrest ,
h 

i
|∇ψ |
∇· µ a⋆pol ∇ψpol = − 8πG
c2 ρpol .

(GR8)

Proof. Varying Sψ in ψrest and integrating

R by parts
(discarding the boundary term), δSrest = dt d3 x −

c4
c2
2
16πG ∇ ψrest − 2 ρrest δψrest ; stationarity for arbitrary
δψrest gives ∇2 ψrest = − 8πG
c2 ρrest , the unscreened Poisson
channel. Varying in ψpol with y = |∇ψpol |/a⋆ , the nonc4
linear term contributes 16πG
µ(y)∇ψpol · ∇δψpol (using
µ(y) = F ′ (y 2 ) and a⋆ = 2a0 /c2 ); integrating by parts,


R
2
c4
δSpol = dt d3 x − 16πG
∇· (µ(y)∇ψpol ) − c2 ρpol δψpol ,
and stationarity gives ∇ · [µ(|∇ψpol |/a⋆ )∇ψpol ] =
− 8πG
c2 ρpol . The nonlinear MOND/RAR operator is thus
attached to optical-polarization energy, not to conserved
rest mass.
Corollary GR.11 (Cold χ clusters with Qχ = 1). After
oscillation onset ρχ ≃ mχ nχ with pχ ≃ 0 and gradient
pressure suppressed on cosmological linear scales, so cosmological χ perturbations are conserved rest-mass dust.
They source ψrest , ∇2 ψrest = −(8πG/c2 )ρχ . The postulate confines the polarization to bound low-acceleration
systems (the Rest-Mass Channel box above), so linear,
unbound perturbations do not polarize the medium; hence
their effective linear-clustering response is Qχ = 1 , recovering cold-CDM-like linear growth and preserving the
CMB third-peak result. The third-peak height is set at
recombination, before any baryon is bound, so it does not
depend on how the polarization of bound systems behaves
once structure forms (Remark GR.12).
Remark GR.12 (Where the µ-law acts). The RestMass Channel box states where the µ-law acts: the
MOND/RAR phenomenology stays confined to the gradient/polarization sector of bound low-acceleration systems. Read literally, the medium is polarized only inside
bound systems. The polarization P then vanishes on
the edge of each bound region, so by Gauss’s theorem
the induced charge ρpol = −∇· P integrates to zero over
the region: the phantom charge is redistributed inside
it, and none of it lies beyond the region. Outside, the

258
field of a non-spherical system keeps the multipoles of
that redistributed charge, which vanish for a spherical
one. Three-dimensional particle-mesh runs of structure
formation with χ and the baryons (the floor composition
of App. AV, every bound baryon polarizing, the bound region cut at r200m or 1.5 r200m of each halo) give a residual
net polarization mass of 0.0003–0.002 of the mean matter
density at z = 0, from haloes whose compensating rims
the grid does not resolve, raise σ8 by a factor 1.001–1.005
and the lensing power at k = 0.1–0.4 h/Mpc by a factor
1.003–1.07 (and by 2–40% at k = 0.7–1.3 h/Mpc), and
leave the CMB untouched (computed). With the compensating rims resolved (halo model), the confinement leaves
the matter power unchanged to 0.2% at k ≤ 0.3 h/Mpc
and raises it only at k ≳ 2 h/Mpc: by 7–56% at k = 2–
10 h/Mpc if bound baryons trace their haloes, and by
under 4% if a third of them do (computed at z = 0.3).
The particle-mesh values above are an upper envelope
set by the unresolved rims; small-scale cosmic shear and
galaxy lensing, with bound-baryon profiles measured in
X-rays and the Sunyaev–Zel’dovich effect, decide the size
of the boost. Every SPARC rotation curve ends inside
0.25 r200m , so the rotation-curve results are kept (computed), and a central galaxy feels almost no external field
from its neighbours, whose net phantom is zero, so the
a0 that SPARC measures stays DFD’s (estimate). In
the rotation-curve fits of Chae et al. this reading gives
a fitted external field of about 0.003 for every central
galaxy, and 0.007–0.013 with the domain-local screen
of App. ZT (computed). Their reported median, 0.044–
0.053 [166, 167], is carried by the least well measured third
of their sample: the other two thirds give 0.009 and 0.002,
the fitted field shows no correlation with environment
density on 4–20 h−1 Mpc scales or with their own field
estimates, and weighted by the per-galaxy errors, with
the 0.02 fitting-function systematic of Chae et al. added
in quadrature, the sample median is 0.012 ± 0.009 and the
mean 0.022 ± 0.007, 1.1σ and 2.8σ from this reading (0.6σ
and 1.8σ with the screen; computed; see also [168, 169]).
The outer declines of NGC 5055 and NGC 5033 are not
produced by this reading. BIG-SPARC, about 4000 galaxies, decides it: this reading predicts no environment trend
in the a0 -free shape of the curves, where MOND with the
fields of Chae et al. predicts a correlation of 0.1–0.3 (estimate). The literal reading is the one cosmology allows:
if the phantom of bound baryons reached beyond their
bound regions, the same runs would give a polarization
mass of 0.47–1.25 of the mean matter density and 2.9–6.4
times the lensing power (computed), against a viability
limit near 0.03 (estimate). The reading has one further
cost. The flat weak-lensing profile of isolated galaxies is
not kept beyond the bound radius, about 0.25–0.6 Mpc for
galaxies of 1010 –1011 M⊙ in baryons: inside it DFD’s 1/R
profile fits the stacked measurement where the galaxies’
χ stays below the µ-law’s demand, and beyond it, with
the χ halo and the surrounding structure included, DFD
falls below the measured points of Brouwer et al.’s looserisolation stack for every halo mass tried, and of Mistele

et al.’s stricter-isolation stack for the lighter halo masses
(computed). The halo model of ΛCDM falls below them
in the same way; the far points are reached only if neighbours admitted by photometric redshifts add signal at the
level Brouwer et al.’s own ΛCDM mock of the selection
finds, so spectroscopically isolated lens samples decide
it. Galaxy groups are not a cost: their hot gas is bound
and has charged constituents, so it polarizes the medium
(Remark GR.9), and on published X-ray masses the µ-law
falls short of groups as it does of clusters, so groups need
χ as clusters do (App. I; computed on published inputs).
Corollary GR.13 (The galactic RAR and the two channels). In galaxies gtotal = grest + gpol : the rest channel gives the Newtonian baseline grest = gN and the
polarization channel the low-acceleration enhancement
√
gpol ∼
a0 gN in the deep-MOND regime. For the
√
baryons, since a0 gN ≫ gN for gN ≪ a0 , the RAR
stays polarization-dominated in galaxy outskirts while the
rest channel carries cosmological χ clustering. The rest
channel also carries the Newtonian pull of any χ inside
a galaxy, and ΛCDM-like χ halos added this way count
the missing mass twice, which SPARC excludes; the RAR
is kept only if χ’s pull does not add inside discs, as under the floor composition of App. AV, adopted in energy
form; how much χ galaxies hold follows at estimate grade
from the adopted energy (ΛCDM-like halos whose inner
parts must be cored; App. AV). Solar-system and strongfield tests sit at g ≫ a0 , where µ → 1 and both channels
are Poisson, so the total ψ is the standard potential and
γ = β = 1, α1 = α2 = 0 (Thm IV.1), the 2PN lightbending, and the compact-star results, which are general
relativity’s (App. WS), are unchanged.
Interpretation and status. The one-operator form
∇· [µ∇ψ] = −(8πG/c2 )ρtot incorrectly screens cold χ rest
mass, making linear χ perturbations over-grow and blocking Qχ = 1. The corrected action separates conserved
rest mass from optical-polarization response — the sourcespace decomposition required by the physical distinction
between conserved rest-energy density and emergent optical strain energy. With Theorem GR.10, DFD’s χ sector
has its particle, relic law, and linear clustering (Qχ = 1,
Cor. GR.11) derived from the corrected action. The abundance normalization is derived too, conditional on the
flux-period identification (Lemma AV.6), the stated multiplicity 158, two stated normalization conventions and the
asserted premise P4′ (Lemma AV.27): the finite SU (2)60
CS-vacuum Casimir expectation gives Ωχ h2 = 0.1193
(−0.6σ from Planck; App. AV Step 5b); in consistent
units the vacuum variance sits 12.45% below the uniformangle value: the continuum sin2 measure lowers it by 15%,
the Sugawara factor K/k raises it by 3.3%, and finiteness
itself moves it by 0.09%; the amplitude is forced given two
stated conventions (Casimir; measure conditional on P4′ ;
k(k +2) from χ’s derived Z2 + Sugawara); the relic bridge
is exact (κrad ), and the grade is conditional-derived on
the named flux-period identification, the stated count 158
and P4′ (App. AV). Status: this is an action completion

259
— the DFD gravitational action — not a derivation from
the single-operator form, which the obstruction of §GR.3
shows cannot yield Qχ = 1.
5.

GR.4

The single DFD growth prediction

With the carrier fixed as χ-matter (Q = 1, GR-like)
on the measured-clock
background (H0early = 67.52, ωχ =
P
mν = 61.5 meV, ns = 1 − 2/60; App. ZS) and
0.11928,
the primordial amplitude As = 2.080 × 10−9 (the value
of 32πα5 , a calibration; App. FK), run forward through
CAMB, the linear growth equation


3 d ln H dD 3 Ωm (a)
d2 D
+
Q D = 0,
Q = 1,
+
−
da2
a
da
da
2 a2
(GR9)
gives σ8 = 0.806, S8 = 0.822 and Ωm = 0.312, and the
growth rates below (computed). Growth is scale independent on linear scales except for neutrino free streaming,
which lowers f by 0.3% between k = 10−3 and 1 h/Mpc.
TABLE CIX. DFD growth prediction (single, χ-matter Q = 1
carrier) on the measured clock, σ8 (0) = 0.806. f σ8 compared
to the 6dFGS, SDSS MGS, BOSS DR12 and eBOSS DR16
measurements; the pull is (DFD−observed)/σobs . Against
these seven points χ2 = 6.4, and 6.3 for Planck ΛCDM (computed).
z

f σ8DFD

f σ8obs

σobs

pull

0.067
0.15
0.38
0.51
0.70
0.85
1.48

0.440
0.455
0.473
0.472
0.460
0.446
0.376

0.423
0.530
0.497
0.458
0.473
0.315
0.462

0.055
0.160
0.045
0.038
0.041
0.095
0.045

+0.30
−0.47
−0.54
+0.35
−0.32
+1.38
−1.91

S8 = 0.822 lies 2.8σ above DES Year 6 (0.789 ± 0.012,
survey errors only), 0.4σ above KiDS-Legacy and 0.8σ
below Planck (computed). The f σ8 fit is comparable to
ΛCDM’s, and the largest residual pull is the high-z QSO
bin. On the one-clock C=1 background (H0 = 72.09,
Ωm = 0.274) the same carrier gave σ8 = 0.820, f (0) =
0.487 and S8 = 0.784; GR.5 and GR.5a record those runs.
6.

GR.5

S8 on the one-clock C=1 background

This subsection and the next record runs on the C=1
background; on the measured clock S8 = 0.822 and the
CMB-lensing amplitude is Aϕϕ = 0.9887 (GR.4, GR.5a).
Corollary GR.14 (Mild low-side S8 match). At the
carried normalization the C=1 background gives S8 =
0.784, which lies mildly on the low (weak-lensing) side
of the S8 tension and matches both galaxy surveys within
∼ 1σ:

Reference

S8

KiDS-1000
0.759 ± 0.024
DES-Y3 (3×2pt)
0.776 ± 0.017
Planck-ΛCDM (CMB) 0.834 ± 0.016

DFD offset (S8 =0.784)
+1.0σ
+0.5σ
−3.1σ

The C=1 background’s lower Ωm = 0.274 together with
the carried σ8 = 0.820 gives a mildly low galaxy-weaklensing S8 = 0.784, consistent with KiDS-1000 (+1.0σ)
and DES-Y3 (+0.5σ): a mild match. This is the same
σ8 = 0.820 that delivers the CMB-lensing pass (GR.5a /
App. CL): because DFD’s linear growth is Q ≡ Geff /G = 1
(LCDM-like), one σ8 sets both the galaxy-lensing S8 and
the CMB-lensing amplitude simultaneously. The galaxylensing match and the CMB-lensing pass are therefore
jointly available at σ8 = 0.820.
a. The value S8 = 0.755. The value S8 = 0.755 is
not independently derived: it is simply S8 = 0.755 algebraically restated from p
the alternative CMB-normalization
σ8 = 0.790 (S8 = σ8 Ωm /0.3). Reaching σ8 = 0.790
requires shrinking As = 32πα5 by 7.2% with no DFD
justification (GR.7 itself flags 0.790 as “asserted, not
independently re-derived”). Moreover, σ8 = 0.790 is
not simultaneously available with the CMB-lensing pass:
at σ8 = 0.790 the CLϕϕ deficit grows to ∼ 11% and
the nine-bin Planck lensing likelihood disfavours DFD
at > 3.6σ (App. CL). Since one σ8 (Q = 1) sets both
arms, the sharper S8 = 0.755 galaxy-lensing number
and the CMB-lensing pass cannot be claimed together.
The S8 = 0.755 value is therefore “an alternative corpus
CMB-normalization (asserted, not re-derived) that would
sharpen galaxy-lensing to S8 = 0.755 but fails CMB lensing at > 3.6σ — not simultaneously available with the
lensing pass.”

7.

GR.5a

The carried σ8 = 0.820 passes Planck
CMB lensing

Corollary GR.15 (CMB-lensing pass at the carried
normalization). Run forward through CAMB on the
frozen DFD background, the amplitude As = 32π α5 ,
a numerical match to the calibrated amplitude, gives
σ8 (0) = 0.820, for which the lensing power spectrum
ratio CLϕϕ,DFD /CLϕϕ,ΛCDM is a flat ∼ 4% below ΛCDM
across L = 8–400 (shape scatter 0.3%: pure amplitude,
no shape penalty); the full nine-bin Planck 2018 lensing
likelihood returns AL = 1.0542 ± 0.0263, a mild +2.06σ
pass (χ2 = 12.66/9, ∆χ2 = +3.9; App. CL). The ∼ 4%
deficit is structural — the one-clock C = 1 background,
H0 = 72.09 with Ωm = 0.274 at every epoch, which the
two-clock postulate does not reproduce (App. ZS), had low
Ωm and Q = 1 growth, so CLϕϕ ∝ Ωm σ82 ran mildly low
— but it is well below any falsification threshold. At the
measured-clock background (H0early = 67.52, ωχ = 0.11928;
App. ZS) the full-convention nine-bin lensing amplitude
is Aϕϕ = 0.9887 (−0.80σ): the mild +2σ flag dissolves.
The numbers above are the C=1-background run.

260
8.

GR.6

Resolving the directional f σ8 ambiguity

The directional RSD signature f σ8 (z, n̂) ∝ (k̂ · ĝ)2
in AC §AC.7 was flagged “regime-contested” because it
presupposes a nonzero cosmological background gradient
ĝ. Theorem AE.1 removes the ambiguity:
Corollary GR.16 (The growth signature is isotropic
on linear FRW scales). Since |∇ψ̄| = 0 on the FRW
background, there is no preferred direction ĝ for true
linear cosmological modes. The directional f σ8 (z, n̂) signature therefore does not exist for linear cosmological
RSD; the linear growth signature is isotropic, governed
by the χ-matter Q = 1 growth law (GR.4). A directional
(k̂ · ĝ)2 modulation survives only near collapsed structures,
where a reconstructed ψ-screen gradient is nonzero (galactic/cluster scales). The directional falsifier of AC §AC.7
is therefore a galactic/cluster-scale falsifier.

9.

GR.7

Status

Growth-Regime Adjudication — Status
Resolved:
• Regime partition (Theorem GR.2): Transition
x = g/a0 = 1.
• Carrier identified (Proposition GR.4): cosmological
linear δm is carried by χ-matter (Q = 1), not by
the AQUAL Geff . Hence no single action-level
Geff (k, a) both spans regimes and fits the
growth data; the growth observables flow from
χ-matter.
• Single prediction (GR.4, measured clock):
σ8 = 0.806, S8 = 0.822 (2.8σ above DES Year 6,
0.4σ above KiDS-Legacy), CMB-lensing
Aϕϕ = 0.9887 (−0.80σ), and the f σ8 (z) table
(χ2 = 6.4 for seven points); the values of
GR.5–GR.5a belong to the C=1 background.
• Directional ambiguity resolved (Corollary GR.16):
linear cosmological growth signature is isotropic;
the (k̂ · ĝ)2 modulation is galactic/cluster-only.
Caveats:
• The f σ8 fit is comparable to ΛCDM’s: χ2 = 6.4 for
seven points against 6.3 for Planck ΛCDM
(computed).
• The value σ8 = 0.820 of GR.5 belongs to the C=1
background and used As = 32πα5 , a numerical
match to the calibrated amplitude (App. FK); on
the measured clock σ8 = 0.806 (GR.4). The
alternative σ8 = 0.790 (S8 = 0.755) is a corpus
CMB-normalization, asserted not re-derived,
requiring a 7.2% As shrink; it would sharpen
galaxy-lensing to S8 = 0.755 but fails CMB lensing
at > 3.6σ and is therefore not simultaneously
available with the lensing pass.
Deciding dark-sector growth item (a proven
obstruction; closable by the Rest-Mass Channel
extension):
• The “one σ8 sets both” pillar (GR.5) rests on
χ-matter clustering Newtonianly (Q = 1) on linear

scales. But Lemma GR.3 puts the same linear R8
modes at x ∼ 2 × 10−3 ≪ 1 (deep-MOND), and
this is a proven structural obstruction, not a
scoping tension: the single-W screen acts on every
spatial gradient, so any linear clustering mode is
screened into the deep-MOND (Q ≫ 1) branch
(“structured ⇔ gradient ⇔ screened”; §GR.3, the
Rest-Mass Channel box). The local-source
(g ≫ a0 ) escape only switches on at δ ≳ 500
(nonlinear collapse), never in the linear regime that
sets σ8 /f σ8 . Reproducing the observed Q = 1
provably requires the Rest-Mass Channel (a second,
µ=1, rest-mass-sourced Poisson operator), which
DFD adopts as a postulate (§GR.3); given it,
Qχ = 1 holds. A future derivation of the postulate
from a deeper principle would demote it to a
theorem. If instead the AQUAL screen leaked into
linear growth, Q would run with z (∼ 20 at z = 0
to ∼ 12 at z = 2), D(z) would not be flat — a
z-dependent f σ8 signature.
Open-problems item F: the regime question is closed
here; the pipeline obligation is open: the production P (k)
pipeline of App. PK has run only on the one-clock C = 1
background.
Scope restriction (App. AC): AC’s operator Mij
(Lemma AC.1) and AE’s optical Q are both retained as
correct.

10.

GR.8

Falsifier

Growth-Regime Adjudication — Falsifier
The adjudication is falsified if any of the following hold:
1. Isotropy of linear RSD broken. A statistically
significant (> 3σ) detection of a (k̂ · ĝ)2 directional
modulation in linear cosmological f σ8 (z, n̂) on
scales free of collapsed structure would contradict
Corollary GR.16.
2. S8 . On the measured clock DFD’s S8 = 0.822 lies
2.8σ above DES Year 6 and 0.4σ above
KiDS-Legacy (GR.4); a weak-lensing S8 that
settles near DES Year 6’s value at higher precision
excludes it.
3. f σ8 excludes the χ-matter Q = 1 law. If the
full Pθg /Pgg pipeline gives f σ8 inconsistent with
the Q = 1 prediction on the measured clock
(Ωm ≃ 0.312, σ8 = 0.806) at > 3σ across ≥ 2
redshift bins, the carrier identification fails. A
detection of a z-running Q(z) (i.e. non-flat D(z))
instead would signal the AQUAL screen leaking
into linear growth — the deciding item of GR.7.
4. Transition mis-located. If galactic/cluster
dynamics require the
√ deep-MOND/Newtonian
transition at g ̸= 2 α cH(z), the partition
condition Eq. (GR2) is wrong.

261
11.

GR.9

Appendix A0Z: Epoch Confrontation of a0 (z)

Reproducibility

The Q = 1.118 comparison, the x(R8 ) table
and the C=1 key values below are produced
by
dfd class/growth regime adjudication.py
(numpy/scipy), which integrates the Q
=
1
and Q
=
1.118 growth ODEs, tabulates
x = g/a0 (z) on R8 , computes σ8 /S8 /f σ8 , and
writes dfd class/fig growth regime.png (Panel A:
Geff /G across regimes with the x = 1 transition and the
deep-MOND linear-scale band; Panel B: f σ8 (z) DFD
vs DESI vs ΛCDM, with an S8 inset showing the mild
low-side placement). Key values as computed on the
one-clock C=1 background (H0 = 72.09, Ωm = 0.274);
the measured-clock values are those of the scope box at
the head of this appendix: x̄AC = 5.853, ⟨Geff ⟩ = 1.118 G,
a0 (0) = 1.197×10−10 m/s2 , x(R8 , δ=1, z=0) = 2.1×10−3 ,
primary (As = 32πα5 ) σ8 = 0.820, S8 = 0.784 (CMBlensing AL = 1.0542, +2.06σ pass); alternative corpus
normalization σ8 = 0.790, S8 = 0.755 (asserted, not rederived, fails CMB lensing > 3.6σ — not simultaneously
available); AC-branch integrated over-growth +54%.

1.

Scope, and the external measurement it builds on

Theorem
√ XIX.3 promotes the acceleration scale to
a0 (z) = 2 α cH(z), and §XIX records its first measurement, over 0.33 < z < 1.44: a theory-neutral re-analysis
of the MUSE–DARK III binned RAR scale [109] fits the
continuous family a0 (z) = A(1 + z)γ and obtains
γ = 0.78 ± 0.15 (statistical)

(A0Z1)

[110], excluding a frozen scale at 5.3σ on the published
√
bin errors and the matter–density scaling a0 ∝ ρ̄m
at 4.8σ (on the exponent’s error; 4.7σ by likelihood
ratio), with Hubble tracking consistent at 2.1σ and
information-criterion-preferred. Adding the local SPARC
z ≃ 0 anchor gives γ = 0.83 ± 0.13 and pushes the
frozen exclusion to 6.6σ. That analysis is published :
MNRAS 551(4) (2026 October), article ID stag1485,
doi:10.1093/mnras/stag1485, published 2026 September 7 (article type Letter). This appendix does not restate
that result. It adds six things: the refit of the same bins
on the one-clock C = 1 background (§A0Z 2); the predefined control showing that the C = 1 background’s
information-criterion margin is not significant on its own
(Stat. A0Z.4); the amplitude tension, stated on both sides
(§A0Z 4); a derived ceiling on any cosmic external-field
term (§A0Z 5); the survey refitted galaxy by galaxy from
its public products (§A0Z 6); and the confrontation with
massive discs at 0.6 < z < 2.5 (§A0Z 7), together with
dated forward falsifiers (§A0Z 8). The published exponent
is a correct fit of the published bins. Counted galaxy
by galaxy, the trend within the survey puts a frozen a0
about 1σ away; anchored to SPARC’s z = 0 value, the
galaxy-level bins still exclude a frozen a0 at about 3σ, and
DFD’s zero-parameter law fits them on the bound-matter
clock. The trend depends on how the molecular gas is
added, and massive discs do not show the predicted rise
beyond z ≃ 1.5 at face value. The epoch law is therefore
contested.
Remark A0Z.1 (The reference is theory-neutral).
Ref. [110] does not mention DFD, adopts a Planck ΛCDM
background (Ωm = 0.307, H0 = 67.7), and is cited here
as an external observational constraint. Its bins are four
quantile bins in redshift from disc–halo decompositions
of 79 star-forming galaxies over 0.33 < z < 1.44; the
fits below treat them as independent, which the source
notes is optimistic because they share the parent analysis’s calibrations. Within each bin the radial points of one
galaxy are correlated, and counted galaxy by galaxy the
uncertainty is several times the published one (computed;
§A0Z 6).
Remark A0Z.2 (Both confidence conventions, as published). The confidence level of the plotted bars is not
stated in Ref. [109], so Ref. [110] carries every fit under both readings. Under the 1σ reading the numbers
quoted above apply. Under the 95% reading the exclusions strengthen — γ = 0 to ≳ 10σ and γ = 3/2 to

262
9.5σ on the exponent’s error (9.2σ by likelihood ratio),
with the exponent family tightening to γ = 0.78 ± 0.08
— while exact E(z) tracking moves into goodness-of-fit
tension (χ2 = 19.9/3, p = 1.8 × 10−4 ), the binned growth
then sitting shallower than E(z) at ∼4σ on the Planck
background. The two exclusions hold under either reading; the consistency of Hubble tracking does not. The
corresponding offset evaluated on the current background
(§A0Z 2) is 4.2σ, where γE = 1.098 (3.4σ on the one-clock
C = 1 background); that is this corpus’s number, not the
source’s.

2.

DFD needs no borrowed background

A theory-neutral analysis must borrow E(z) from somewhere. DFD need
√ not: its own background supplies E(z),
so a0 (z) = 2 α cH(z) is a zero-parameter curve up to
one overall amplitude. The fits of Table CX were run
on the C = 1 background (H0 = 72.0897, Ωm = 0.2749;
App. CL, App. GR); the CMB-anchored channels read
H0early = 67.52 ± 0.09, Ωm ≃ 0.31 (App. ZS), which moves
this background. Under the two-clock postulate the absolute a0 scale tests the postulate (Falsifier ZS.14, item
iv); a Euclid/DESI lensing RAR decides it once baryonic
masses are known to about 2% (App. ZS). The C = 1,
Ωm = 0.2749 row of Table CX belongs to the one-clock
C = 1 background. At Ωm ≃ 0.312 the measured-clock
refit gives A = 1.394 ± 0.032, χ2 = 5.36/3, AICc = 9.36,
γE = 1.098, and a 2.1σ exponent offset. Hubble tracking
remains viable, without the low-Ωm AICc improvement.
TABLE CX. Weighted-least-squares fits to the four binned
RAR-scale measurements of Ref. [109] as tabulated in Table 1
of Ref. [110], with the plotted bars read as 1σ. AICc =
χ2 + 2k + 2k(k + 1)/(N − k − 1) at N = 4. DFD’s row uses
its derived Ωm ; only the overall amplitude is free.
Model

k

χ2 /ν

Status A0Z.4 (On the one-clock C = 1 background the
improvement over Planck is not significant). A predefined
false-positive control was run: drawing Ωm at random
from [0.10, 0.50] and refitting, the probability that a random background beats DFD’s σ-distance is P = 0.426.
A ∆AICc of −1.18 on four points is therefore not
evidence in itself. What is not reproducible by chance
is the absence of freedom: see Rem. A0Z.5. Independently, applying a virialised-regime external-field correction moves the effective exponent away from the measurement, from 1.73σ to 2.27–2.76σ on the C = 1 background.
Remark A0Z.5 (DFD has no tuning freedom in γE ). Ωm is
not fitted: it is fixed by H0 and ωb h2 = 0.022273 with the
derived ωχ h2 . At the C = 1 background (H0 = 72.0897,
ωχ h2 ∈ [0.1182, 0.12086], i.e. Ωm ∈ [0.27030, 0.27542]) the
induced γE band is [1.03224, 1.04031], of width 0.0081 —
about 19 times narrower than one σ of the measurement.
MOND must assume a0 (z) and borrow E(z); DFD fixes
both from its measured clock, and therefore cannot move.
At the measured clock (H0 = 67.52, ωχ = 0.1192806;
App. ZS) Ωm ≃ 0.312 and the recomputed effective slope
is γE = 1.098. The no-tuning-freedom structure is unchanged; only the numerical advantage of the C = 1
background disappears.
3.

The zero-parameter local value

At z = 0, on bound-matter clocks (Postulate ZS.11), the
same law gives a prediction with no measured coefficient,
√
a0 (0) = 2 α cH0loc = 1.1966 × 10−10 m s−2 , (A0Z2)
against the canonical SPARC value g † = [1.20 ±
0.02 (stat) ± 0.24 (sys)] × 10−10 of Refs. [10, 16]: within
the published systematic uncertainty. No galactic parameter is adjusted anywhere in Eq. (A0Z2). On the early
clock the same law gives 1.1208 × 10−10 , −3.96σstat and
−0.33σ on the systematic uncertainty.

AICc

a0 = A (frozen)
1 28.83/3 32.83
a0 = A(1 + z)3/2
1 23.21/3 27.21
a0 = a + bz (survey’s own)
2 1.10/2 17.10
a0 = A(1 + z)γ
2 1.16/2 17.16
a0 = A E(z), Planck Ωm = 0.307 1 5.18/3 9.18
a0 = A E(z), DFD current
1 5.36/3 9.36
Ωm ≃ 0.312
a0 = A E(z), DFD C = 1
1 4.00/3 8.00
Ωm = 0.2749 (provenance)

Status A0Z.3 (Measured clock: viable, without the C = 1
AICc edge). On the measured-clock background the refit
gives A = 1.394 ± 0.032, χ2 = 5.36/3, AICc = 9.36, and
γE = 1.098, +2.1σ from the measured γ = 0.78 ± 0.15.
This is essentially the Planck-background row and remains
the preferred one-parameter description relative to the
frozen and matter-density alternatives. The lower AICc
8.00, amplitude 1.4497 ± 0.0333, and γE = 1.0395 are the
C = 1, Ωm = 0.2749 row, not a DFD advantage.

4.

The amplitude tension

Exponent-based conclusions are exactly invariant under any redshift-independent rescaling of a0 ; amplitude
conclusions are not.
Status A0Z.6 (Amplitude, both sides). Against the
measured-clock fitted amplitude A = 1.394±0.032, DFD’s
early-clock a0 (0) = 1.1208 × 10−10 lies below the survey
anchor by 8.5σstat (about 2σ counted galaxy by galaxy),
corresponding to a coherent stellar-mass shift of about
+0.095 dex. The gap is the same on either clock: the
survey’s masses are fitted to its rotation curves, so on distances computed with H0loc its amplitude and the boundmatter a0 = 1.1966 × 10−10 rise together, and the boundmatter value lies 0.094 to 0.103 dex (8.5 to 9.2σstat ) below
it (computed). The accepted survey and SPARC systematic budgets remain much wider than their statistical
bars, so this is a mass-scale-conditional comparison, not

263
an unconditional failure. The theory-free survey-versuscanonical-SPARC offset remains about 17% and 0.8σ on
SPARC’s systematic-dominated budget. By the same
currency a frozen a0 requires +0.220 to +0.354 dex per
bin, 2.3 to 3.8 times more, and a redshift tilt that no
common-mode mass rescaling supplies. The survey’s own
stated requirement to reconcile all bins with a canonical
frozen a0 is +0.2 to +0.45 dex, which its authors argue
is unsupported by independent checks. A fuller baryon
census supplies a tilt of that kind (§A0Z 6). The survey’s
stellar term is a disc mass fitted to the rotation curve;
above z ≃ 0.9 it sits about 0.4 dex below the stellar mass
from the galaxies’ spectral energy distributions, and with
those masses the refitted trend falls. With the molecular
gas the survey does not count, the trend depends on how
the gas is added: in proportion to the survey’s own disc
masses, within its stated 0.2 dex systematic, the rise stays
and DFD’s zero-parameter law fits it; tied to SED-based
masses through the scaling relations of Ref. [170], the
trend is about flat. On either census the amplitude of
Hubble tracking lies 0.02 to 0.09 dex below DFD’s earlyclock zero-parameter value (0.18 dex with both changes
together), where on the published census it lies 0.095 dex
above (computed). The published exponent is a correct
fit of the published bins; DFD is exposed to the absolute stellar-mass scale, and the measured redshift trend
is exposed to the baryon census.

5.

A cosmic ceiling on any external-field term

Theorem A0Z.7 (Cosmic external-field ceiling). Let
gbg (z) be the Newtonian acceleration at the Hubble radius
RH = c/H(z) sourced by the mean matter density inside
it,√measured in units of DFD’s own epoch scale a0 (z) =
2 α cH(z). Then
xbg (z) ≡

Ωm (z)
gbg (z)
√ ,
=
a0 (z)
4 α

(A0Z3)

in which G,
√ c and H(z) cancel identically. Consequently,
with τ = x,
p
Ωm (z)
α−1/4
= 1.710719
for all z,
τbg (z) =
≤
2
2 α1/4
(A0Z4)
with equality only in the matter-dominated limit Ωm (z) →
1.
Proof. ρ̄m (z) = Ωm (z) · 3H(z)2 /8πG and gbg =
4π
3H 2
c
1
= 4π
3 Gρ̄m RH √
3 G · Ωm 8πG · H =√2 Ωm (z) cH(z). Dividing by 2 α cH(z) gives Ωm (z)/4 α; G, c and H have
cancelled (symbolic residual identically zero, free symbols
{Ωm , α} only). Ωm (z) is monotone increasing to 1 in a
flat background, so τbg is bounded by α−1/4 /2.
Remark A0Z.8 (A ceiling, not a value). Equation (A0Z3)
is Lemma GR.3 (App. GR) evaluated at Rδ = RH ; the
identity is exact, since that lemma gives x = 2.437 × 10−3
on R8 at Ωm ≃ 0.312, whose product with RH /R8 is

xbg (0) = 0.913 (0.8045 on the one-clock C = 1 background). The saturated value is not realised. Theorem AE.1 (App. AE) gives aFRW
= 0 identically on
ext
the homogeneous background, and an independent check
forbids saturation directly: a saturated floor at z = 0
would cap ν at 1.625 (1.6887 on the one-clock C = 1
background), whereas SPARC rotation curves require ν
of 5–60. xbg is therefore an upper bound on any cosmic
external-field term at any epoch, not its value.

6.

The survey refitted galaxy by galaxy

The four bins of Ref. [109] are fits of the radialacceleration relation to the radial points of the galaxies
in each redshift quartile. The survey’s baryonic term is
a stellar disc whose mass is fitted to the rotation curve
together with a model dark halo, plus an atomic-gas term
of constant surface density; molecular gas is not counted.
The survey’s public products give each galaxy’s decomposition, so every galaxy can be rebuilt, the bins refitted
with errors counted galaxy by galaxy, and the census
changed. The rebuilt rotation curves match the release’s
own derived velocities and dynamical masses to within
0.03 dex in the median (computed).
Status A0Z.9 (The survey’s rise at galaxy level). A sample of 79 galaxies selected with the survey’s cuts gives
bins with the published shape and a level about 12%
higher, and an exponent γ = 0.77 ± 0.21 against the
published 0.78 ± 0.15 (computed). Resampling whole
galaxies instead of radial points, the bin uncertainties are
4 to 6 times the point errors, because the radial points
of one galaxy are correlated and each bin holds about 20
galaxies. At galaxy level the exponent’s 16–84% range
is −0.10 to 1.67, and within the survey a frozen a0 lies
about 1σ away, against 5.3σ on the published bin errors;
anchored to SPARC’s z = 0 value, as Ref. [110] does,
the galaxy-level bins still exclude a frozen a0 at about
3σ: on SPARC’s statistical bar the exclusion is 3.2 to
4.7σ, and DFD’s zero-parameter law fits them on the
bound-matter clock (χ2 = 3.8 to 9.0 for five points) but
not on the early clock (χ2 = 19.5 to 23.9, mostly from the
z = 0 point); on SPARC’s systematic bar the exclusion is
2.3 to 3.5σ and the law fits on either clock (χ2 = 3.8 to
9.0). This is a comparison of amplitudes across two surveys that inherits their relative mass and distance scales
(computed). The trend also depends on how the molecular gas is added. In proportion to the survey’s own disc
masses, within its stated 0.2 dex systematic, the rise stays,
γ = 1.02±0.22 on point errors, and DFD’s zero-parameter
law fits as well as the best power law (fitted amplitude
1.16 against DFD’s early-clock 1.12); tied to SED-based
masses through the scaling relations of Ref. [170] it is
about flat, γ = 0.06 ± 0.21 (−1.03 to 0.96 at galaxy level);
and with stellar masses from the galaxies’ spectral energy
distributions in place of the fitted disc masses it falls,
γ = −1.04 ± 0.22 (−2.32 to −0.32). Those SED masses
exceed the kinematic 95% limit in about a third of the

264
high-z galaxies, so the flat and falling variants are not
established. The atomic-gas term, which the data leave
unconstrained in 91% of the galaxies, moves γ between
0.5 and 1.3 over its physical range (computed). At galaxy
level the SED-based censuses prefer a frozen a0 to tracking, by about 1σ with SED-tied molecular gas and about
2σ with the SED masses (computed). Above z ≃ 0.9 the
fitted disc masses of 34 of 55 galaxies reach the lower edge
of the halo model’s prior on the stellar-to-halo mass ratio,
against 13 of 34 below, and in the two upper bins they
sit about 0.4 dex below the SED masses. The prior edge
clips the low tail of the likelihood and raises the median
disc masses by about 0.08 dex, so the offset is kinematic,
not an artefact of the prior (computed). The same pattern holds on three sample definitions. DFD’s early-clock
zero-parameter amplitude lies within 0.02–0.09 dex of the
tracking amplitude fitted on the fuller censuses (0.18 dex
with both changes together), where the published census
puts it 0.095 dex low (computed).
The published exponent is a correct fit of the published
bins. The refit adds the galaxy-level errors and the census
beneath them: on present data the survey alone does not
separate Hubble tracking from a frozen a0 ; anchored to
SPARC it excludes a frozen a0 at about 3σ, a comparison
across two surveys that DFD’s zero-parameter law fits on
the bound-matter clock. Direct molecular-gas masses for
these galaxies, which fix how the gas scales, would make
it decisive.

7.

Massive discs at 0.6 < z < 2.5

The survey of §A0Z 1 and §A0Z 6 observes low-mass
discs. Rotation curves of massive star-forming discs reach
higher redshift. For a galaxy that obeys the SOURCE law,
Eq. (BS8), the dark-matter fraction at a radius
where gobs
 p

is measured is fDM = 1 − gbar /gobs = exp − gbar /a0 ,
so each published pair of dark fraction and circular velocity at the effective radius Re returns
(1 − fDM ) gobs
,
(ln fDM )2
with no further modelling.
a0 =

gobs =

Vc2 (Re )
,
Re

Status A0Z.10 (The massive discs at face value). The 100
discs of Ref. [111], whose baryon census counts molecular
gas through the scaling relations of Ref. [170], give median
a0 = 1.9, 1.4 and 1.2 × 10−10 m s−2 at median z = 0.8,
1.5 and 2.2, where DFD’s law predicts 1.8, 2.6 and 3.7 on
the early clock and 1.89, 2.82 and 3.95 on bound-matter
clocks (computed). DFD’s value is met at z ≃ 0.8; the
rise that tracking requires beyond z ≃ 1.5 is not seen
on that census, while fits of the rotation curves that
leave the baryonic mass free neither show nor exclude
it (computed on the public data cubes). The exponent
fitted within the sample is γ = −0.86+0.46
−0.48 on statistical
errors. It is fragile: without the discs with Vrot /σ0 < 2.1,
where the disc–halo degeneracy is largest, it is −0.20+0.45
−0.40 .

DFD’s exponent over these redshifts, γE = 1.29, lies
4.5σ away on statistics alone (likelihood ratio) and about
1.7σ away once a ±30% survey-wide scale of the pressuresupport correction and a ±0.1 dex redshift tilt of the
baryonic masses are marginalised (computed; the widths
are estimates), although the discs’ own data disfavour the
calibrations that would rescue DFD, a doubled pressure
support or z ≃ 2 baryons 0.2 dex lighter, by ∆(−2 ln L) =
18–40 (computed). On the TOTAL branch, Eq. (BS9), the
z ≃ 2.2 value is 2.4, at the registered falsification threshold
of §A0Z 8. The discs sit where the rise should show: at
z ≃ 2.2 their baryonic pull at Re is below a0 (z) (median
0.5–0.9 a0 (z)), where DFD’s rise adds about 0.07 dex to
the rotation speed. On the census of SED stellar masses
and the gas of Ref. [170] they rotate 0.1–0.17 dex slower
than DFD predicts, 6–9σ on statistical errors and 1.5–2.2σ
when the census’s own ±0.2 dex calibration is counted
(computed with two mass models). Ref. [171] drew a
similar face-value conclusion from the discs of Ref. [172],
which all but exclude an a0 about four times today’s at
z ≃ 2, where DFD’s is about three times today’s.
Remark A0Z.11 (Two baryon censuses). Ref. [109] counts
a disc mass fitted to the rotation curve and a modelled
atomic-gas term, and treats molecular gas as a systematic
of about 0.2 dex; the massive discs above count molecular
gas through the scaling relations of Ref. [170]. Counted
the same way, with the gas tied to SED-based masses,
the survey’s own trend is about flat, while with the gas in
proportion to the survey’s own disc masses the rise stays
and DFD’s law fits it (Stat. A0Z.9); which scaling applies
is open. On the SED-tied census neither data set shows
the rise that tracking predicts, although at galaxy level
the survey does not exclude it (computed). The exponent
published in Ref. [110] is a correct fit of the published
bins; the census and the errors behind the bins are what
remain open. The KMOS3D baryonic Tully–Fisher zero
points [173] and the refits of 41 of the massive discs by
a second pipeline [174] share the parent sample and its
calibrations, so they are not independent confirmations.
The zero-point evolution is missed by a frozen a0 and by
Newtonian baryons as well, so it does not yet test a0 (z)
(computed).
Status A0Z.12 (Environment at fixed redshift: a prediction). Under the domain-local screen of App. ZT, a
galaxy’s a0 is set by the expansion rate of its own domain,
and an overdense domain lowers it (Cor. ZT.5, with the
declared exact closure of Status ZT.8). Massive discs at
z ≃ 1–2 sit in strongly clustered haloes. In domains of
RD = 10 h−1 Mpc, the smallest the local test allows for
the linear coupling, the screen lowers their a0 by about
10% at the average environment of such haloes, to 3.3
at z ≃ 2.2 against the measured 1.2 (estimate). Reaching the measured values needs δenv ≃ 2 at z ≃ 1.5 and
≃ 3 at z ≃ 2.2. The matter overdensity on that scale
has a standard deviation of 0.34 at z ≃ 1.5 and 0.27
at z ≃ 2.2, so such domains are fluctuations of about 6
and 11 standard deviations (3.5 and 5.3 for a lognormal
field) and are essentially absent (computed). The densest

265
structures known at these redshifts, the z = 1.61 cluster
Cl 0332−2742, the Hyperion proto-supercluster and the
LATIS protoclusters, have matter overdensities of about
0.8–1.3 on 10 h−1 Mpc, which lower a0 by 20–34%, not
by the factors of 2–20 that the discs at z ≃ 2.1–2.5 show
(computed on published values). The measured environments agree: in the environment catalogue of the same
deep fields (Fossati et al. 2017; local density in 0.75 Mpc
apertures), the a0 inferred for 46 of the massive discs does
not correlate with local density at fixed redshift (rank correlation −0.10), and at z ≃ 1.8–2.6 discs in dense and in
less dense surroundings show the same deficit (computed
on published values). The screen therefore does not reconcile the massive discs beyond z ≃ 1.5; at z ≃ 0.8, where
DFD’s global value is already met, it needs domains no
denser than average. At RD = 20 h−1 Mpc every shift is
about three times smaller. The low-mass discs of §A0Z 6,
in less clustered haloes, are lowered by 6–7% at their
average environment, which is small beside the census effects found there (estimate). The prediction stands and is
falsifiable: at fixed redshift, a0 in high-redshift discs falls
as the density of their 10 h−1 Mpc domain rises, by about
10% at the average environment of massive haloes and by
up to about a third in the densest known structures. The
test sets matter-density maps of the deep fields on that
scale against per-galaxy a0 ; the kinematics already exist.
Status A0Z.13 (What decides the high-redshift test). The
z ≃ 2.2 value is set by two calibrations. It reaches the
registered line a0 (z=2) = 2.5 if the z ≃ 2 baryonic masses
are about 0.1 dex lower than those fitted in Ref. [111]
(about 0.2–0.3 dex in molecular gas), and DFD’s value
if they are about 0.2 dex lower (about 0.45 dex in gas);
against the census of SED stars and scaling-relation gas
DFD’s value needs them 0.3–0.5 dex lower (fits of the
public rotation curves with the baryonic mass free also
put the masses below the census, at DFD’s a0 and at a
frozen one alike; computed); or if the pressure-support
correction is 1.8 or 2.4 times the standard self-gravitatingdisc correction. No published prescription is that large:
the others in use are smaller at Re , and DFD’s own
vertical gravity makes the correction about 10% smaller
(computed; the gas equivalents and the vertical-gravity
factor are estimates). The external field of DFD’s own
large-scale structure is nearly constant in units of a0 (z)
and does not supply the difference (estimate), and neither
does the domain-local screen, whose required domains are
absent from the measured environments (Stat. A0Z.12).
Three kinds of data decide the test for both samples:
direct molecular-gas masses (CO, [C i] or dust), to 0.1 dex
for the z ≃ 2 discs; kinematic masses not limited by a halomodel prior, with one pressure-support correction for all
samples; and per-galaxy analyses with galaxy-level errors,
of the MUSE–DARK public products and of the massive
discs, with a0 (z) imposed in the fit; for the massive discs,
fits that leave the baryonic mass free need no census,
but inside 3Re a heavier χ halo or a stronger pressure
correction mimics the rise, so they decide it only with
an independent bound on the χ halo or rotation curves

beyond 3Re (computed on the public data cubes).
8.

Dated falsifiers, registrable now

TABLE CXI. a0 (z) in 10−10 m s−2 . DFD’s zero-parameter column carries no fitted amplitude; it is the epoch law evaluated
at the C = 1 registration background (the provenance row
of Table CX), retained unaltered as the dated registration,
and the amplitude column carries the registration amplitude
A = 1.4497. Bound-matter re-evaluations are given in the
registered-predictions box below.
z

DFD DFD measured A EPl survey
(0-par) (×Afit ) γ = 0.78
linear

1.0 2.05
1.5 2.68
2.0 3.42
2.5 4.23
3.0 5.12
4.0 7.09

2.48
3.25
4.14
5.13
6.21
8.59

2.46
2.93
3.38
3.81
4.23
5.03

2.49
3.28
4.20
5.22
6.32
8.76

2.59
3.39
4.18
4.98
5.77
7.36

Status A0Z.14 (Registered predictions). At its C = 1
registration background and with no fitted amplitude,
DFD predicts a0 (z=2) = 3.42 and a0 (z=3) = 5.12
(units 10−10 m s−2 ); with the registration amplitude imposed, 4.14 and 6.21. [These are the values as registered. Re-evaluated on bound-matter clocks (Eq. (A0Z2),
Ωm ≃ 0.312, distances computed with H0loc ), the zeroparameter values are 3.61 and 5.44, and 4.49 and 6.75 at
A = 1.486, the survey amplitude on the same distances;
the falsification windows below are unchanged and both
routes lie inside them.] The measured exponent family
extrapolates to 3.38 and 4.23, and the survey’s linear
form to 4.18 and 5.77. The forms separate by about
50% at z = 3 and 75% at z = 4. Ref. [110] makes
the same argument — that the viable functional forms
diverge rapidly beyond the measured range, so that z ≳ 2
kinematics discriminate among them — and the columns
above are this corpus’s own evaluations of each form on
DFD’s background and amplitude conventions. Kills: a
measured a0 (z=2) < 2.5 or > 5.5, or a0 (z=3) < 3.5 or
> 7.5, falsifies the DFD epoch law on either amplitude
convention, provided the measurement counts molecular
gas in its baryon census, states its pressure-support correction and carries galaxy-level errors. On their published
calibrations the massive discs of §A0Z 7, which meet these
three provisos, give a0 ≃ 1.2 at z ≃ 2.2, below the z = 2
line at face value (2.4, at the line, on the TOTAL branch);
the line is reached if their baryonic masses are about
0.1 dex lower or their pressure-support correction is 1.8
times the standard one (Stat. A0Z.13); the domain-local
screen gives values this low only in domains about 6 to
11 standard deviations denser than average, which the
measured environments do not show (Stat. A0Z.12). Instruments: rotation curves of massive discs already reach
the regime at z ≃ 0.6–2.5 [111]; JWST and ALMA kine-

266
matics extend it to z ≳ 3, and SKA H i kinematics to
the outer discs. The census, the error treatment and the
environment decide the test, not the reach (Stat. A0Z.13).

Appendix AF: Galaxy Bias in DFD: a
Frame-Consistent Derivation from the σ8 Closure

This appendix addresses the apparent “factor of 80
galaxy bias problem” that arises when the σ8 closure of
Appendix AE is read naively. The closure equation
Ωb δb Q(δb ) = Ωm σ8,obs

(AF1)

implies δb = 0.0104 at R8 versus σ8,obs = 0.811, a ratio
F8 ≡ σ8 /δb = 77.9. A direct concern is that matching
observed galaxy clustering then requires bias b ∼ 146
relative to baryons for LRG-like tracers, factors of ∼78
larger than published LCDM bias values bLCDM ∼ 1.9.
The resolution is that DFD permits two equally valid
bias-frame conventions related by F8 , fixed by the closure
with zero free parameters. The “bias multiplier” is not
a free parameter; it is a derived consequence of the closure equation. Galaxy formation in the well-depth frame
produces the same numerical bias as LCDM; the barebaryon-frame number is larger by F8 purely by definition.
Like the closure it starts from, this appendix belongs to
the baryon-only construction of Appendix AE, in which
the baryons are the only clustering matter; it is not DFD’s
physical growth carrier. That carrier is the cold χ relic
with Qχ = 1 under the adopted Rest-Mass Channel,
and unbound linear perturbations grow with Geff = G
(App. GR). The bias frames, the multiplier F8 and the
falsifiers of Sec. AF 5 are statements about the construction.

1.

AF.1

Setup: bias-frame ambiguity in modified
gravity

In LCDM, galaxy bias b is defined as the ratio
Pgg (k)
b2 =
,
Pmm (k) ≡ ⟨|δm (k)|2 ⟩,
(AF2)
Pmm (k)
where δm is the total-matter density contrast at scale 1/k.
In LCDM, total matter is dominated by cold dark matter;
δm ≃ δCDM on linear scales, with σ(δm , R8 ) = σ8 .
In the construction the baryons are the only clustering
matter, so the analogous definition admits two natural
choices for the reference field:
(B)

B-frame (bare baryons):: Pmm (k) = ⟨|δb (k)|2 ⟩ where
δb is the baryon density contrast.
(W )

W-frame (well-depth / convergence):: Pmm (k) =
⟨|Ωb δb (k)Q[δb ]/Ωm |2 ⟩ where Q[δb ] is the deepMOND amplification factor of Eq. (AF1).
The amplitudes of the two reference fields differ by
exactly F8 at scale R8 :
(W )

σ(Pmm , R8 )
(B)
σ(Pmm , R8 )

=

(Ωb /Ωm ) Q δb
Ωb Q
=
= F8 ≃ 77.9,
δb
Ωm
(AF3)

267
by the σ8 closure Eq. (AF1). Both frames are internally
consistent; they describe the same physical universe with
different choices of normalization.
Galaxy bias takes correspondingly different values in
each frame:
s
s
Pgg
Pgg
(W )
(B)
,
b
≡
,
b(B) = F8 b(W ) .
b
≡
(B)
(W )
Pmm
Pmm
(AF4)

2.

AF.2

Galaxy formation operates in the
W-frame

field:
b(ν) = 1 +

ν2 − 1
,
δc

where δc ≃ 1.686 is the linear-theory spherical-collapse
threshold. Since (i) galaxy formation depends on the
well-depth field, (ii) the well-depth field has σ = σ8 in
both LCDM and DFD by closure, and (iii) the astrophysical threshold (cooling, fragmentation) is fixed by atomic
physics (epoch-invariant in DFD per Sec. XIX), the value
of ν at which a given tracer is selected is the same in both
cosmologies.
Therefore:
(W )

bDFD (ν) = bLCDM (ν).
Galaxy formation requires gas to fall into a gravitational
potential well, cool radiatively, fragment, and form stars.
The relevant physical input is the local gravitational well
depth, not the bare matter density:

AF.3

• Cooling time ∝ ρ−1 T 1/2 Λ−1

• Star formation efficiency ∝ tcool /tff
where ρwell is the effective mass density sourcing the
gravitational acceleration. In LCDM, ρwell = ρmatter
(predominantly CDM). In DFD, ρwell is the effective
matter-equivalent density that produces the observed
gravitational acceleration; on R8 scales, this is precisely (Ωb /Ωm ) Q δb times ρcrit Ωm , identical to LCDM’s
Ωm δm ρcrit by the closure Eq. (AF1).
a. Consequence: same astrophysics, same threshold.
The peak-bias relations of Mo & White [175] and ShethTormen [176] relate galaxy bias to the height ν at which
a tracer is selected relative to the rms of the underlying
a.

Numerical predictions.

Theorem: DFD galaxy bias from the
closure

Theorem AF.1 (Galaxy bias from σ8 closure). Under the postulates of DFD, the closure of Appendix AE
(Theorem AE.1 and Eq. (AF1)), and the W-frame galaxy
formation principle Eq. (AF6), the linear galaxy bias of
any tracer in the bare-baryon frame is
Ωb Q(δb )
(B)
bDFD = F8 bLCDM ,
F8 =
= 77.9, (AF7)
Ωm
on R8 scales, with F8 fixed by the closure and no free
parameters.
Proof. By Eq. (AF4), b(B) = F8 b(W ) in any cosmology
(W )
with the two-frame structure. By Eq. (AF6), bDFD =
bLCDM since galaxy formation responds to the well-depth
field whose statistical amplitude σ8 is matched by construction in both cosmologies. By Eq. (AF3) and the
closure Eq. (AF1), F8 = (Ωb /Ωm )Q is determined to the
value 77.9 at the closure point.

For the survey tracer set, the DFD W-frame and B-frame biases are:

Tracer
BGS / low-bias
ELG-like
DESI/BOSS LRG
LRG high-bias
eBOSS QSO z ∼ 1.5

(W )

(B)

bDFD = bLCDM bDFD = F8 bLCDM σgalaxy
1.20
1.50
1.88
2.10
2.40

The W-frame values match published LCDM bias measurements; the B-frame values are equivalent under
Eq. (AF4) but referenced to bare-baryon density con-

(AF6)

For LRG-like tracers with observed bLCDM ≃ 1.88, the
DFD W-frame bias is also 1.88.

3.

• Free-fall time ∝ (Gρwell )−1/2

(AF5)

93.5
116.8
146.4
163.6
187.0

trast.

0.97
1.22
1.52
1.70
1.95

268
4.

AF.4

Cross-frame consistency: what each probe
measures

5.

AF.5

Falsifiable predictions

Galaxy Bias — Falsifiers
Different observational probes naturally measure bias
in different frames. The construction is consistent if and
only if the same F8 = 77.9 emerges across all such probes.
Galaxy clustering (auto-correlation):: Pgg (k) measured directly. Bias inferred via b = σgalaxy /σ8fid
where σ8fid is the fiducial cosmology’s σ8 amplitude.
This implicitly uses the W-frame.
Galaxy-galaxy lensing (GGL):: Cross-correlation
⟨γt δg ⟩ between background-galaxy shear γt (which
traces κDFD ∝ Qδb ) and foreground galaxy density.
Amplitude ∝ b(W ) × σ82 . W-frame native.
Galaxy-CMB lensing cross-correlation:: Planck
CMB lensing κCMB traces the same κDFD field
via the high-redshift integrated optical depth.
Cross-correlation amplitude ∝ b(W ) × σ82 . W-frame
native.
Lyman-α forest cross-correlation:: The Lyα flux F
traces baryon overdensity directly: δF ∝ −A δbβF for
fluctuating-Gunn-Peterson. The cross-correlation
⟨δg δF ⟩ amplitude is then ∝ b(B) × σb × σF – Bframe native. The QSO-Lyα forest cross-correlation
from eBOSS DR14 [177] can in principle test b(B)
directly.
kSZ amplitude:: The kinetic Sunyaev-Zel’dovich signal traces the integrated baryon
√ momentum ρb vpec .
From Sec. AE 10, vpec ∝ δb in deep MOND.
ACT/SPT/DESI kSZ measurements [100, 178] provide a partial test.

1. Universal F8 . The conversion factor
F8 = σ8 /σb = 77.9 must be the same for all tracer
types (BGS, ELG, LRG, QSO) and at all redshifts.
Cross-survey measurement of F8 at ±10%
inconsistency level falsifies the construction.
2. Lyman-α baryon amplitude. At z = 2–5, the
linearly extrapolated baryon density contrast from
Lyα flux power spectrum should be
σb (R8 , z) = D(z)σb (R8 , 0) ∼ 0.002–0.004, vastly
smaller than LCDM’s σm (R8 , z) ∼ 0.17–0.34.
Measurement of σb ∼ σ8 in Lyα forest at this scale
falsifies the construction.
3. kSZ amplitude. As in Sec. AE 11, the
baryon-only construction predicts a kSZ amplitude
about 0.9
√ of LCDM’s at R8 scales, from its
vpec ∝ δb scaling; a detection of
LCDM-amplitude kSZ at high precision falsifies the
construction. DFD’s growth carrier, the cold χ
relic with Qχ = 1, predicts no such suppression on
linear scales (App. GR).
4. Scale dependence of F . On scales other than
R8 , F varies as F (R) ∝ R−1/4 (deep MOND with
neff ∼ −2). Strong deviation in observed
scale-dependent bias ratio (B-frame vs W-frame)
from this prediction falsifies the construction.

6.

AF.6

Scale dependence

The conversion factor F8 was derived at the sphericaltop-hat
scale R8 = 8h−1 Mpc using Q(δb ) = (1 +
p
1 + 4/q)/2 in the deep-MOND regime where q =
gN /a0 ≪ 1. On scales R ̸= R8 , the deep-MOND boost
varies:
1
1
Q(R) ≃ p
=p
.
q(R)
(Ωb /2)(HR/c)δb (R)/(a0 /cH)
(AF8)
Using δb (R) ∝ R−(n+3)/2 for the linear baryon spectrum
(post-recombination) and neff ≃ −2 at R8 scales:
Q(R) ∝ R−1/4 ,

F (R) ∝ R−1/4 .

(AF9)

Numerically:
• At R = 100 Mpc (k = 0.01 h/Mpc): F ≃ 46
• At R = R8 = 11.85 Mpc (k = 0.1 h/Mpc): F = 77.9
• At R = 1 Mpc (k = 1 h/Mpc): F ≃ 145
This is a mild scale-dependence: F varies by about a
factor 3 across two decades of scale, consistent with approximately scale-independent linear bias on quasi-linear
scales as observed in BOSS/eBOSS/DESI.

269
7.

AF.7

Open extensions and limitations

This appendix closes the linear-bias question on R8
scales: F8 is a derived quantity, not a free parameter.
Several extensions remain program-grade:
1. Spherical-collapse threshold δcDFD . The peakbias formula Eq. (AF5) uses δc = 1.686 from LCDM
spherical-collapse dynamics. DFD’s deep-MONDamplified collapse may give a different δcDFD . The
W-frame argument Eq. (AF6) is robust to this
because it operates on the operational definition
b = σg /σ8 , but a derivation of the halo mass function in DFD requires solving the collapse problem.
2. Halo bias and assembly bias. Mass-dependent
bias b(Mhalo ) and assembly-bias dependencies on
local environment are well-developed in LCDM Nbody simulations but require DFD implementation.
The framework here gives the leading-order behavior; mass-dependent corrections are program-grade.
3. Non-linear bias. On scales k > 0.2 h/Mpc, nonlinear bias terms b2 , bs2 , b3 become important
and require perturbation-theory or simulation-based
modeling.
4. Velocity bias and FoG. Galaxy peculiar velocities
can differ from the underlying matter velocity field,
introducing additional “velocity bias.” This is small
in LCDM (∼ 1%) but unverified in DFD.
5. Direct b(B) measurement. The B-frame bias
prediction b(B) ∼ 146 for LRG has not been directly
measured. Galaxy-Lyα cross-correlation analyses
currently use LCDM-fiducial cosmology; a DFD
re-analysis would test the prediction.

8.

AF.8

Summary

Galaxy Bias in the Construction — Status
Within the construction:
• Two-frame structure of DFD galaxy bias:
b(B) = F8 b(W ) with F8 = 77.9 from σ8 closure.
(W )
• W-frame bias matches LCDM: bDFD = bLCDM
because galaxy formation responds to the
well-depth field whose amplitude is σ8 in both
cosmologies.
• No free parameters: F8 fixed by closure with zero
degrees of freedom.
• Falsifiable cross-probe consistency: galaxy-galaxy
lensing, galaxy-CMB lensing cross-correlation, kSZ
amplitude, and Lyman-α forest auto-correlation
must all yield consistent F8 = 77.9.
Still program-grade:
• DFD spherical-collapse threshold δcDFD for halo
abundance matching.
• Mass-dependent halo bias b(M ) and assembly bias.
• Non-linear bias terms in the perturbation-theory
expansion.
• Direct b(B) measurement via galaxy-Lyα
cross-correlation in DFD-fiducial framework.
Resolution of Open Theorem Obligation #4
(Sec. AE 12): The “galaxy-bias derivation” is closed at
the level of linear bias on R8 scales. The ∼80× multiplier
between observed galaxy clustering and bare-baryon
density contrast is a derived consequence of the σ8
closure, not a free parameter. Higher-order extensions
(scale-dependent bias, halo-bias function, assembly bias)
remain program-level work.

270
Appendix AG: Master Claims Confrontation Matrix

This appendix consolidates every quantitative empirical claim in this monograph into a single catalog. The
intent is to let a referee or independent reader assess at a
glance which claims are theorem-grade, which are derived
under stated auxiliary assumptions, and which are conjectured or empirical-fit-grade; what each claim predicts;
what is observed; the current agreement; and the explicit
threshold that would falsify the claim.
This appendix introduces no new physics. Every entry
cross-references a theorem, derivation, table, or section
already present in the monograph.

1.

Tier Definitions

T (Theorem):: Result follows from a formal proof using
only the core DFD axioms (n = eψ , a = (c2 /2)∇ψ, the
derived µ-response of App. N/App. BS, with µ = x/(1+x)
its weak-response form) plus stated mathematical lemmas.
D (Derived):: Result follows from the core axioms plus
an explicit auxiliary closure framework whose assumptions
are displayed (spectral action, Chern-Simons truncation
kmax = 60, manifold X = CP 2 × S 3 , generation count
Ngen = 3, PDE-validated numerical results).
C (Conjectured):: Result is plausible from DFD principles but not yet derived from the core axioms or a
fully-displayed closure framework. Open program items.
E (Empirical):: Result is a phenomenological match
using physically-motivated correction factors at literaturebounded values, where the closure depends on those corrections.
validated (Validated):: An observational test that has
already been performed and is consistent with DFD at
stated σ.
pending (Pending):: An observational test where the
experimental capability exists but no published measurement yet decides the claim.
contested (Contested):: An observational test that
has been performed, where the published measurements
disagree at face value, some consistent with DFD and
some not, and the verdict turns on how the data are
calibrated and how their errors are counted.
open (Open):: A predicted observable for which no comparable measurement currently exists.
The group entry uses the E label, here for an amplitude
read from the data it is compared with; the cluster entry’s
amplitude follows from the derived abundance and ΛCDMlike assembly (estimate; Appendix I). The galactic entries
of Block 3 are the µ-law on the baryons; with χ present
they hold under the floor composition when the ΛCDMlike χ halos of the adopted energy are cored, which is

an estimate (App. AV). The tier-grading rule is that a
claim’s tier is set by its weakest link, not its strongest.
2.

Block 1: Fundamental Constants and Microsector

• Fine-structure constant α−1 from microsector
closure (Sec. X). DFD: the closed form, exact under two stated choices, gives α−1 = 137.03599985;
the integers alone give 137.03307 (21 ppm), and the
bracket coefficient c = 1/13 brings it to 0.0049 ppm, 32
times the uncertainty of the CODATA 2022 value. SM:
free parameter. Observation: 137.035999177(21) (CODATA 2022). Agreement: 0.0049 ppm with the stated
c = 1/13, inside the 0.1 ppm falsifier band, which is
this row’s stated level. Falsifier: |∆α−1 | > 0.1 ppm
not from QED loops. Tier: D (coefficient provenance
open, Rem. X.4; the identification of the hyperchargetrace coefficient with 1/e2 is not derived, §X I). Status:
validated .
√
• Higgs VEV from α8 2π scaling (App. Z). DFD:
v = 246.09
√ GeV; the exponent 8 is derived and the
prefactor 2π is fitted, with a no-go on deriving it. SM:
free parameter. Observation: 246.22 GeV. Agreement:
0.05% with the fitted prefactor. Falsifier: |∆v/v| > 1%.
Tier: D (exponent). Status: validated .
• Strong CP: θ̄ = 0 at tree level, loop-bounded
(App. G 5). DFD: θ̄ = 0 at tree level, with θYM = 0
fixed by the spectral action given that nothing is added
to it for the gauge fields, loop-induced ≲ 10−16 . SM:
needs axion or fine-tuning. Observation: |θ̄| < 10−10
(nEDM). Agreement: six orders of magnitude below the
bound. Falsifier: detection of |θ̄| > 10−15 from QCD
alone. Tier: D (conditional on nothing being added
to the spectral action for the gauge fields, and on the
det-orthogonal CP offset, Postulate Y.10 of App. AO,
on the branch of the offset connected to ξ = 0). Status:
validated .
• Fermion mass hierarchy
√ (9 masses) (Sec. XIX).
DFD: mf = Af αnf v/ 2 with integer nf , 1.42%
leading-order mean error; the top and charm are derived,
and the other seven species rest on seven data-selected
prefactors (App. FM). SM: 9 free parameters. Observation: PDG values. Agreement: 1.42% mean (leading
order). Falsifier: individual mass off by > 10% for any
f . Tier: D. Status: validated .
• Weinberg angle from gauge partition (App. Z 1).
DFD: sin2 θW = 3/13 = 0.2308. SM: free parameter. Observation: 0.23129(4) (MS, MZ ). Agreement: 0.23%; 3/13 is the bare angle, and its radiative
relation to MZ is not computed (App. Z). Falsifier:
|∆ sin2 θW / sin2 θW | > 1% at tree level. Tier: D. Status: validated .
• CKM Cabibbo angle from CP 2 cohomology
(App. AO). DFD: λ = 31α = 0.2262. SM: free pa-

271
rameter. Observation: 0.22501(68) (PDG 2024). Agreement: 0.54% (1.8σ). Falsifier: |∆λ| > 0.005. Tier: D
(given the selected count of Table CXXVIII). Status:
validated .
p
• Gatto–Sartori–Tonin closure |Vus | p
=
md /ms
(App.
AY, Thm AY.1). DFD: |Vus |/ md /ms =
p
312 α/7 = 1.0009 (i.e. ms /md = 1/(7α) = 19.58).
SM: unexplained. Observation: ms /md = 19.9(±1)
(FLAG). Agreement: 0.09% closure, < 0.5σ on ms /md .
Falsifier: ms /md outside [17, 23]. Tier: D (given the
selected count of Table CXXVIII and the data-selected
prefactors Ad , As , App. FM). Status: validated .
• TM1 solar mixing angle (App. AY, Thm AY.3).
DFD: cos2 θ12 cos2 θ13 = 23 exact, sin2 θ12 = (1 −
9α)/(3(1 − 3α)) = 0.3184. SM: free parameters. Observation: sin2 θ12 = 0.307(+12
−11 ) (NuFIT 6.0); JUNO’s
first result 0.3092 ± 0.0087. Agreement: +0.95σ and
1.1σ. Falsifier: sin2 θ12 ∈
/ [0.30, 0.34] or non-maximal
θ23 . Tier: D. Status: validated .
• PMNS matrix structure (Sec. XIX). DFD: trimaximal TM1 (App. X); sin2 θ13 = 3α a stated conjecture.
SM: free parameters. Observation: global fit values.
Agreement: θ23 = 45◦ , in soft ∼2σ tension with NuFIT 6.0 (octant unresolved). Falsifier: off-diagonal
element > 20% off. Tier: D. Status: validated .
• Hubble–Planck closure: √ GℏH02 /c5 = Cα57
(App. O). DFD: H0 /MP = C α28.5 , exponent derived and coefficient measured (the vacuum coefficient R = ΩΛ C = 0.6042 ± 0.0027; today’s earlyclock value C = 0.87784 ± 0.00266 from BAO+θ∗
and 0.8772 from plik-lite), giving the cross-channel
H0early = 67.52 ± 0.09 km/s/Mpc concordant across five
CMB-anchored channels. ΛCDM: free quantity, no theoretical prediction. Observation: Planck 67.36 ± 0.54;
SH0ES 73.49 ± 0.93 (2025). Agreement: 0.3σ from
the acoustic/CMB determination; local probes read
the bound-matter clock, H0loc = α28.5 /tP = 72.09
(Postulate ZS.11), 0.2σ from the Cepheid-free mean
71.87 ± 0.96 and 1.5σ below SH0ES 2025. Falsifier:
bright sirens or mass-sheet-robust time delays settling
at 67.5, or early-ruler and local ladders converging with
distance (Falsifier ZS.14). Tier: D (exponent); coefficient measured. Status: validated on the acoustic side;
the local side rests on the stated two-clock postulate, at
the cost of a laboratory-unit age of 12.92 Gyr against
ages of the universe of 13.5–13.8 Gyr inferred from globular clusters with a ΛCDM formation delay (2.1–2.6σ
with statistical and systematic errors in quadrature, one
programme).
• Cosmological constant scale: (H0 tP )2 = Cα57 .
DFD: ρΛ /MP4 ∼ 10−122.7 from the α57 exponent
(10−122.9 with the measured normalization). ΛCDM:
“10122 problem”. Observation: matches observed ΩΛ H02 .
Agreement: exponent-level identity; the measured O(1)

prefactor of the observed density enters via the clock
dictionary (Rem. AP.6), and on bound-matter clocks
the closure scale (3/8π)α57 carries none (App. AU) and
the identity is microsector-scoped (Rem. AU.3). Falsifier: the exponent 57 fails to match the hierarchy. Tier:
D for the exponent; C measured. Status: validated .
• Clock coupling: kα = α2 /(2π) (App. P). DFD:
kα = 8.5 × 10−6 (unscreened); λα ≈ 2.1 × 10−8 (pure-α
residual, before screening). SM: 0 (LPI exact). Observation: the Yb+ E3/E2 gravitational-LPI limit has
improved to |(c2 /α) dα/dΦ| ≲ 3 × 10−9 (Filzinger et
al. 2023, −2.4(3.0) × 10−9 ; superseding the earlier
3.2×10−8 ). Agreement: consistent; screened at the local
acceleration the residual is 2λα Σ(⊕) ≈ 1.5 × 10−13 per
unit ∆Φ/c2 , four orders below this bound, which does
not test it. Falsifier: a same-ion E3/E2 gravitationalLPI null below the screened residual. Tier: D. Status:
open.
• Majorana scale: MR = MP α3 (App. P). DFD:
MR ≈ 4.7 × 1012 GeV. SM: free seesaw parameter.
Observation: not separately observable. Falsifier: mν
pattern inconsistent with MR α3 structure. Tier: D.
Status: open.
• Proton stability: τp = ∞ (App. G 7). DFD: exact (no
decay channel for a single nucleon), if baryon number is
violated only by the electroweak anomaly, which changes
it only in steps of three. DFD’s black holes radiate at
Hawking’s rate (Cor. AP.36), so a gravitational violation
is not excluded; its rate is not computed, and a dimensional estimate gives τp ≳ 1042 yr, beyond Hyper-K.
The soliton reading below is not used. On that reading
baryon number is the internal winding integer, which is
conserved while the winding field stays continuous. By
Derrick’s scaling argument an energy with at most two
derivatives, which is all the printed action gives that
field, has no finite-size minimum in the unit-winding sector, and such textures can collapse and unwind in finite
time (Shatah 1988; Turok and Spergel 1990). A quartic
(Skyrme-type) term prevents the collapse, and with it
a minimum-energy unit-winding soliton exists (Esteban 1986; Kapitanski and Ladyzhenskaya 1983). The
printed action does not contain that term, so the soliton
is posited, not derived. SM: the same anomaly selection
rule. Observation: Super-K bounds. Agreement: exact.
Falsifier: p → e+ π 0 detection with τp < 1040 yr (a discriminating test against GUT-type ∆B = 1 mediators).
Tier: D (conditional). Status: pending.
3.

Block 2: Gravitational Tests (PPN, Pulsars, GW,
Strong-Field)

• PPN γ (Sec. IV). DFD: γ = 1 exactly. GR: γ = 1.
Observation: Cassini 1.000021(23). Agreement: exact.
Falsifier: |γ − 1| > 10−5 . Tier: T. Status: validated .
• PPN β (Sec. IV, Padé Thm AA.1). DFD: β = 1

272
exactly. GR: β = 1. Observation: LLR β − 1 =
(−4.5 ± 5.6) × 10−5 [121]. Agreement: exact. Falsifier:
|β − 1| > 10−3 . Tier: T. Status: validated .
• All 10 PPN parameters. DFD: match GR (the
preferred-frame set to within the bounded residual of
Thm IV.1). GR: baseline. Observation: all bounds satisfied. Agreement: within all bounds (preferred-frame
residual ≲ 4 × 10−8 ). Falsifier: any PPN parameter
differs from GR at solar-system precision. Tier: T.
Status: validated .
• Mercury perihelion advance. DFD: 42.98′′ /century.
GR: 42.98′′ /century. Observation: 42.98(4)′′ /century.
Agreement: exact. Tier: T. Status: validated .
• Shapiro time delay. DFD: matches GR. GR: matches
GR. Observation: Cassini consistent. Agreement: exact.
Tier: T. Status: validated .
anomalous
quadrupole
Q2
• Solar-system
(App. NL). DFD: Q2 = 0 exactly: under the
environmental-confinement axiom the Sun’s own field is
Newtonian inside the Galaxy (App. NL), and the action
completion also gives Q2 = 0 at first order (Thm NL.8).
GR: 0. Observation: DE440 (1.6 ± 1.8) × 10−27 s−2 [30];
Cassini (3 ± 3) × 10−27 [29]. Agreement: −0.89σ
(DE440), −1.00σ (Cassini). Falsifier: a confirmed
|Q2 | > 10−29 s−2 of MOND-like origin. Note: the
local per-point reduction of the same response gives
30.6–31.8 × 10−27 at the bound-matter-clock a0 , i.e.
+16.1 to +16.8σ (DE440), and is excluded; the same
obstruction gives MOND itself an 8.7σ RAR-vsquadrupole tension [179]. Tier: T (obstruction) + D
(value). Status: validated .
sky-orientation
modulation
• Wide-binary
(App. NL, Falsifier NL.20). DFD: exactly 0% even
(cos2 θ) modulation of the force boost at all separations
and mass ratios, by Thm NL.10 (vobs /vN inherits
≲ 7 × 10−4 , zero for equal-mass pairs); the odd cos θ
term is ∝ (m1 − m2 ) and vanishes for equal-mass pairs;
local pairs, all inside the Milky Way’s frozen zone,
show no modulation of either parity and no boost
(Axiom NL.14). MOND/AQUAL: 6.9% peak-to-peak
at 104 AU, 46.6% at the gint = gext resonance. Observation: not yet performed; Gaia eDR3 has 1.1 × 106
pairs at > 99% bound probability [53]. Agreement: —.
Falsifier: any detected modulation, even or odd, or any
boost, in local pairs; any even modulation in any pair.
Tier: T. Status: open.
• Oort spike: a Newtonian Sun (App. NL, Status NL.12). DFD: under Axiom NL.14 the Sun’s own
field is Newtonian; a 1-Gyr integration of 2 × 104
comets in the Galactic tide gives a median original
1/a = 4.0 × 10−5 AU−1 , 77% in [0, 0.75] × 10−4 , and
no hyperbolic originals; every printed reading without
the axiom fails, with 46–100% hyperbolic originals or
an energy budget up to about eighteen times the al-

lowance. MOND: excluded (Vokrouhlický et al. 2024).
Observation: CODE, centroid 3–4 × 10−5 AU−1 , 4.6%
hyperbolic. Agreement: consistent (a postdiction: the
axiom was stated to pass this test). Falsifier: a confirmed wide-binary boost near the Sun. Tier: D. Status:
validated .
• Lensing mass = dynamical mass at every acceleration, except for bodies in weak hosts (App. NL,
Cor. NL.17). DFD: identical to O(v 2 /c2 ), as in GR,
by the one-function metric at first post-Newtonian order; bodies in weak hosts also feel the completed law’s
reciprocal force, which light does not, and for them
the two differ by it. RAQUAL: differs (its conformal
coupling leaves lensing on the Einstein-frame potential).
TeVeS: equal only by adding a vector field. Observation:
strong- and weak-lensing mass–dynamics comparisons
consistent. Falsifier: a confirmed lensing/dynamical
mass ratio ̸= 1 at fixed radius in a relaxed system. Tier:
T. Status: validated .
• Light deflection. DFD: reproduces the leading (1PN)
GR deflection in the declared physical branch. GR:
matches GR. Observation: VLBI consistent. Agreement: exact at 1PN; the static 2PN deflection coefficient is GR’s 15π/4 (§IV; App. AP, Thm AP.39). Tier:
T (1PN). Status: validated .
• Frame dragging (Lense-Thirring). DFD: matches
GR. GR: matches GR. Observation: Gravity Probe
B consistent. Agreement: exact. Tier: T. Status:
validated .
• GW propagation speed: cT = cγ exactly (Sec. V;
App. WS). DFD: cT = cγ on every background: light
and gravitational waves share one metric. GR: cT = c.
Observation: GW170817: |cT /c − 1| < 10−15 . Agreement: exact. Falsifier: any TT–EM cone split. Tier: T.
Status: validated .
• GW170817 tidal deformability and EM counterpart timing. DFD: matches GR. GR: matches GR.
Observation: consistent with GR + DFD. Agreement:
exact. Falsifier: inconsistency with GR-equivalent at
LIGO sensitivity. Tier: T. Status: validated .
• ppE GW phase deviations. DFD: all deviations
zero at leading order. GR: all zero. Observation: all
bounds satisfied. Agreement: exact. Falsifier: nonzero
δ φ̂b in any GW band. Tier: D. Status: validated .
• Binary pulsar orbital decay (App. E 2). DFD:
matches GR (B1913+16, J0737-3039). GR: matches
GR. Observation: all consistent. Agreement: exact.
Falsifier: a decay rate that departs from the GR value
by more than 3σ of its measurement error for any pulsar.
Tier: D. Status: validated .
• No-wormhole theorem (App. R, Thm AA.2). DFD:
no traversable wormholes. GR: allows with exotic matter. Observation: —. Agreement: exact theorem. Fal-

273
sifier: exhibition of DFD wormhole solution. Tier: T.
Status: open.

4.

Block 3: Galactic Dynamics

• µ(x) = x/(1 + x) from S 3 composition (App. N,
Thm N.8). DFD: one µ-response at all galactic scales,
the weak-response form of the derived exponential law
(App. BS). ΛCDM: dark matter halos. Observation:
SPARC RAR. Agreement: < 5% residuals. Falsifier:
SPARC residual scatter exceeds 0.15 dex floor. Tier:
T. Status: validated .
√
• a0 = 2 α cH0 from scaling stationarity (App. N,
Thm N.14). DFD: 1.1966 × 10−10 m/s2 on boundmatter clocks (Postulate ZS.11), 1.1208 × 10−10 on
the early clock. MOND: free parameter. Observation:
McGaugh+ 2016 extraction (1.20 ± 0.02stat ± 0.24sys ) ×
10−10 . Agreement: within the systematic error on
bound-matter clocks. Falsifier: a precision
absolute a0
√
measurement inconsistent with 2 α cH0 after theory
and observational systematics are propagated. Tier: D;
physical normalization from the bound-matter clock.
Status: validated .
• SPARC RAR head-to-head (Sec. VII). DFD: beats
Newton in all 16 galaxies tested; nopt = 1.15 ± 0.12
centered on n = 1. ΛCDM: DM halo per galaxy. Observation: McGaugh+ 2016. Agreement: n = 1 confirmed;
standard MOND n = 2 disfavored. Falsifier: best-fit
n exits [1.0, 1.5] at 95% CL on full sample. Tier: D.
Status: validated .
• Baryonic Tully-Fisher: Mbar ∝ vf4 (Sec. VII). DFD:
exact slope. ΛCDM: DM halo prediction. Observation:
McGaugh+ slope 3.94 ± 0.08. Agreement: matches.
At high redshift DFD’s a0 (z) lowers the zero point,
by 0.48 dex in Mbar at fixed vf at z = 2 in the deep
regime (derived). The KMOS3D zero point rises by
0.17 ± 0.06 dex from z ≃ 0.9 to 2.3 [173]; a frozen
a0 and Newtonian baryons miss that rise as well, and
the sample shares its galaxies and calibrations with
the massive-disc rotation curves of App. A0Z, so the
zero-point evolution does not yet test a0 (z) (computed).
Falsifier: BTFR slope outside [3.7, 4.3] at high significance. Tier: T. Status: validated .
• Zero-parameter amplitude–shape relation (ν, u)
(App. AT 7). DFD: the mass discrepancy ν = gobs /gbar
and the log-slope variable u = −2 d ln ν/d ln gbar obey
a closed relation with a0 eliminated identically — u =
(ν − 1) ln[ν/(ν − 1)] for the exponential occupation
response, u = 2(ν − 1)/(2ν − 1) for the closure form
µ = x/(1 + x) of Thm N.8 — so this is a galactic
test that uses no acceleration scale at all. ΛCDM: no
relation between the two (halo shape is a free function).
Observation: SPARC, 8 equal-count gbar bins, 2316
points from 113 galaxies. Agreement: χ2 /dof = 1.08,

p = 0.37, largest pull +2.26σ; the n ≥ 2 (“standard”
and steeper) members of the same family are excluded
by shape alone at 2∆ ln L = +14.97 and +30.89 with
Υ⋆ marginalised (+14.07 with Υ⋆ free). Stated limit:
these data do not separate the two DFD-admissible
forms from each other (+0.43), and provably cannot —
every a0 -free shape statistic on SPARC is bounded at
2∆ ln L ≤ +3.99. Distance and inclination errors are
exactly horizontal in the (ln ν, u) plane (Thm. AT.18),
so only Υ⋆ needs marginalising. Falsifier: the binned
locus off the curve by > 5σ; any bin robustly at ν < 1;
a measured u > 1 asymptote; or Υ[3.6] > 0.58 M⊙ /L⊙
measured independently (Falsifier AT.22, currently ≃
2.5σ clear). Tier: T (the relation and the systematics
vectors); D (the confrontation). Status: validated .
• Wide binary boost at 104 –2 × 104 AU (Sec. VII).
DFD: γ = 1.000 exactly for local pairs (Axiom NL.14);
pairs beyond R ≃ 16 kpc keep the boost (γ = 1.70–
1.97 at 104 AU on the forward law, R = 20–100 kpc);
an isolated-pair 41.6% velocity figure is not a DFD
prediction. MOND: γ ≃ 1.4. Observation: Chae 2023
Gaia γ ≃ 1.43 ± 0.06; Chae et al. 2026; Yoon et al.
2025; Banik 2024 (−0.3σ) and Saad & Ting 2026 (0.0σ)
null. Agreement: with the null analyses; +7.2σ from
Chae (2023), +4.3σ from Chae et al. (2026). Falsifier:
a confirmed boost of the Chae type in local pairs. Tier:
D (conditional on the axiom). Status: contested .
• The confinement step at R ≃ 13–16 kpc (App. NL).
DFD: bound subsystems inside the edge are Newtonian
and those outside boosted; a 1 M⊙ pair’s relative force
changes by a factor 1.59 at 104 AU (1.85 at 2 × 104 AU)
across it. MOND: a smooth decline. Observation: halo
pairs not yet (Gaia DR4/DR5); the line-of-sight dispersion profiles of 28 outer-halo globular clusters beyond
16 kpc (Baumgardt catalogue, version 4) give a boost
0.33 ± 0.08 of the predicted one, about 5σ below it with
the error scaled for the fit’s χ2 /dof = 2.9 and about 9σ
on the 18 clusters least affected by binaries. Agreement:
the inner clusters are consistent with Newtonian dynamics, as predicted; the outer-cluster boost is contradicted.
Falsifier: a boost inside the edge, no step, or halo pairs
outside the edge with a boost well below the predicted
one. Tier: D. Status: open for the halo pairs.
• Classical dwarf spheroidals. DFD: on the forward
form of the printed law of satellites, at populationsynthesis mass-to-light ratios, the µ-law alone falls 0.2–
0.3 dex (3–4σ) short of the eight Milky Way dispersions
(computed), nearly all of it in the four that the Galaxy’s
field dominates and that are not tidally safe; the four
tidally safe dwarfs fit the µ-law alone to about 1σ (computed); χ subhalos, which the floor law adds, or tidal
disequilibrium can supply the difference. ΛCDM: DM
halo per dSph. Observation: measured dispersions.
Agreement: the tidally safe dwarfs fit on the µ-law
alone; the others need χ or are out of equilibrium. Falsifier: inconsistent dynamical M/L in any single isolated

274
classical dSph at > 5σ. Tier: D. Status: open.
• Ultra-faint dwarf M/L extreme values. DFD: systematic effects (binary contamination, tidal heating)
inflate inferred M/L by factors of 10–100. ΛCDM:
dark matter. Observation: UF dSph M/L ∼ 100–1000.
Agreement: rests on the systematics estimate of Table XV; χ subhalos, which DFD forms (Ext. Deriv.
App. AH), can also raise the dynamical M/L where χ
exceeds the phantom. Falsifier: UF dSph kinematics
confirmed pristine yet retain an anomalous M/L that
no χ subhalo can supply. Tier: E. Status: open.
√
• Galactic-scale extension a0 (z) = 2 α cH(z)
(App. N, Thm. XIX.3; App. A0Z). DFD: a0 ∝ H(z)
with no free exponent on DFD’s own background,
1.8, 2.6 and 3.7 × 10−10 m s−2 at z = 0.8, 1.5 and
2.2 on the early clock and 1.89, 2.82 and 3.95 on
bound-matter clocks (Postulate ZS.11; effective exponent γE = 1.10 over 0.33 < z < 1.44 and 1.29 over
0.6 < z < 2.5). MOND: static a0 . Observation: two
kinds of data, in tension at face value. Low-mass discs
at 0.33 < z < 1.44 (the MUSE–DARK binned RAR
scale [109], fitted in Ref. [110]) give a0 = A(1 + z)γ with
γ = 0.78 ± 0.15 (0.83 ± 0.13 with the SPARC z ≃ 0
√
anchor), excluding a frozen a0 at 5.3σ and a0 ∝ ρm
at 4.8σ (on the exponent’s error; 4.7σ by likelihood
ratio) on the published bin errors; with the bars read
as 1σ, Hubble tracking fits the bins with χ2 = 5.36/3
at A = (1.394 ± 0.032) × 10−10 m s−2 , 2.1σ from the
measured exponent. Refitted galaxy by galaxy from the
survey’s public products, the published rise reproduces
(γ = 0.77 ± 0.21), but the galaxy-level uncertainties are
several times the published bin errors, because the radial points of one galaxy are correlated, and within the
survey a frozen a0 lies about 1σ away at galaxy level;
anchored to SPARC’s z = 0 value the galaxy-level bins
still exclude a frozen a0 at about 3σ, and DFD’s zeroparameter law fits them on the bound-matter clock. The
trend also depends on how the molecular gas is added:
tied to the survey’s own disc masses the rise stays and
DFD’s law fits it, tied to SED-based masses it is about
flat (γ = 0.06 ± 0.21; −1.03 to 0.96 at galaxy level), and
with stellar masses from spectral energy distributions it
falls (γ = −1.04±0.22; −2.32 to −0.32), although those
SED masses exceed the kinematic 95% limit in about a
third of the high-z galaxies; the survey’s unconstrained
atomic-gas term moves γ between 0.5 and 1.3 over its
physical range; above z ≃ 0.9 most fitted disc masses
sit at the edge of the halo model’s prior, and their
offset from the SED masses is kinematic, not an artefact of the prior. DFD’s zero-parameter amplitude lies
0.095 dex below the tracking amplitude on the published
census and 0.02–0.09 dex above it with either fuller census (0.18 dex with both changes together; computed).
Massive discs at 0.6 < z < 2.5 with molecular gas
counted [111] give a0 = 1.9, 1.4 and 1.2 × 10−10 m s−2
at median z = 0.8, 1.5 and 2.2: DFD’s value is met
at z ≃ 0.8, and the rise beyond z ≃ 1.5 is not seen

on that census, while fits that leave the baryonic mass
free neither show nor exclude it. Their within-sample
exponent is γ = −0.86+0.46
−0.48 on statistical errors (−0.20
without the discs with Vrot /σ0 < 2.1); DFD’s exponent
lies 4.5σ away on statistics alone (likelihood ratio) and
about 1.7σ away once a ±30% pressure-support scale
and a ±0.1 dex redshift tilt of the baryonic masses are
marginalised (computed), although the discs’ own data
disfavour the calibrations that would rescue DFD, a
doubled pressure support or z ≃ 2 baryons 0.2 dex
lighter, by ∆(−2 ln L) = 18–40 (computed). DFD’s
domain-local screen (App. ZT) lowers a0 in overdense
domains: by about 10% for massive z ≃ 2 discs at
the average environment of their haloes, and by 20–
34% in the densest known structures; it would match
these discs only in 10 h−1 Mpc domains about 6 to 11
standard deviations denser than average, and their measured environments show no such trend (computed on
published values). Agreement: contested. DFD’s value
is met at z ≲ 1 by the massive discs and, in amplitude,
by the survey with either fuller census; the survey’s
rise is about 1σ within the survey at galaxy level and
about 3σ anchored to SPARC, with DFD’s law fitting
on the bound-matter clock, and it depends on how the
molecular gas is added; beyond z ≃ 1.5 the massive
discs, which sit where the rise should show, do not show
it: on the published census they rotate 0.1–0.17 dex
slower than DFD predicts (6–9σ on statistical errors,
1.5–2.2σ with the census’s calibration counted), and
the registered falsification threshold a0 (z=2) < 2.5 is
met at face value. The threshold is reached if the z ≃ 2
baryonic masses are about 0.1 dex lower than the fitted
ones (about 0.2–0.3 dex in molecular gas), and DFD’s
value if they are about 0.2 dex lower (about 0.45 dex in
gas; 0.3–0.5 dex against the census), or with a pressuresupport correction 1.8 or 2.4 times the standard one,
which no published prescription gives (DFD’s own vertical gravity makes it about 10% smaller). Falsifier:
on a baryon census that counts molecular gas, with
galaxy-level errors and a stated pressure-support correction, high-z kinematics giving γ more than 3σ from
γE = 1.10 over 0.33 < z < 1.44, or a0 (z=2) outside
[2.5, 5.5] × 10−10 m s−2 . Tier: D. Status: contested .
• No sub-Newtonian µ-law response, at any acceleration (App. BS, Thm BS.7). DFD: ν = gobs /gbar =
Z1 > 1 structurally — the mode’s ladder is bounded
below by the ground state, so detailed balance offers
no de-excited level and ν ≥ 1 is a theorem rather than
a property of the fitted µ. MOND: the same by construction of µ ≤ 1, but as a modelling choice, not a
proof. ΛCDM: no constraint — a baryon-deficient halo
is permitted. Observation: no confirmed gobs < gbar
system. Falsifier: any system with Mdyn /Mbar < 1 at
> 3σ after baryon-census systematics; equivalently a
converged Υ[3.6] > 0.58 M⊙ /L⊙ (Falsifier AT.22, which
rests on this theorem and is branch-independent for
that reason). Tier: T. Status: validated .

275
• Solar-system safety is structural, not screened
(App. BS, Cor. BS.14, Table CXXXVI). DFD: at
Saturn’s orbit (a = 9.5826 AU) the mode exponent
is τ = 733.60 (at the empirical SPARC benchmark
a0 = 1.20 × 10−10 m s−2 ), so the mode entropy is
S/k = 1.86 × 10−316 and its heat capacity C/k =
1.36 × 10−313 ; at Earth’s surface τ = 2.859 × 105 and
S/k = 2.34 × 10−124147 . There is no occupied state
to boost: solar-system agreement is quantum freezeout of a single mode, with no parameter chosen to
achieve it and no fifth-force suppression, chameleon or
Vainshtein mechanism invoked. Beyond about 103 AU
the Sun’s own mode is not frozen by its depth; there the
Sun is Newtonian by the environmental-confinement
axiom (App. NL; Oort-spike row). ΛCDM/GR: agreement by construction. Scalar-tensor alternatives: agreement requires an explicit screening mechanism with its
own scale. Observation: Cassini/ephemeris bounds on
anomalous acceleration at Saturn. Agreement: the predicted anomaly is ∼ 10−318 of a0 , unobservably below
any bound. Falsifier: a detected anomalous acceleration
at Saturn’s orbit exceeding 10−300 a0 , or, for an isolated
system, any demonstration that the deep-MOND and
Newtonian regimes require separate closure assumptions rather than the two limits of one C/k. Tier: T.
Status: validated .
• Momentum balance: the Local Group barycentre (App. AV). DFD: zero drift of the matter barycentre
against the cold local Hubble flow (Noether’s theorem
under the adopted energy). ΛCDM: zero; the printed
total-ψ rule: 15–20 km/s. Observation: 6.9 ± 3.8 km/s
towards M31 along the Milky Way–M31 line, the component along it of the (24 ± 4) km/s motion towards
Sculptor found by Karachentsev et al. (2009), with the
error of the barycentre’s position included. Agreement:
1.8σ. Falsifier: a drift of 15 km/s or more at 5 km/s
precision or better, with tides modelled. Tier: D (conditional on the adopted energy). Status: pending.
• χ’s equivalence principle beyond the filled zone
(App. AV). DFD: no acceleration difference where χ fills
the demand (for a Milky Way modelled with 6×1010 M⊙
of baryons, to about 208 kpc for 2×1012 M⊙ ); δ = 0.08–
0.34 at Fornax for a 1.0–1.2 × 1012 M⊙ halo, with the
leading arm enhanced. ΛCDM: δ = 0. Observation:
Sagittarius |δ| ≲ 0.10; Fornax β ≲ 0.2. Agreement:
consistent for a χ halo of 1.5 × 1012 M⊙ or more in that
model (abundance matching gives 1.7 × 1012 ). Falsifier:
an acceleration difference where χ fills the demand, or
the fifth-force sign. Tier: D. Status: pending.
• Vertical force at the Sun (App. AV). DFD: with the
measured local stars and gas, 2πG × (66–69) M⊙ pc−2
at 1.1 kpc where χ comes close to the µ-law’s demand
(local χ density at least about 0.0068 M⊙ pc−3 for a
round χ halo), and 84 ± 4 from the µ-law alone. ΛCDM:
67 for the same baryons. Observation: 68 ± 4, 74 ± 6
and 63.6 at 1.1 kpc; 67 ± 6 and 68.5 ± 1.0 at 1.0 kpc.

Agreement: within 0.7σ with χ at the rotation-curve
density (0.0079 ± 0.0008); the µ-law alone is 3.1–3.2σ
high. Falsifier: a local χ density below 0.006 M⊙ pc−3
from the rotation curve beyond the Sun and stellar
streams, with the local stars and gas at 47 ± 3 M⊙ pc−2 .
Tier: D. Status: pending.
5.

Block 4: Cluster Scale

• Universal µ from galaxies to clusters (App. I).
DFD: n = 1 at cluster scale, no scale-dependent µ.
ΛCDM: DM halo. Observation: the 16-cluster sample of App. I. Agreement: consistent once χ supplies
the missing mass, whose amount the clusters measure
(0.84 ± 0.12 of the cosmic mix when χ adds its Newtonian pull); the cluster masses alone do not test n.
Falsifier: cluster RAR with strong-lensing-resolved subhalos shows n ̸= 1 at > 3σ. Tier: D. Status: pending.
• PDE-validated Jensen factor JPDE = 1 + 0.39fsub
from 3D AQUAL solver. DFD: J ≃ 1.07–1.12
for fsub = 0.15–0.30, and J ≃ 1.04 for the baryonic
substructure of clusters (fsub ≃ 0.10) to which the µlaw responds, robust across M200 ∈ [3, 12] × 1014 M⊙ .
ΛCDM: not applicable. Observation: validated against
algebraic-MOND for spherical NFW within 1%. Agreement: computed (PDE). Falsifier: high-resolution
multigrid AQUAL solver gives J outside [1.05, 1.15]
at fsub = 0.20. Tier: D. Status: open.
• Cluster χ budget (App. I). DFD: under the µlaw alone the sixteen clusters fall short (Obs/DFD
1.57 ± 0.08 relaxed, 1.99 ± 0.16 merging, at a0 =
1.20 × 10−10 m/s2 and at the bound-matter a0 =
1.1966 × 10−10 m/s2 ); the shortfall is the derived χ,
4.5 ± 0.6 Mb = 0.84 ± 0.12 of the cosmic mix when χ
adds its Newtonian pull and the relaxed hydrostatic
masses carry a bias b = 0.21, or the ΛCDM budget
(1.28–1.51 of the cosmic mix inside r500 ) under the floor
composition of App. AV. A no-χ five-factor correction
stack Ci = B JPDE T M P is not supported (its baryonic items sum to 1.26, and its projection factor has
the wrong sign for lensing masses). ΛCDM: dark matter halos. Observation: 16-cluster sample. Agreement:
consistent: predicted 1.21 ± 0.05 of the cosmic mix inside r500 from the derived abundance and ΛCDM-like
assembly (estimate), measured 1.28 ± 0.04 (no hydrostatic bias) to 1.51 ± 0.04 (b = 0.21) on errors of the
mean; with the calibration uncertainties the measured
masses are 1.10 (0.97–1.23) and 1.30 (1.13–1.47) times
the predicted ones (App. I). Falsifier: a cluster sample
whose masses the µ-law on its baryons already supplies,
leaving no room for χ. Tier: D (estimate). Status:
validated .
• Bullet Cluster lensing offset (Sec. VII). DFD: the
collisionless χ carries the lensing peak; without χ DFD
gives 24–39% of the lensing mass inside 250 kpc, and

276
χ at 0.6–0.9 of the cosmic mix fits it when it adds
its Newtonian pull (1.1–1.2 under the floor composition); 129 kpc main cluster offset. ΛCDM: DM halo
offset. Observation: observed 155 kpc. Agreement: 83%
match. Falsifier: lensing peak coincides with gas at any
reasonable resolution. Tier: D. Status: validated .
• Galaxy groups: χ as in clusters (App. I). DFD: on
published X-ray masses the µ-law alone falls short of
groups as it does of clusters: the median Obs/DFD is
1.22–1.36 at r500 for the 23 Chandra groups of Sun et al.
(2009) and 1.79 for the 16 groups of Gastaldello et al.
(2007), at the bound-matter a0 with distances computed
with H0loc ; under the floor composition 20 to 22 of the
23 are Newtonian plus χ at 1.2–1.7 of the cosmic mix,
as clusters are (1.28–1.51). ΛCDM: dark-matter halos.
Observation: Sun et al. (2009), Gastaldello et al. (2007),
Virgo (Simionescu et al. 2017), Fornax (Su et al. 2017).
Agreement: consistent with ΛCDM-like assembly of the
derived abundance, whose group-scale baryon depletion
depends on feedback; the stellar masses are estimates.
Falsifier: a sample of groups with securely measured
baryon fractions whose median Obs/DFD at r500 falls
below 1, the µ-law on their baryons alone (single groups
below 1 occur within the stellar-mass uncertainty: 3
of the 23 with the higher stellar calibration). Tier: E.
Status: validated .
• CMB peak ratio from baryon loading and the
derived χ-matter (App. J; App. AV). DFD: R = 2.34
from the analytic baryon-loading estimate; with the
derived cold χ-matter added, CAMB on the measured
clock gives 2.20 (the peak positions follow the clock measured from BAO and θ∗ ) (Thm. AV.31), while baryons
alone give H2 /H1 = 0.39. ΛCDM: R ≃ 2.21 with
ΩCDM ∼ 0.27. Observation: R ≃ 2.21 (Planck 2018
best fit). Agreement: the χ-matter spectrum within
0.6σ on the measured clock; the analytic estimate 6%
high. Falsifier: the peak ratio requires a cold component beyond the derived χ abundance at > 3σ. Tier:
D. Status: validated .
• CMB third-peak height from χ-matter (App. AV).
DFD: H3 /H2 ≈ 1.0 from the cold χ field with mχ =
32.005 eV and Ωχ h2 = 0.1192806 (−0.60σ; computed
on the flux-period reading that the measured abundance selects; no fitted density), obtained from the fluxperiod/motivated-multiplicity chain and finite SU (2)60
CS-vacuum Casimir expectation with their named assumptions preserved; Qχ = 1 is conditional on the RestMass Channel. ΛCDM: same height from a postulated
cold component. Observation: H3 /H2 ≈ 0.98 (Planck
2018). Agreement: H3 /H2 = 0.977 and H3 /H1 = 0.444
on the measured clock, within 0.6σ. Falsifier (decisive):
the ∆ψ ×κ cross-spectrum carries the third-peak height,
a halo-recoil DM detection appears in a current-reach
channel, or a dark-matter photon line appears at 16.0 eV
(77.5 nm) or 32.005 eV (38.74 nm) at any strength (χ
has no photon coupling, E = 0; App. AV). Tier: D for

the carrier, conditional on the stated mass/abundance
premises and the Rest-Mass Channel with its premise
that linear perturbations do not polarize the medium
(App. GR). Status: validated .
• RAR intrinsic scatter ≲ 0.037 dex (App. AV, Remark AV.38). DFD: σRAR ≲ 0.037 dex (the halo-shape
floor alone gives 0.013–0.025 dex) if the ΛCDM-like χ halos of the adopted energy are cored, as SPARC requires
under the floor
√ law (App. AV), because the derived constant a0 = 2 α cH0 forces the a0 -drift channel to zero
(conditionally on the DLSC axiom, App. ZT, a domainlocal environment modulation ∆a0 /a0 = −(f /3) δenv
reopens this channel at 0.015–0.027 dex for RD = 10–
20 h−1 Mpc — inside this intrinsic budget, and a candidate explanation for part of it). ΛCDM: ∼ 0.06–0.08 dex
(the a0 -universality is a fine-tuning puzzle requiring pergalaxy feedback tuning). Observation: joint inference of
the relation and the galaxy parameters on SPARC gives
0.034 ± 0.001 (stat) ± 0.001 (sys) dex, given the SPARC
error model [169]; a 10% uncertainty on the baryonic
velocity components sets the preferred value to zero.
Agreement: below the bound and the ΛCDM range
under the SPARC error model; above the halo-shape
floor alone, inside the budget only with the domainlocal channel. Falsifier: true intrinsic RAR scatter
>∼ 0.06 dex (Stone’s 0.11 confirmed at the intrinsic
level) that is not the one-sided upward lift of χ halos
denser than the phantom falsifies DFD. Tier: D. Status:
pending (medium confidence).

6.

Block 5: Cosmology

• CMB acoustic scale on the measured clock
(Apps. J and ZS). DFD: 100θ∗ = 1.04110 and ℓ1 ≃ 220.4
with no global ψ-lensing map. Observation: 100θ∗ =
1.04109 ± 0.00031. Agreement: +0.03σ; θ∗ is one of
the data that measure C, so the test is its concordance
with BAO (χ2 = 10.52/13 on 13 pre-DESI points; DESI
DR2 BAO alone sits 3.4σ above the θ∗ -only clock). Falsifier: the frozen-input acoustic forward solution fails
at precision after the measured clock and stated inputs
are propagated. Tier: D given measured C. Status:
validated .
• Native linear-clustering S8 on the measured
clock. DFD: R8 = 11.848 Mpc, σ8 ≃ 0.8063,
Ωm ≃ 0.312, S8 ≃ 0.822. Observation: KiDSLegacy 0.815+0.016
−0.021 (about +0.45σ using its upper error),
with the joint external-probe result 0.814+0.011
−0.012 (about
+0.74σ), and DES Year 6 0.789 ± 0.012 (2.8σ below, survey errors only; App. GR). Tier: D conditional on the
adopted Qχ = 1 Rest-Mass Channel. Status: validated .
• Distance duality: DL /(1 + z)2 DA = 1 (DDR violation = 0). DFD: exact identity. ΛCDM: exact identity
(in standard FLRW). Observation: DL /(1 + z)2 DA =
1.01 ± 0.02 (Sec. XVI), consistent with no violation;

277
DESI DR2 BAO with Pantheon+ or DES-Dovekie supernovae and cosmic chronometers find no violation
within 1σ [180]. Falsifier: detection of DDR violation
> 0.01 at any z. Tier: T. Status: validated (shared
with ΛCDM).
• DESI BAO: evolving dark energy hint. DFD:
w = −1 exactly (one constant vacuum term, App. AU;
no dark-energy clustering), ΛCDM-identical in the distance sector. ΛCDM: static Λ. Observation: DESI
DR2 dataset-dependent preference for w(z) ̸= −1 [95].
Agreement: DFD stands with ΛCDM on this test
and inherits the same tension; not a DFD confirmation. Falsifier: DESI DR3 (or successors) confirming
(w0 , wa ) ̸= (−1, 0) at > 5σ. Tier: C. Status: pending.
• CMB-foreground correlation: ∆ψ(n̂) correlates
with structure. DFD: a local, anisotropic correlation with foreground structure at the amplitude of
the conditional model of App. LS; there is no global
screen (App. ZS). ΛCDM: the integrated Sachs–Wolfe
cross-correlation, which DFD shares (AISW = 1.00,
Sec. XVI I), with no ∆ψ screen and no local anisotropic
term. Observation: not yet measured. Falsifier: no
correlation found at predicted σψ across multiple frequencies. Tier: D. Status: open.
res
• ESPRESSO ∆α/α at z ∼ 1. DFD: ξLPI
× (+2.0 ×
−6
res
10 ), conditional on the undetermined residual ξLPI
(Eq. (253)). SM: 0 (no evolution). Observation:
+1.3 ± 1.3 × 10−6 . Agreement: none to state until
res
res
ξLPI
is determined. Falsifier: a determined ξLPI
whose
prediction is inconsistent with the measured ∆α/α at
> 3σ. Tier: D. Status: open.

7.

Block 5a: Growth–Lensing Consistency (EG ,
ACT DR6 × BOSS)

DFD’s forward prediction for the geometry-vs-dynamics
ratio is the GR-form statistic evaluated at DFD’s derived
matter fraction: with gravity-universal growth (Qχ = 1)
and the ψ-screen cancelling between the lensing numerator
and the clustering denominator of the ratio estimator,
EG (z) = Ωm,0 /f (z) at the current Ωm ≃ 0.312, scaleindependent on linear scales. The Ωm = 0.274 lowernormalization signature belongs to the one-clock C = 1
background; the result at the measured clock is effectively
the Planck-ΛCDM normalization.
• EG growth–lensing statistic (ACT DR6 CMBlensing × BOSS) [97]. DFD: EG = 0.402 at z =
0.555 (CMASS) and 0.453 at z = 0.316 (LOWZ), scaleindependent. ΛCDM (Planck values, the analysis’ own
reference): 0.401 and 0.452. Observation: 0.31+0.06
−0.05
(CMASS) and 0.49+0.14
(LOWZ),
scale-independent
−0.11
within errors. Agreement: the current DFD and Planck
predictions are numerically indistinguishable at the
displayed precision and both are consistent with the

data. Named conditionality: the ψ-screen cancellation
is established for the ratio form EG = Ωm,0 /f and
remains to be verified against the specific DR6 estimator
definition. Falsifier: an EG measurement at z ≈ 0.5
with precision sufficient to exclude the shared GR-form
value, or a detected scale-dependence of EG on linear
scales. Tier: D. Status: validated (no current DFD
edge).
The screen cancels in the ratio and the current
measured-clock Ωm is Planck-like, so the present data
provide a consistency check, not a two-normalization discriminator.
8.

Block 6: Laboratory Tests (Clocks, Cavities,
Matter Waves, Antimatter)

α
• Cross-species clock LPI: KA = kα SA
speciesdependent. DFD (extended non-metric channel, coefficient not derived from the minimal action): speciesdependent at the indicative scale ∼ 10−15 ; minimal
DFD: zero (Thm. XI.2). SM: exactly zero (Einstein equivalence principle). Observation: current null
at 10−17 in pure-α channel. Agreement: consistent
(screened). Falsifier: cross-species LPI null at < 10−15
on cross-species channel involving strong sector. Tier:
D. Status: open.

• Pure-α residual λα ≈ 2.1×10−8 in same-ion E3/E2
(Sec. XI). DFD (extended channel, ϵH estimate, not
derived from the minimal action): λα ≈ 2.1 × 10−8 ;
minimal DFD: zero. SM: 0. Observation and agreement:
as for the clock-coupling row (Block 1), whose residual
this is; the screened value lies four orders below the
Filzinger et al. 2023 bound, which does not test it.
Falsifier: a same-ion E3/E2 null below the screened
residual. Tier: D. Status: open.
• Composition/strong-sector residual λN,e,s ≈ 4.2 ×
10−7 . DFD (extended channel, ϵH estimate, not derived
from the minimal action): 4.2 × 10−7 in cross-species
channels; minimal DFD: zero. SM: 0. Observation:
BACON-II Al+ /Sr/Yb at ∼ 8 × 10−18 . Agreement:
consistent (screened). Falsifier: cross-species LPI null
at < 5 × 10−7 at multi-σ. Tier: D. Status: open.
res
• Cavity-atom residual ξLPI
. DFD: screened residual
after geometric cancellation. SM: ξ = 0. Observation:
not yet at sensitivity. Falsifier: next-gen cavity-atom
test reaches predicted sensitivity and finds null. Tier:
D. Status: open.

• Kennedy–Thorndike null: BKT = 0 at fixed
ψ (Sec. XI, Cor. XI.3). DFD: the co-located cavity/atom ratio is independent of the laboratory’s velocity relative to the preferred (CMB) frame, identically at fixed ψ; preferred-frame and tidal remainders
≲ 10−22 , the former common-mode (Rem. XI.4). SM:
0 (local Lorentz invariance). Observation: βRMS −

278
αRMS − 1 = −1.7(4.0) × 10−8 (sidereal), weighted mean
−4.8(3.7) × 10−8 [61]; earlier null at 2 × 10−13 in fractional frequency [62]. Agreement: −0.4σ (sidereal),
−1.3σ (weighted mean). Falsifier: a boost-correlated
cavity/atom modulation above 10−22 that survives systematics. Tier: T. Status: validated .
• Th-229 nuclear clock anomaly. DFD (extended
strong channel; ks a matching coefficient, not derived from the minimal action): screened annual halfamplitude about 350 Hz at αs (MZ ), 0.9–1.5 kHz at
the nuclear-scale αs (Sec. XI F); minimal DFD: zero.
SM: 0. Observation: the 2026 reproducibility data of
Ooi et al. exclude the unscreened amplitude and leave
a 26 Hz–1 kHz window. Falsifier: Th-229 clock at
∼ 10−19 stability finds null in strong-sector channel.
Tier: D. Status: pending.
• Matter-wave phase at 1PN: HDFD = HGR identically (Sec. XIII). DFD: no DFD-specific T 2 /T 3 /T 4
term through 1PN (any static potential), nor at 2PN,
where the static metric is GR’s (App. WS). SM: identical (same γ = β = 1 metric). Observation: consistent with all Solar-System PPN data (§IV C). Falsifier:
none beyond γ = β = 1; a species-dependent (Katom )
residual above the corpus’s own 10−14 Eötvös bound
(§XV G) would indicate new non-metric coupling. Tier:
T. Status: validated .
• Antimatter gravity: aH̄ = aH at metric level.
DFD: equal. SM: equal (Einstein). Observation:
ALPHA-g consistent. Agreement: exact. Falsifier:
antimatter shows a ̸= g at metric. Tier: T. Status:
validated .
• Antimatter species-dependent residual σA
(Sec. XV C). DFD: non-metric residual at 10−7 –10−8 .
SM: 0 exactly. Observation: not yet at sensitivity. Falsifier: ALPHA-g + species-resolved measurements show
null at predicted level. Tier: D. Status: open.
• LPI test ξ = 0 vs DFD ξ ̸= 0 (cavity-atom). DFD:
nonzero residual ξ res , the cavity-atom residual listed
earlier in this block. SM: ξ = 0 exactly. Observation: current bound |ξ| < 10−5 . Agreement: consistent
(screened). Falsifier: cavity-atom ξ null at predicted
screened residual level. Tier: D. Status: open.
9.

Block 7: Strong-Field, Hyperbolicity, and Open
Items

• No singularity reached in absolute time
(App. WS). DFD: its maximal slicing never reaches
the singularity of a collapsing star, so the BKL regime
lies beyond every finite absolute time (for maximal
slices proved for spherical collapse and crushing singularities and argued for generic rotating collapse; that
DFD’s slices stay maximal during collapse is argued).
GR: predicts BKL chaos near spacelike singularities.

Observation: untested regime (inside horizons). Falsifier: a derivation that DFD’s slices reach a curvature
singularity at finite absolute time. Tier: D. Status:
open.
• Hyperbolicity for small perturbations on smooth
backgrounds. DFD: the linearised evolution on any
static background is well-posed in its energy space,
with conserved energy, no growing modes and finite
propagation speed, including at points where ∇ψ = 0
(Theorem III.9). GR: well-posed IVP. Observation:
—. Agreement: exact theorem. Falsifier: numerical
instability for any DFD perturbation evolution. Tier:
T. Status: open.
• Full hyperbolicity for nonlinear strong-field dynamics. DFD: locally well-posed wherever ∇ψ ̸= 0, for
the exact law and its approximant, with characteristic
speeds up to 1.017 c and every characteristic advancing
the absolute time (Theorem III.5); smooth solutions
can form gradient singularities in finite time, as fluids form shocks; nonlinear evolution through points
where ∇ψ = 0 is an open question of degenerate hyperbolic equations. GR: well-posed (with constraint
propagation). Observation: —. Falsifier: DFD evolution code finds genuine ill-posedness for a configuration
with ∇ψ ̸= 0. Tier: T (local). Status: open.
• Uniqueness of CP 2 ×S 3 (App. AB, Thm AB.1). DFD:
unique compact manifold (up to diffeomorphism) satisfying the seven vacuum axioms (V1), (V2), (V4), (V4′ ),
(V5), (V7), (V10) — including simple connectedness
of Mg and positive Ricci on Mc ; the axiom set is satisfiable (explicit model exhibited). Observation: —.
Agreement: exact theorem. Falsifier: alternative compact manifold satisfies the same seven axioms with
different topological invariants. Tier: T. Status: open.
• Higgs quartic λH = 1/8 (dimension count; App. Z).
DFD: λH = 1/8 at the electroweak matching scale. SM:
input parameter. Observation: running quartic 0.1260
at mt . Agreement: mH = 124.6 GeV at mt , 0.40%
(2.7σ) low; exact at 182 ± 4 GeV. Falsifier: a derived
electroweak matching scale outside 175–189 GeV, the
range in which the running quartic is within 2σ of 1/8.
Tier: C. Status: open.
• PDE uniqueness theorem for ψ rest frame
(Sec. XVIII). DFD: unique up to gauge. GR: boost
invariance. Observation: not yet proven. Falsifier:
demonstration of multiple inequivalent rest frames in
any DFD configuration. Tier: C. Status: open.
• Standard quantum unitarity preserved (Mott
problem) (Sec. XVII). DFD: no modification to QM.
SM: standard QM. Observation: all QM tests consistent.
Agreement: exact. Falsifier: QM modification detected
at any precision. Tier: T. Status: validated .
• Diosi-Penrose collapse rate vs DFD (Sec. XVIII).
DFD: no anomalous collapse. DP: mass-scaling decoher-

279
ence. Observation: MAQRO will discriminate. Falsifier:
MAQRO confirms DP scaling at predicted level. Tier:
D. Status: open.
10.

Summary statistics

The Master Claims Confrontation Matrix contains 78
distinct claims spanning fundamental constants, gravitational tests, galactic and cluster dynamics, cosmology,
and laboratory probes.
a. Aggregating by tier.
• Theorem-grade (T): 26 claims, representing the central structural identities of the theory (PPN parameters,
µ-derivation, the no-wormhole theorem, GW propagation, BTFR, antimatter gravity at metric level, manifold
uniqueness, matter-wave 1PN identity, the linear and
local well-posedness of the field equation). (Proton
stability is graded conditional — see its entry.)
• Derived (D): 47 claims, including the microsector
quantities (α−1 , the exponent of the Higgs VEV,
fermion masses with their stated prefactors, CKM,
PMNS, H0 closure, cluster JPDE ), the cluster χ budget
at estimate grade, and the environmental-confinement
entries, which are conditional on that axiom.
• Conjectured (C): 3 claims, all explicitly open program
items (λH = 1/8, ψ rest-frame uniqueness, the DESI
evolving-w confrontation).
• Empirical (E): 2 claims, the group χ budget and the
UF-dwarf M/L bound.
b.

Aggregating by status.

• Validated (validated ): 45 claims, where current observational data supports DFD at the stated agreement
level. This includes essentially all established gravitational tests, the closed-form value of α−1 and the
exponent of the Higgs VEV, the SPARC galactic dynamics tests, the Oort spike, the cluster and group χ
budgets, and the CMB peak structure (the peak ratio
and the third-peak height on the measured clock).
• Pending (pending ): 8 claims, where the experimental capability exists or is imminent but no published
measurement yet decides the claim: proton stability,
the Local Group barycentre, χ’s equivalence principle
beyond the filled zone, the vertical force at the Sun, the
intrinsic scatter of the radial-acceleration relation, the
universal µ from galaxies to clusters, the DESI evolvingdark-energy hint and the Th-229 nuclear clock.
• Open (open): 23 claims: 21 with no comparable
observational test currently available, the ultra-faint
dwarfs, and the classical dwarfs, four of which, exposed
to the Galaxy’s tides, need χ subhalos, which DFD does
not yet predict in detail, or tidal disequilibrium. The
21 include strong-field BKL absence, the wide-binary
orientation null, the confinement step for wide binaries

at R ≃ 13–16 kpc (the outer-halo globular clusters
contradict its boost) and the screened clock residuals,
which lie orders of magnitude below current bounds.
• Contested (contested ): 2 claims, the epoch extension
a0 (z): within the survey the rise that low-mass discs at
z < 1.44 show on the published bins is about 1σ counted
galaxy by galaxy and depends on how the molecular
gas is added, although anchored to SPARC’s z = 0
value the galaxy-level bins still exclude a frozen a0 at
about 3σ and DFD’s zero-parameter law fits them on
the bound-matter clock, and massive discs at z ≃ 2
with molecular gas counted do not show the rise; and
the local wide-binary boost, where the Gaia analyses
disagree and DFD predicts none.
11.

Reading guide for referees

A referee should focus principal scrutiny on the following entries:
1. Theorem-grade structural claims: Verify that the
proofs cited (e.g., Theorem N.8 for µ-derivation, Theorem AA.1 for the Padé identity, Theorem AB.1 for
CP 2 × S 3 uniqueness) are mathematically sound and
use only the stated assumptions. The theorems are
independent of the rest of the theory; failure of any
single one would force a substantial revision.
2. Derived microsector results: Verify that the closure framework (spectral action, Chern-Simons on S 3 ,
kmax = 60 truncation) is internally consistent and reproduces the listed quantities. The microsector framework is the closure that promotes “conjectures” to
“derived,” so its assumptions matter.
3. Cluster χ budget (D, estimate): Under the µ-law
alone the sixteen clusters fall short (relaxed 1.57 ± 0.08;
mergers 1.99 ± 0.16, at a0 = 1.20 × 10−10 m/s2 and at
the bound-matter a0 = 1.1966 × 10−10 m/s2 ), and the
shortfall is the derived χ at 0.84 ± 0.12 of the cosmic
mix (additive composition, hydrostatic bias b = 0.21).
This is the most measurement-dependent claim in the
matrix. A referee may legitimately demand: (a) the
per-cluster baryonic masses and apertures of App. I, (b)
the hydrostatic bias used to equalize the relaxed and
merging classes, (c) the composition of χ with the µ-law,
which moves the budget from 0.84 to 1.28–1.51 of the
cosmic mix, and (d) any no-χ closure, which the fivefactor stack does not supply. The PDE-validated JPDE
is computed; for baryonic substructure it is ≃ 1.04.
4. Pending and open experimental tests: Crossspecies clock LPI, Th-229 nuclear clock, BACON-II
next iteration, precision EG result, Yb+ same-ion
E3/E2 next-generation (the clock tests probe the extended non-metric channel, Thm. XI.2). These will
determine whether the corresponding sectors survive
or are falsified within the next 1–3 years. They are the
decisive tests, not the historical data.

280
12.

Use of this matrix

Appendix AN: The Optical-Metric Master Equation:
Unified Light and Matter Motion

This matrix is intended for three audiences:
• Referees: To provide a single catalog summarizing
the falsifiability structure of the theory. Failure of any
T-class entry forces structural revision; failure of any
pending entry would falsify a specific prediction. The
matrix is designed to be a referee’s first stop, not a
substitute for the detailed sections.
• Experimenters: To identify which open or pending
tests offer the highest discriminating power between
DFD and competing frameworks. The Yb+ E3/E2
same-ion null at |(c2 /α) dα/dΦ| ≲ 3 × 10−9 is currently
the sharpest in the laboratory regime; in cosmology
the sharpest pending tests are those of the two-clock
postulate: bright sirens, mass-sheet-robust time delays
and the absolute a0 scale (Falsifier ZS.14).
• Independent theorists: To distinguish the theoremgrade structural identities from the auxiliary closureframework derivations. The theorem-grade items (PPN,
µ-derivation, manifold uniqueness, no-wormhole) carry
forward independent of the closure framework; alternative closure frameworks for the microsector would
change the D entries but not the T entries.
Every empirical claim carries a tier label and a falsifier,
so structural claims are not mistaken for hidden firstprinciples proofs and auxiliary closure derivations are not
mistaken for empirical fits.

The two foundational motion laws of DFD, the one-way
light speed c1 = c e−ψ = c/n and the matter acceleration
a = (c2 /2)∇ψ, are not independent postulates. This
appendix shows that both are limits of a single equation
of motion: the massive wave equation on one optical
metric, with light as its null sector and matter as the
slow limit of its timelike sector. The factor 1/2 in the
acceleration law, which might appear fitted, is forced by
the rest-energy structure of that equation.

1.

The isotropic optical metric

Light propagation fixes only the conformal class of a
metric: null cones encode the local speed but not an
overall scale. Within that class we adopt the isotropic
representative
ds̃2 = − c2 e−ψ dt2 + eψ δij dxi dxj ,
2 −ψ

(AN1)

with g̃00 = −c e , g̃ij = e δij , and inverse g̃ 00 =
−eψ /c2 , g̃ ij = e−ψ δ ij . Its null condition ds̃2 = 0 gives
|dx/dt| = c e−ψ = c/n, so Eq. (AN1) agrees with the
eikonal light metric ds̃2 = −c2 dt2 /n2 + dx2 on null structure; the two are conformally related and yield identical
light cones.
Equation (AN1) is distinguished within its conformal
class by its timelike sector. In the weak field g̃00 /c2 ≈
−(1 − ψ) = −(1 + 2Φ/c2 ) identifies the potential Φ =
− 21 c2 ψ, reproducing the effective Newtonian potential of
the main text and the matter acceleration. The spatial
part gives g̃ij ≈ (1 + ψ)δij = (1 − 2γΦ/c2 )δij with Φ =
− 21 c2 ψ, hence the parametrized post-Newtonian value
γ = 1 and the GR light-deflection coefficient 4GM/c2 b.
Equation (AN1) is therefore, to first order in ψ, the unique
representative whose full geodesic structure, null and
timelike, reproduces simultaneously the refractive index,
the matter acceleration, and γ = 1. (The bare eikonal
form −c2 dt2 /n2 + dx2 encodes only the light sector; its
timelike geodesics do not by themselves yield Φ = −c2 ψ/2,
which is why the isotropic representative Eq. (AN1) is
the natural carrier of the unified motion law.)

2.

ψ

The master wave equation and its eikonal limit

The fundamental object for motion is the wave equation
on Eq. (AN1),


p


m2 c2
˜
˜ g̃ = √1 ∂µ −g̃ g̃ µν ∂ν ,
□g̃ − 2
Ψ = 0,
□
ℏ
−g̃
(AN2)
which covers the full wave regime. The ray, or particle,
regime follows from the eikonal substitution Ψ = A eiS/ℏ ,
whose leading order in ℏ is the mass shell g̃ µν ∂µ S ∂ν S +
m2 c2 = 0. With the inverse metric of Eq. (AN1) this

281
becomes the single dispersion relation
E 2 = c2 e−2ψ |p|2 + m2 c4 e−ψ

(AN3)

This relation contains both sectors and is the eikonal of
Eq. (AN2), not an independent ansatz.
Theorem AN.1 (Unified light–matter motion; Derived). Light and matter are the null (m = 0) and
timelike (m > 0) sectors of the single dispersion relation
Eq. (AN3). The massless sector gives the group velocity
vg = ∂E/∂|p| = c e−ψ = c/n; the slow massive sector
gives a = (c2 /2)∇ψ, with the factor 1/2 arising from the
e−ψ/2 rest-energy term rather than being assumed.
Proof. Setting m = 0 in Eq. (AN3) gives E = c e−ψ |p|
and vg = c e−ψ = c/n. For m > 0,
r
e−ψ |p|2
2 −ψ/2
1+
E = mc e
m2 c2
(AN4)
2
|p| −3ψ/2
2 −ψ/2
+
.
≈ mc e
e
2m
In the weak field e−ψ/2 ≈ 1 − ψ/2, so to leading order
E ≈ mc2 − 21 mc2 ψ + |p|2 /2m, giving the nonrelativistic
Hamiltonian HNR = |p|2 /2m + U with U = − 12 mc2 ψ.
Hence a = −∇U/m = (c2 /2)∇ψ, the DFD matter acceleration law.

3.

The gravitational field equation comes from varying ψ;
the matter motion law comes from varying the trajectory.
These are different variations of different fields, and the
matter law is not obtained from δS/δψ.
We are explicit about the one input Eq. (AN2) does
not supply: the origin of the mass m. The dispersion
relation and the geodesics are identical for any value of m,
whatever its source; m enters through the matter action.
Three statements bound the options. First, the appealing
reading of matter as a static finite-energy soliton of ψ
itself is excluded for the full response, not only where µ →
1: for the approximant and both branches of the exact
law the flux obeys xF ′ (x)R< 2F (x) at every
x > 0, so


Derrick’s scaling derivative
3H −F |∇ψ| d3 x is positive
for every non-trivial configuration; Rindependently, testing
the vacuum equation with ψ gives F (|∇ψ|) |∇ψ| d3 x =
0, hence ∇ψ = 0. Second, a scalar Kaluza–Klein tower on
the internal
√ manifold is excluded (every such mode has
mass ≥ α MP ≈ 1018 GeV) and is moreover non-chiral.
Third, the fermion-mass content of DFD resides instead
in the spectral-action Dirac/Yukawa sector on CP 2 × S 3
(Appendices Y and X), where the light fermions are the
chiral zero modes acquiring mass through Higgs overlap,
dressed by powers of α. That sector is chiral and subelectroweak precisely because it is not a Laplacian tower.
Equation (AN2) is thus the master equation of DFD’s
gravitational-optical sector, complete within that domain
and deliberately silent on the origin of matter.

Gravity as the gradient of the squared light speed

The two sectors share one exact statement. For a
particle momentarily at rest the all-orders acceleration is


clight = c e−ψ
a = − 14 ∇ c2light ,
(AN5)
since − 41 ∂ψ (c2 e−2ψ ) = 12 c2 e−2ψ , which reduces to
(c2 /2)∇ψ in the weak field. Equation (AN5) is exact
and mechanism-agnostic: matter falls down the gradient
of the squared local speed of light. It is the optical statement of gravitation, and it makes the unification of the
two sectors manifest.

4.

Relation to the action, and the boundary on
matter

Equations (AN2)–(AN5) descend, together with the
field equation for ψ and the radiative sector, from a single
action by varying distinct fields, a fork rather than a
linear chain:


δS
8πG
= 0 ⇒ ∇· µ(|∇ψ|/a∗ )∇ψ = − 2 ρ
δψ
c
[Eq. (21), App. N],
δS
= 0 ⇒ geodesics of g̃ (Theorem AN.1),
δxµ
δS
= 0 ⇒ □ hTT = source, cT = cγ (App. WS).
δhTT

5.

Status and falsifiability

The unification carries empirical content. Because light
and matter share one metric but couple to different invariants of it (null versus timelike), DFD keeps Etherington’s
distance duality DL = (1 + z)2 DA exact; for co-located
clocks the shared metric predicts a universal response
(Thm. XI.2), and any species-dependent local-positioninvariance signature belongs to the extended non-metric
sector (Sec. XI). The light-bending coefficient is the GR
value 4GM/c2 b by construction (γ = 1), so solar-system
tests are passed, while the deep-field sector reproduces
galaxy rotation curves through Eq. (21). The single most
distinctive statement is Eq. (AN5): gravitation is the
gradient of the squared local speed of light.

282
Appendix AO: The CP Sector: Strong-CP Protection,
the CKM No-Go, and Euler-Projected Closure

This appendix is the consolidated CP-sector theorem package. The δCP ≈ 68◦ localization estimate (App. K), the
conjectural unitarity-triangle relation γ = 4π/11, and
the post-hoc “19 + 5/2” apex correction recorded at candidate grade in the companion volume (App. AH) are
replaced by the results below; the tree-level, loop-bounded
strong-CP result of App. L is restated here in its anomaly
form. The apex value selected below, ρ̄ = 43
2 α = 21.5α,
coincides with the “Path A” candidate of the companion
volume, which the pre-strengthened axioms do not force
(the no-forcing anti-theorem below): there it is an unforced candidate, here it is derived as a theorem-grade
consequence of an explicitly added selection principle, the
Euler-projection Postulate E.1 of §AO 8. Consistent with
that anti-theorem, the postulate is a strengthening of the
corpus, not a consequence of the pre-strengthened axioms.
This appendix isolates the CP sector of Density Field
Dynamics (DFD) as a self-contained theorem package
within the unified DFD manuscript. The purpose is
to state exactly which parts of the Cabibbo–Kobayashi–
Maskawa (CKM) matrix are forced by the existing DFD
geometry, which parts require a strengthened CP-sector
postulate, and what quantitative predictions follow once
that postulate is adopted. We first prove that the minimal real DFD Yukawa kernel has vanishing CKM CP
violation: if the up- and down-sector overlap kernels are
real, then the CKM matrix is real orthogonal and the Jarlskog invariant vanishes. We then formulate the minimal
det-orthogonal CP offset: a conjugation-odd Hermitian
Berry perturbation that generates weak-sector CP violation while preserving a quark determinant that is positive
on the branch of Postulate Y.10 and hence protecting
the strong-CP angle at tree level. The anomaly side of
strong-CP protection is stated as a conditional theorem:
when the CP mapping-torus Dirac spectrum is paired by
an even-dimensional chirality symmetry, the Dai–Freed
anomaly phase is trivial; with the quark determinant positive and θYM = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields, no axion is
required at tree level inside the DFD branch.
The CKM magnitude skeleton is built from line-bundle
cohomology on the CP 2 factor and from the real dimension of X = CP 2 × S 3 . The resulting integers are
λ = 31α,

A = 108α,

η̄ = 49α.

The remaining apex ambiguity is treated explicitly. The
pre-strengthened DFD corpus contains an anti-theorem:
existing axioms do not force the choice between the
raw even-cohomology branch ρ̄ = 22α and the half-shift
branch ρ̄ = 21.5α. This appendix therefore introduces
Postulate E.1, the Euler-projection postulate, as a new
necessary selection principle in the strengthened DFD–
SD branch. It states that the physical CP-even apex

coordinate is the minimizer of

2
1
D2
Iρ (r) =
+ χ(CP 2 )r,
r−
2
2

r = ρ̄/α,

where D = dimR (CP 2 × S 3 ) = 7 and χ(CP 2 ) = 3. Stationarity gives
 2

7
43
ρ̄ =
−3 α=
α.
2
2
Thus the strengthened CP sector predicts


43
(λ, A, ρ̄, η̄) = 31, 108, , 49 α,
2
with α−1 = 137.03599985. Numerically this gives
λ = 0.2262179284, A = 0.7881140731,
ρ̄ = 0.1568930794,
γ = arctan(98/43) = 66.3093◦ ,

η̄ = 0.3575702739,
β = 22.9823◦ ,

αUT = 90.7084◦ ,

|Vus | = 0.2262164,

|Vcb | = 0.0403311,

|Vub | = 0.0036554,

|Vtd | = 0.0083576,

with exact-standard-parameterization Jarlskog invariant
JCKM = 2.9731 × 10−5 .
The appendix closes by separating theorem-grade consequences inside the strengthened DFD–SD branch from
the still-testable status of Postulate E.1, and by giving a
Fubini–Study overlap motivation for the postulate without claiming that the pre-strengthened axioms already
prove it.
1.

Purpose and scope

The CKM sector has two distinct problems. The first
is the magnitude problem: why the quark weak-mixing
matrix has the observed hierarchical shape. The second
is the phase problem: why a nonzero weak CP-violating
phase exists while the strong-CP parameter remains experimentally consistent with zero. The Standard Model
parametrizes both facts but does not derive the numerical
CKM parameters from more primitive structure. DFD
attempts to explain the hierarchy by tying the flavor
geometry to the finite internal space
X = CP 2 × S 3 ,

D := dimR X = 7,

(AO1)

and to an α-locked finite microsector.
This appendix is written with a strict status discipline.
We use the following terminology.
Definition AO.1 (Theorem-grade inside a branch). A
statement is theorem-grade inside a named branch if
it follows from explicitly stated axioms, definitions, and
postulates of that branch by ordinary mathematical proof,
with no phenomenological fitting hidden in the proof.
Definition AO.2 (Postulate-grade selection rule). A
selection rule is postulate-grade if it is introduced as a

283
new structural assumption because it is not forced by
the older axioms. Consequences of the postulate may be
theorem-grade, but the postulate itself remains a falsifiable strengthening of the branch.

admissible. This is exactly an absence-of-forcing result.
This appendix resolves the ambiguity only after adding
the Euler-projection postulate in Section AO 8. This is
the correct branch statement:

The central caveat is the following.
Anti-Theorem AO.3 (No CKM apex forcing in the
pre-strengthened corpus). In the pre-strengthened DFD
axioms, the CKM apex branches
43
ρ̄A =
α,
ρ̄B = 22α
(AO2)
2
are not uniquely separated by an existing internal symmetry. Therefore the pre-strengthened corpus cannot claim
a theorem selecting one branch over the other.

Proof. The older DFD CKM construction fixes the cohomological skeleton
λ = 31α,

A = 108α,

η̄ = 49α,

(AO3)

but the real-axis apex coordinate admits more than one
natural assignment. The raw even-cohomology assignment
gives 22α, while the half-shift or Euler-pulled assignment
gives (43/2)α. Without a rule that distinguishes raw even
cohomology from a projected complex apex, both are

DFD–SD plus Euler projection gives


43
(λ, A, ρ̄, η̄) = 31, 108, , 49 α.
2

(AO4)

It is not claimed that the original DFD axioms alone
already forced the apex.

2.

CKM and CP notation

The charged-current weak interaction couples up-type
and down-type quarks through
g
LW = √ ūLi γ µ Vij dLj Wµ+ + h.c.,
(AO5)
2
where
V = VCKM = Uu† Ud

(AO6)

is the mismatch between the unitary transformations that
diagonalize the up- and down-type quark mass matrices.
In the Wolfenstein convention,




1 − λ2 /2
λ
Aλ3 (ρ − iη)
 + O(λ4 ).
−λ
1 − λ2 /2
Aλ2
VCKM = 
3
2
Aλ (1 − ρ − iη) −Aλ
1

We use the barred unitarity-triangle coordinates (ρ̄, η̄),
which absorb higher-order Wolfenstein corrections in standard global-fit conventions. At the accuracy relevant for
the integer derivation, the branch is specified by


43
(λ, A, ρ̄, η̄) = 31, 108, , 49 α.
(AO8)
2
The CP-violating rephasing invariant is the Jarlskog invariant


∗
JCKM = Im Vij Vkl Vil∗ Vkj
,
(AO9)

θ̄ = θYM + Arg det(Mu Md ).

3.

DFD microsector and cohomological inputs

Axiom AO.4 (Internal flavor space). The DFD
CP/flavor microsector is modeled by
X = CP 2 × S 3 ,

D = dimR X = 7.

(AO13)

(AO10)

(AO11)

Axiom AO.5 (Fine-structure normalization). The DFD
microsector fixes the dimensionless coupling α used below.
Numerically,

Here
s12 = λ,
s23 = Aλ2 ,
p
η̄
s13 = Aλ3 ρ̄2 + η̄ 2 ,
δ = arctan .
ρ̄
The strong-CP parameter is

ing θ̄ = 0 or a value beneath experimental bounds. The
DFD branch in this appendix accomplishes this by placing
weak CP violation in a det-orthogonal Berry offset.

The CP 2 factor carries the holomorphic line-bundle ladder
O(k). The S 3 factor supplies the three-dimensional generation/phase geometry and the real internal dimension
contributing to D = 7.

which, in the standard parametrization, is
JCKM = c12 c23 c213 s12 s23 s13 sin δ.

(AO7)

α−1 = 137.03599985,

(AO12)

A viable CP sector must generate JCKM ̸= 0 while preserv-

α = 0.007297352528493264.
(AO14)
This appendix treats α as already fixed by the independent

284
For any indices i, j, k, l, the product

DFD α-lock sector.
Lemma AO.6 (Line-bundle dimension on CP 2 ). For
k ≥ 0,

 (k + 1)(k + 2)
h0 CP 2 , O(k) =
.
(AO15)
2

∗
Vij Vkl Vil∗ Vkj

is real because V
vanishes:

z0a0 z1a1 z2a2 ,

a0 + a1 + a2 = k,

ai ≥ 0.

(AO16)

The number of nonnegative integer triples summing to k
is the stars-and-bars count


k+2
(k + 1)(k + 2)
=
.
(AO17)
2
2
Equivalently, Hirzebruch–Riemann–Roch gives the same
result from
Z
χ(CP 2 , O(k)) =

ch(O(k))Td(T CP 2 )


Z
3
kH
2
=
e
1+ H +H ,
2
CP 2
CP 2

where

R

(AO19)

The first important theorem is negative. DFD’s minimal real Kähler/Yukawa sector cannot generate CKM
CP violation by itself.
Theorem AO.7 (Real Yukawa kernels imply JCKM = 0).
Let Mu , Md be real quark mass matrices whose Hermitian
squares
(AO21)

are real symmetric with nondegenerate spectra. Let
OuT Hu Ou = diag(m2u , m2c , m2t ),

OdT Hd Od = diag(m2d , m2s , m2b )

(AO22)
with Ou , Od ∈ O(3). Then
VCKM = OuT Od ∈ O(3)

(AO23)

is real, and hence every Jarlskog invariant vanishes:
JCKM = 0.

(AO24)

Proof. A real symmetric matrix is diagonalized by a real
orthogonal matrix. Therefore the weak-basis mismatch is
the product of two real orthogonal matrices:
V = OuT Od ∈ O(3).

This corollary is the reason the strengthened DFD–SD
branch introduces a det-orthogonal Berry offset rather
than pretending the real kernel already contains CP violation.

Strong-CP protection

θ̄ = θYM + Arg det(Mu Md )

The minimal real-kernel no-go theorem

Hd = Md MdT

Corollary AO.8 (Need for a conjugation-odd datum).
Any DFD branch with nonzero CKM CP violation must
contain an additional conjugation-odd structure beyond
the minimal real Yukawa kernel.

The strong-CP problem is the requirement that

H 2 = 1. The H 2 -coefficient is

Hu = Mu MuT ,

Thus JCKM = 0.

5.

The specific section counts used throughout are
k
0 1 2 3 4 5 6 7
(AO20)
h0 (O(k)) 1 3 6 10 15 21 28 36

4.

(AO27)

(AO18)

CP 2
2

3k
(k + 1)(k + 2)
k
+
+1=
.
2
2
2

(AO26)

= V . Therefore its imaginary part

∗
Im(Vij Vkl Vil∗ Vkj
) = 0.

2

Proof. Holomorphic sections of O(k) over CP are homogeneous polynomials of degree k in three complex
homogeneous coordinates [z0 : z1 : z2 ]. A basis is given
by monomials

∗

(AO25)

(AO28)

remain zero or extremely small. In the present branch,
weak CP is allowed only if it is det-orthogonal: it may
rotate eigenvectors and generate CKM phase, but it may
not create a complex determinant phase.
Definition AO.9 (Det-orthogonal mass deformation).
Let M (ξ) = M (0) +∆M (ξ) be a smooth one-parameter deformation of an invertible mass matrix. The deformation
is det-orthogonal at ξ = 0 if
d
log det M (ξ) = 0.
(AO29)
dξ ξ=0
It is exactly det-orthogonal on an interval if Arg det M (ξ)
is constant on that interval.
Lemma AO.10 (Antisymmetric imaginary perturbations
are Hermitian). Let S = S T be a real symmetric matrix
and let T = −T T be real antisymmetric. Then
M (ξ) = S + iξT

(AO30)

is Hermitian for real ξ.
Proof. Since S † = S T = S and (iξT )† = −iξT T = iξT ,
one has M (ξ)† = M (ξ).
Theorem AO.11 (Determinant phase protection). Let
Mf (ξ) = Sf + iξTf , for f = u, d, be the up- and downsector mass matrices of Postulate Y.10 below, with Sf =
SfT real symmetric positive definite and Tf = −TfT real
antisymmetric. Then, for sufficiently small |ξ|,


Arg det Mf (ξ) = 0,
Arg det Mu (ξ)Md (ξ) = 0.
(AO31)
Consequently the determinant contribution to θ̄ is exactly
zero on that interval.

285
Proof. Each Mf (ξ) is Hermitian. Its eigenvalues are real.
Since Sf is positive definite and eigenvalues depend continuously on ξ, there exists ϵf > 0 such that Mf (ξ) remains
positive definite for |ξ| < ϵf . Hence all eigenvalues are
positive on this interval and
det Mf (ξ) > 0.

(AO32)

Arg det Mf (ξ) = 0.

(AO33)

Therefore

Proof. If Dψ = λψ with λ ̸= 0, then
D(Γψ) = −ΓDψ = −λ(Γψ).

(AO43)

Thus every nonzero eigenvalue is paired with an eigenvalue of opposite sign and the same multiplicity. The eta
function
X
η(s) =
sgn(λ)|λ|−s
(AO44)
λ̸=0

Taking the product for f = u, d gives the second claim.

therefore vanishes term by term after analytic continuation to s = 0. Hence η(D) = 0, so the Dai–Freed phase is
exp(0) = 1.

Theorem AO.12 (Trace form of first-order det-orthogonality). Let M (ξ) = M (0) + ξ∆M + O(ξ 2 ), with M (0)
real symmetric positive definite and ∆M = iT , T = −T T
real antisymmetric. Then
d
log det M (ξ) = 0.
(AO34)
dξ 0

Corollary AO.15 (No axion required in this branch).
If the bare Yang–Mills angle is θYM = 0 (fixed by the
spectral action given that nothing is added to it for the
gauge fields; App. L), the det-orthogonal weak CP offset
preserves Arg det(Mu Md ) = 0, and the CP anomaly phase
is trivial, then

Proof. Jacobi’s determinant identity gives


d
log det M (ξ) = Tr (M (0) )−1 ∆M .
dξ 0

θ̄ = 0
(AO35)

Since (M (0) )−1 is real symmetric and T is real antisymmetric,


Tr (M (0) )−1 T = 0.
(AO36)
Indeed,
Tr(ST ) = Tr((ST )T ) = Tr(T T S T ) = Tr(−T S) = − Tr(ST ),

(AO45)

inside the DFD–SD branch at tree level, without introducing an axion. Loop corrections with the StandardModel chiral Yukawa structure shift Arg det(Mu Md ) only
through the degree-thirteen Jarlskog structure (for Hermitian tree-level mass matrices with the Standard-Model chiral couplings the shift vanishes identically through Yukawa
degree eleven; the onset is the multi-loop effect established
in Refs. [149, 150]); the induced angle is at the StandardModel level, θ̄ ≲ 10−16 [149, 150], six orders of magnitude
below the neutron-EDM bound.

(AO37)
so Tr(ST ) = 0. Multiplying by i gives zero as well.
6.

The anomaly side of strong-CP protection can be stated
as follows.
Definition AO.13 (CP mapping torus). Let σCP be the
DFD CP involution on X. The CP mapping torus is
TCP = (X × [0, 1])/(x, 0) ∼ (σCP (x), 1).

(AO38)

Since dimR X = 7,
dimR TCP = 8.

(AO39)

Theorem AO.14 (Even-dimensional spectral pairing
implies no CP anomaly). Assume the CP-twisted Dirac
operator DTCP on the mapping torus admits a chirality
operator Γ such that
Γ2 = 1,

ΓDTCP Γ = −DTCP ,

(AO40)

and that the gauge twisting preserves this pairing. Then
the nonzero spectrum is symmetric under λ 7→ −λ, the
eta invariant vanishes,
η(DTCP ) = 0,

(AO41)

and the Dai–Freed CP anomaly phase is trivial:


iπ
η(DTCP ) = 1.
(AO42)
ACP = exp
2

Det-orthogonal geometric CP and nonzero CKM
phase

Postulate AO.16 (Y.10: det-orthogonal geometric CP
offset). The strengthened DFD–SD branch contains one
conjugation-odd Berry coordinate ξ. In a generation basis
adapted to the real slice RP 2 ⊂ CP 2 , the up- and downsector mass matrices (Yukawa kernels) have the form
Mf (ξ) = Sf + iξTf + O(ξ 2 ),

f = u, d,

(AO46)

where Sf is real symmetric positive definite, Tf is real
antisymmetric, the offset direction is the A5 -Galois-odd
oriented eigendirection of the Berry bundle, and the
physical offset lies on the branch connected to ξ = 0,
where det Mf > 0. Consequently the Hermitian squares
Hf = Mf Mf† = Mf2 have the form Hf (ξ) = Sf2 +
iξ (Sf Tf + Tf Sf ) + O(ξ 2 ) with Sf Tf + Tf Sf real antisymmetric, which is the form used in the theorem below. Existence: for Sf with simple spectrum s1 < s2 < s3 , the family Mf (ξ) = eiξAf Sf e−iξAf with (Af )ab = (Tf )ab /(sb −sa )
for a ̸= b (real symmetric) in the eigenbasis of Sf is Hermitian, has ξ-independent eigenvalues and determinant,
and equals Sf + iξTf + O(ξ 2 ).

286
Theorem AO.17 (Y.10 generates weak CP generically).
Assume nondegenerate quark spectra and Y.10. If the first
nonzero variation of the commutator invariant
C(ξ) = det[Hu (ξ), Hd (ξ)]

(AO47)

is
nA = Ngen h0 (O(7)) = 108.

(AO53)

A = 108α.

(AO54)

Thus

at ξ = 0 is nonzero, then the CKM Jarlskog invariant is
nonzero for sufficiently small nonzero ξ:
2

JCKM (ξ) = Cξ + O(ξ ),

C ̸= 0.

(AO48)

Proof. For three generations, Jarlskog’s determinant identity gives
Y
Y
det[Hu , Hd ] = 2i JCKM
(m2ui − m2uj ) (m2dk − m2dl ),
i<j

k<l

(AO49)
up to the sign convention fixed by the ordering of masses
and phases. The mass-difference products are nonzero
by nondegeneracy. Therefore det[Hu , Hd ] ̸= 0 if and only
if JCKM ̸= 0. If the first nonzero derivative of C(ξ) at
ξ = 0 is linear and nonzero, then C(ξ) ̸= 0 for sufficiently
small nonzero ξ, hence JCKM (ξ) ̸= 0. The coefficient C
is the corresponding derivative divided by the nonzero
mass-difference product.
Remark AO.18. The theorem is deliberately generic. It
does not claim that any antisymmetric offset creates CP
violation. It says the Y.10 offset creates CP violation
exactly when the oriented up/down misalignment has a
nonvanishing commutator volume. This is the correct
mathematical condition.

7.

Cohomological CKM magnitude skeleton

The following three integers are selected counts
(App. FK), stated before the Euler-projection postulate
is introduced.
Theorem AO.19 (Cabibbo ladder, given the selected
count). The DFD primitive 1 ↔ 2 flavor-connector count
is
nλ = h0 (O(2)) + h0 (O(3)) + h0 (O(4)) = 31.

(AO50)

Thus
λ = 31α.

Proof. The heavy-sector doorway samples the full real
internal dimension
D = dimR (CP 2 × S 3 ) = 7.

(AO55)

The line-bundle count at k = 7 is
8·9
h0 (O(7)) =
= 36.
2
There are three generation projectors, so

(AO56)

nA = 3 · 36 = 108.

(AO57)

Thus A = 108α.
Theorem AO.21 (CP-odd height count, given the selected count). The CP-odd unitarity-triangle height count
is
nη = D2 = 49.

(AO58)

η̄ = 49α.

(AO59)

Thus

Proof. The CP-odd coordinate measures an oriented area
displacement off the totally real RP 2 slice. In the DFD
internal product space X, the primitive dimension controlling a full oriented displacement is
D = dimR X = 7.

(AO60)

An area coordinate is bilinear in independent internal
directions, so the maximal dimension-square count is
nη = D2 = 72 = 49.

(AO61)

Multiplication by the α-locked Berry amplitude gives
η̄ = 49α.
Remark AO.22 (Why this is not enough). The preceding
results fix λ, A, and η̄. They do not select the CP-even
real-axis apex coordinate ρ̄.

(AO51)

Proof. The minimal off-diagonal flavor connector excludes
k = 0, the scalar identity channel, and k = 1, the generation projector triplet. The primitive nearest-neighbor
ladder therefore begins at k = 2. The CP 2 flavor window closes at k = 4, the real dimension of the CP 2 base.
Hence

8.

Euler-projected selection of the CP-even apex

Multiplying by the α-locked amplitude unit gives λ =
31α.

Postulate AO.23 (E.1: Euler projection of the CP-even
apex). In the strengthened DFD–SD branch, the physical
value of the CP-even apex coordinate ρ̄ is not a free parameter and is not given by the raw even-cohomology sum.
It is the unique real-axis projection of the ideal complex
apex that minimizes the reduced geometric action encoding
the competition between:

Theorem AO.20 (Heavy-sector doorway, given the selected count). The DFD 2 ↔ 3 heavy-sector scaling count

1. the ideal A5 -covering-space apex scale set by the full
internal dimension D = 7, and

nλ = h0 (2) + h0 (3) + h0 (4) = 6 + 10 + 15 = 31. (AO52)

287
2. the topological localization cost of confining the CPeven coordinate to the physical CP 2 microsector,
measured by χ(CP 2 ) = 3.
Equivalently, with
r := ρ̄/α,

D := dimR (CP 2 × S 3 ) = 7,

χ := χ(CP 2 ) = 3,

Equivalently, the standard maximal torus action on CP 2
has three isolated fixed points,
p1 = [1 : 0 : 0],

p2 = [0 : 1 : 0],

p3 = [0 : 0 : 1],
(AO67)
whose Poincare–Hopf indices sum to 3.

(AO62)
the physical value of r is the unique minimizer of the
strictly convex reduced geometric action

2
D2
1
+ χ(CP 2 )r.
r−
Iρ (r) =
(AO63)
2
2

Theorem AO.27 (Euler-projected CKM apex). Assuming Postulate AO.23, the DFD–SD CP-even apex coordinate is

Status AO.24 (Status of E.1). Postulate E.1 is a new structural assumption of the strengthened DFD–SD branch.
It is not derived from the pre-strengthened DFD axioms
and is not yet derived from explicit Berry/Yukawa overlap integrals. Indeed the per-connector α unit cannot
follow from CP 2 × S 3 geometry alone: the executed overlap integrals are coupling-blind rationals (Gram δab /3,
connector moment 1/12, Verlinde hop 2 cos(π/62)), and
App. O’s derived per-mode α is multiplicative (it would
give α31 , not 31α) — so E.1 is an independent postulate,
not an App. O extension. Its consequences, including
ρ̄ = 43α/2, are theorem-grade inside the strengthened
branch once E.1 is accepted. Two further derivation routes
are closed: the compact one-boson-exchange sum on CP 2
yields α × (rational/π) with a cutoff-divergent O(1) (the
4π split g 2 = α vs. g 2 = 4πα being conventional, cf. the
App. O coupling-normalization remark), and holonomy
on simply-connected CP 2 is topologically trivial (a unitmodulus phase, never an α-sized amplitude). No structural no-go, however, blocks E.1’s eventual derivation:
the live path is a selection rule forcing inter-generation
connectors to couple through the internal gauge field (exchange, which scales as α1 per connector and is additive
across connectors) rather than by coupling-blind direct
overlap.

For D = 7, this gives

Remark AO.25 (Meaning of the two terms). The
quadratic term anchors r to the ideal A5 covering-space
apex. The linear term is the Euler localization cost of
confining the CP-even coordinate to the physical CP 2
microsector. Since χ(CP 2 ) = 3, it represents one unit of
α-normalized real-axis localization cost per topological
generation anchor.

(AO64)
2

Proof. The integral cohomology of CP is
H 0 (CP 2 ; Z) = Z,
H 2 (CP 2 ; Z) = Z,
4

(AO65)

2

H (CP ; Z) = Z,
and all odd cohomology groups vanish. Therefore
χ(CP 2 ) = b0 − b1 + b2 − b3 + b4 = 1 + 1 + 1 = 3. (AO66)


D2
− χ(CP 2 ) α.
2

(AO68)

43
α = 21.5α.
2

(AO69)

ρ̄ =

Proof. The reduced action (AO63) is strictly convex because
d2 Iρ
= 1 > 0.
dr2

(AO70)

Its unique stationary point is therefore its unique global
minimizer. Compute
dIρ
D2
=r−
+ χ(CP 2 ).
dr
2
Set the derivative to zero:
D2
r∗ =
− χ(CP 2 ).
2
Since r = ρ̄/α,
 2

D
ρ̄ = r∗ α =
− χ(CP 2 ) α.
2
For D = 7 and χ(CP 2 ) = 3,
49
49 − 6
43
r∗ =
−3=
=
.
2
2
2
Thus
ρ̄ =

a.

Lemma AO.26 (Euler characteristic of CP 2 ).
χ(CP 2 ) = 3.


ρ̄ =

43
α.
2

(AO71)

(AO72)

(AO73)

(AO74)

(AO75)

Fubini–Study overlap motivation for Postulate E.1

Postulate E.1 is structurally natural from the overlap
geometry, but this appendix does not claim that this
motivation eliminates E.1 as an independent strengthenedbranch postulate. The point of the following argument is
to show why the linear Euler term has the right geometric
form.
The three real generation anchors are naturally associated with the three torus fixed points of CP 2 :
p1 = [1 : 0 : 0],

p2 = [0 : 1 : 0],

p3 = [0 : 0 : 1].
(AO76)

288
By Poincare–Hopf localization,
X
index(pi ) = χ(CP 2 ) = 3.

9.

(AO77)

Final Wolfenstein branch and numerical
reconstruction

i

Let
(0)

Gij (r) = ⟨wi (r) | wj (r)⟩

(AO78)

denote the real unperturbed Gram matrix for generation
wavefunctions localized near these fixed points. If the
leading diagonal localization factor in α-normalized realaxis coordinate r is
⟨wi (r) | wi (r)⟩ ∼ e−r ,

(AO79)

and if off-diagonal overlaps are negligible or symmetrycontrolled at leading order, then
det G(0) (r) ≈

3
Y

⟨wi (r) | wi (r)⟩ ∼ e−3r .

(AO80)

i=1

Taking minus the logarithm gives
− log det G(0) (r) ∼ 3r = χ(CP 2 )r.

(AO81)

This is precisely the linear Euler-localization term in
(AO63).
Remark AO.28 (Why this remains motivation rather than
proof). A full derivation of E.1 from microscopic overlaps
would have to prove the exact normalization
⟨wi (r) | wi (r)⟩ = e−r

(AO82)

in the same r = ρ̄/α coordinate, show that the Gram
determinant factorizes exactly or with a controlled correction, and derive the quadratic covering-space term from
the A5 Berry/Yukawa action. Until those steps are supplied, the Fubini–Study argument motivates E.1 but does
not replace it.
Corollary AO.29 (Unitarity-triangle angle). The
strengthened DFD–SD branch predicts


 
 
49
98
η̄
= arctan
= arctan
.
γ = arctan
ρ̄
43/2
43
(AO83)
Numerically,
γ = 66.3093◦ .

(AO84)

Proof. Using η̄ = 49α and ρ̄ = (43/2)α, the factor α
cancels:
η̄
49α
98
=
=
.
ρ̄
(43/2)α
43
Taking the arctangent gives the result.

Theorem AO.30 (Strengthened DFD–SD CKM branch).
In the DFD–SD branch with Euler projection,


43
(λ, A, ρ̄, η̄) = 31, 108, , 49 α.
(AO86)
2
Proof. Combine the Cabibbo ladder theorem, the heavysector doorway theorem, the CP-odd height theorem, and
the Euler-pulled apex theorem.

(AO85)

With α−1 = 137.03599985, this gives
α = 0.007297352528493264,

(AO87)

λ = 31α = 0.2262179283832912,

(AO88)

A = 108α = 0.7881140730772725,
43
α = 0.1568930793626052,
ρ̄ =
2
η̄ = 49α = 0.3575702738961699.

(AO89)

The unitarity-triangle angles are
 
η̄
γ = arctan
= 66.3093105972◦ ,
ρ̄


η̄
β = arctan
= 22.9822978300◦ ,
1 − ρ̄
αUT = 180◦ − β − γ = 90.7083915728◦ .

(AO90)
(AO91)

(AO92)
(AO93)
(AO94)

For an exact-standard reconstruction, define
s12 = λ,

(AO95)
2

(AO96)

3

(AO97)

s23 = Aλ ,
iδ

s13 e = Aλ (ρ + iη),
q
cij = 1 − s2ij ,

(AO98)

where the unbarred apex (ρ, η) follows from the theorem’s
barred (PDG-comparable) apex by the exact standardparameterization inversion
√
1 − A2 λ4 (ρ̄ + iη̄)

,
ρ + iη = √
(AO99)
1 − λ2 1 − A2 λ4 (ρ̄ + iη̄)
a conversion of modulus (ρ + iη)/(ρ̄ + iη̄) = 1.026040.
Holding the theorem’s barred values exact, this gives
s23 = 0.0403313839226825,

(AO100)

s13 = 0.00365535209821,

(AO101)

◦

δ = 66.3426441103 .

(AO102)

The standard CKM matrix is


c12 c13
s12 c13
s13 e−iδ
V = −s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 eiδ s23 c13  .
s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ c23 c13


(AO103)

289
The predicted magnitudes are


0.9740702 0.2262164 0.0036554
DFD
|VCKM
| = 0.2260915 0.9732708 0.0403311 .
0.0083576 0.0396246 0.9991797

Current direct/global determinations of γ are not yet
sharp enough to decisively separate the two. The strengthened DFD–SD branch chooses Path A not because current
data slightly prefer it, but because the Euler-projection
action selects it.

(AO104)
∗
The apex realized by this matrix, −(Vud Vub
)/(Vcd Vcb∗ ),
reproduces the theorem values (ρ̄, η̄) = (43/2, 49) α to
|residual| = 1.1 × 10−31 , and its tree-level angle is γ =
66.3093106◦ , in exact agreement with the boxed falsifier.
At leading Wolfenstein order the main entries are
|Vus | = 0.2262179284,

ρ̄
γ
|Vub |
|Vtd |
Jlead

(AO105)

2

|Vcb | = Aλ = 0.0403313839,
(AO106)
p
|Vub | = Aλ3 ρ̄2 + η̄ 2 = 0.0035625840,
(AO107)
p
3
|Vtd | = Aλ (1 − ρ̄)2 + η̄ 2 = 0.0083554489. (AO108)
These leading-order barred expressions omit the O(λ2 /2)
barred→unbarred conversion carried by the exact reconstruction above; the exact-matrix values are the ones
quoted in the boxed matrix.
Theorem AO.31 (Jarlskog invariant). The strengthened
DFD–SD CKM branch predicts
JCKM = 2.9731 × 10−5

(AO109)

using the exact standard-parameterization expression.
Proof. The exact expression is
JCKM = c12 c23 c213 s12 s23 s13 sin δ.

(AO110)

Substituting the values above gives
JCKM = 2.9731063041 × 10−5 .

(AO111)

At leading Wolfenstein order,
JCKM ≈ A2 λ6 η̄.

(AO112)

Using the DFD integers,
A2 λ6 η̄ = (108α)2 (31α)6 (49α)
= 1082 · 316 · 49 · α9 = 2.9764712924 × 10−5 .
(AO113)
The difference between the exact and leading values is
the expected higher-order trigonometric correction.

10.

Quantity Path A: 43α/2 Path B: 22α

Comparison of the apex branches

The Euler-projected branch is not the only mathematically possible branch before the postulate is added. The
two relevant branches are:
43
Path A: ρ̄A =
α = 0.1568930794,
(AO114)
2
Path B: ρ̄B = 22α = 0.1605417556.
(AO115)

a |V

ub | and |Vtd | are quoted at leading (barred-Wolfenstein)
order for both paths; the exact Path-A values are in the
boxed matrix above.

The small numerical separation is why the anti-theorem
matters. The theory needed a structural selection rule,
not a statistical preference.
11.

Falsifiability and experimental targets

The branch makes sharp predictions:
γ = 66.3093◦ ,

◦

γA = arctan(98/43) = 66.3093 ,

(AO116)

◦

(AO117)

γB = arctan(49/22) = 65.8209 .

β = 22.9823◦ ,

JCKM = 2.9731 × 10−5 .

(AO118)
The most direct discriminator is the angle γ. The DFD–
SD value is essentially coincident with the current CKMfitter indirect global-fit central value, γCKMfitter ≈ 66.3◦ (a
difference of ∼ 0.01◦ ) — the best consistency check against
the full CKM unitarity structure — and it lies well inside
present direct tree-level determinations, which are the
cleaner future falsifier because they are less contaminated
by loop/new-physics assumptions: the LHCb combination
◦
γ = (63.8+3.5
−3.7 ) (arXiv:2404.18789) places DFD–SD about
◦
2.5 away, comfortably within 1σ, while the charm/beauty
global analysis γ = (65.7 ± 2.5)◦ (arXiv:2409.06449) is
only 0.61◦ (0.24σ) away. The decisive test is whether
future LHCb and Belle II direct determinations converge
near 66.3◦ as uncertainties approach the 1◦ scale.
The strongest falsifier is therefore:
γDFD–SD = 66.3093◦ .

(AO119)

If future tree-level determinations converge far from this
value with sub-degree uncertainty, the Euler-projected
DFD–SD CKM branch is falsified. The magnitudes are
tested as well: λ = 31α sits 1.8σ above the global fit, and
the kaon-decay determinations of |Vus | lie 3 to 6 standard
deviations below 31α; |Vcb | = 0.0403 agrees with the
exclusive determination and lies 3.7σ below the inclusive
one; against the global fit the four values give χ2 = 10.1
for four parameters (computed).
12.

They give

0.1568931
0.1605418
66.3093◦
65.8209◦
a
0.0035626
0.0035761a
a
0.0083554
0.0083248a
−5
2.9765 × 10
2.9765 × 10−5

Scope and order of the claims

The claims of this appendix obey the following discipline.

290
η̄
43
α, 49α
2

49α

γ = 66.309◦
0




β = 22.982◦
1

43
α
2

ρ̄

FIG. 16. Euler-projected DFD–SD unitarity triangle. The apex is pulled from the ideal covering-space value D2 /2 by the Euler
cost χ(CP 2 ) = 3, giving ρ̄ = (49/2 − 3)α = (43/2)α.

1. Pre-strengthened DFD does not force the apex
branch; the anti-theorem stating this is kept explicit.
2. No unconditional claim makes the raw 22α apex
theorem-grade.
3. Postulate E.1 is a necessary strengthening of the
DFD–SD branch for CKM closure; it is not claimed
to follow from the pre-strengthened axioms.
4. The consequences of that postulate are theoremgrade inside the strengthened branch.
5. The CKM integer list is


43
(λ, A, ρ̄, η̄) = 31, 108, , 49 α.
2

(AO120)

6. The minimal real-kernel no-go theorem is stated
before Y.10 is introduced.
7. The determinant-phase protection theorem is stated
before the weak CKM phase is calculated.
8. The main falsifier is γ = 66.3093◦ .
13.

Final theorem ledger
14.

Conclusion

The CP sector of DFD separates what the prestrengthened axioms prove from what the strengthened
branch adds. The real kernel cannot generate CKM CP
violation. That is a theorem. The det-orthogonal Berry
offset supplies a minimal weak-CP mechanism while preserving strong-CP protection. The cohomological magnitude skeleton fixes λ = 31α, A = 108α, and η̄ = 49α. The
remaining apex ambiguity is resolved only after adding
Postulate E.1, the Euler-projection rule of the strengthened DFD–SD branch, whose reduced action has a unique
minimizer at
 2

7
43
ρ̄ =
− χ(CP 2 ) α =
α.
(AO121)
2
2
This yields the final strengthened-branch CKM solution


43
(λ, A, ρ̄, η̄) = 31, 108, , 49 α,
(AO122)
2

with
γ = 66.3093◦ ,

JCKM = 2.9731 × 10−5 .

(AO123)

The result is not an unconditional proof from the older
DFD axioms. It is a theorem-grade closure inside the
strengthened DFD–SD branch with Euler projection.

291
TABLE CXII. Final CP-sector theorem ledger for the strengthened DFD–SD branch.
Claim

Result

Minimal real kernel
Strong-CP determinant
protection
CP anomaly

Status

Cabibbo ladder

Real Yukawa kernels imply J = 0.
Det-orthogonal offsets of a positive-definite Sf preserve
Arg det(Mu Md ) = 0 on the branch connected to ξ = 0.
Even-dimensional chirality pairing gives η = 0, Dai–Freed phase
= 1.
One conjugation-odd Berry coordinate supplies weak CP while
preserving determinant phase.
λ = [6 + 10 + 15]α = 31α.

Postulate + theorem
consequences
Selected count

Heavy-sector doorway

A = 3h0 (7)α = 108α.

Selected count

Y.10 CP offset

2

CP-odd height
Apex selection

η̄ = 7 α = 49α.
ρ̄ = (49/2 − 3)α = 43α/2.

CKM prediction

γ = 66.3093◦ , J = 2.9731 × 10−5 .

Appendix AP: Corpus-Closure Theorems: the α57
Hierarchy, the MOND Scale, and Rigor Discipline

This appendix is the consolidated corpus-closure theorem
package for the gravitational/MOND sector. It collects
the α57 cosmological clock
√ hierarchy, the action-derived
MOND prefactor a0 = 2 α cH0 , and the H0 -free acceleration invariant a20 Gℏ/c7 = 4Cα58 (C the clock coefficient,
1 on bound-matter clocks, Postulate ZS.11). The minimal
real-kernel J = 0 statement below is consistent with, and
cross-references, the CP-sector treatment of Appendix AO.
This appendix presents the core closure chain of Density
Field Dynamics (DFD). The primary closed result is the
acceleration invariant
r
√
a20 Gℏ
c7
58
29
= 4C α ,
a0 = 2 C α
,
7
c
Gℏ
which predicts the galactic MOND/RAR acceleration
scale with a derived exponent; galaxies read bound-matter
clocks, where C = 1 (Postulate ZS.11), so it carries
no measured coefficient (the clock normalization, Remark AP.6). The appendix gives the proof chain: a
Spinc /index mode count 60 − 3 = 57, a primed Gaussian determinant producing α57 , the finite spectral-action
clock dictionary GℏH02 /c5 = Cα57 , a gauge-emergence
proof of ka = 3/(8α), an S 3 stationarity theorem selecting
√
Ξ∗ = 3/2, and the action-level derivation a0 = 2 α cH0 .
1.

Purpose and standard of proof

The central target is the acceleration invariant
Π2 ≡

a20 Gℏ
= 4C α58 ,
c7

(AP1)

with C the clock coefficient, 1 on bound-matter clocks
(Postulate ZS.11) and measured on the early clock (Re-

Theorem
Theorem
Conditional theorem

Selected count
Postulate E.1 + theorem
consequence
Falsifiable prediction

mark AP.6). This is equivalent to
r
√
c7
29
a0 = 2 C α
.
Gℏ

(AP2)

Eq. (AP1) could be read as numerology: the exponent
58 might appear to have been inferred from the observed
acceleration scale. The proof below derives the exponent
as
58 = (60 − 3) + 1.

(AP3)

The 60 − 3 = 57 comes from a finite microsector determinant; the extra
√ +1 comes from the squared action-derived
prefactor 2 α in the scalar acceleration crossover.
Definition AP.1 (Strict theorem-grade status). A result
is called strict theorem grade in this paper only if:
1. every coefficient has a non-adjustable origin;
2. every dimensionful identification is stated as either
a theorem or a declared dictionary axiom;
3. no observational value is used to choose an exponent
or integer;
4. branch-dependent calculations are not mixed;
5. contradictory results are either resolved, retired, or
demoted to archived consistency-check notes.
Remark AP.2 (Hard scoping rule). A corpus can meet this
standard without every speculative sector being a theorem.
What makes it meet the standard is that theorem-grade
claims are proved and non-theorem claims are not mislabeled as theorems.

292
2.

No result may be theorem-grade if one of its coefficients is
described as plausible, calibrated, motivated, or empirically
selected.

Notation and DFD primitives

We use
c = reference two-way light speed,

(AP4)

G = Newton constant,

(AP5)

ℏ = reduced Planck constant,

(AP6)

α = fine-structure constant,

(AP7)

H0 = cosmological clock scale,

(AP8)

a0 = galactic acceleration crossover.

(AP9)

The Planck time, length, and acceleration are
r
r
r
Gℏ
Gℏ
c7
c
c2
,
ℓP =
,
aP =
tP =
=
=
.
5
3
c
c
Gℏ
tP
ℓP
(AP10)
The DFD scalar density field is denoted by ψ. In the
weak/static optical sector,
n(x) = eψ(x) ,

(AP11)

and the physical acceleration field is normalized as
c2
g = ∇ψ.
(AP12)
2
This factor 1/2 is not optional in the closure theorem: it is
the conversion between scalar-gradient scale and physical
acceleration.

3.

Corpus-level rigor axioms

Axiom AP.3 (Branch discipline). A calculation performed in one DFD branch cannot be quoted as a theorem in another branch unless an embedding theorem
identifies the two branches. In particular, the isotropic
exponential exterior (the parent strain’s σ set to zero),
the PPN-closed physical metric, a nonminimal horizonclosure model, and a quantum-corrected vacuum action
are distinct until proved equivalent.
Axiom AP.4 (Status discipline). This appendix assigns
every claim exactly one of the following statuses:
Status
Theorem

Meaning

Derived
from
declared
axioms/dictionary
with
no
fitted
continuous parameter.
Proposition Mathematically derived but dependent
on a declared dictionary map.
Model pre- Computed inside one explicitly named
diction
model branch.
PhenomData-matching or effective description
enological
with stated correction budget.
closure
Conjecture Structurally motivated but not proved.
Program
Calculation, simulation, or formal
item
proof not yet completed.
Retired
Contradictory, false, duplicated, or
branch-incompatible.

Axiom AP.5 (Finite spectral-action clock dictionary).
The finite DFD spectral action identifies the primed internal Gaussian determinant ratio with the unique dimensionless cosmological clock hierarchy, up to an O(1)
normalization coefficient C:
GℏH02
Z′
= C α′ ,
C = O(1). (AP13)
5
c
Z1
This dictionary is the bridge between finite internal topology and the infrared de Sitter clock scale: it fixes the
exponent of the hierarchy. The coefficient C is not fixed
by the dictionary. It is today’s value of the early clock,
and the time-independent measured input is the vacuum
coefficient R = ΩΛ C (Remark AP.6). If this axiom is not
adopted, the α57 exponent statement must be downgraded
from theorem to proposition.
(H0 tP )2 =

Remark AP.6 (What is derived vs. what is measured).
The derivation order is clean and non-circular: the exponent 57 = 60 − 3 is pure Hirzebruch–Riemann–Roch
arithmetic with no cosmological input, and H0 first appears downstream as a-posteriori validation. What the
exponent theorem does not supply is the O(1) coefficient
C in (H0 tP )2 = C α57 . An eight-route, enumerationcomplete no-go proves C is not derivable from the printed
corpus: every printed equation connecting the vacuum deposit to a rate is exponent-only, convention-importing, or
C-covariant (the determinant ratio transmits a coordinate
but cannot fix one); the homogeneous ψ-sector Hamiltonian vanishes identically, and the background obeys the
Friedmann law of the Einstein–Hilbert term (§XIX E),
which converts the vacuum density into the clock rate
but fixes neither; and the printed spectral route cannot
land the measured value for any regulator in its forced
class. Nor can C share the normalization of G. In the
spectral action α is fixed by its own trace (§X C), G by
f2 Λ2 (Proposition WS.5) and every vacuum term by f4 Λ4 ,
f2 Λ2 TrDF2 and f0 TrDF4 , while the determinant ratio α57
carries no moment. With G held fixed, admissible plateau
cutoffs move f4 /f22 over [1/2, ∞), so no relation fixed by
G fixes C. If the vacuum is the α57 -suppressed a0 term,
a reading that is a conjecture, R = ΩΛ C = 6π(f4 /f22 )/N ;
for N = 60 the measured R requires f4 /f22 = 1.923±0.009
with the internal masses neglected (1.81 with the printed
internal spectrum), a heavy-tailed cutoff that no printed
condition selects, and every compact Riesz cutoff misses
by 109σ or more and the sharp cutoff by 168σ (computed). The requirement depends on N : a trace over the
4096 states of the α carrier would need 131. C therefore
plays the role of the cosmological constant of general relativity: the theory carries G, which sets the unit, and
one measured vacuum coefficient. The measured input
is therefore one continuous number, measured once, as
ΛCDM measures H0 . Because C = (HtP )2 /α57 changes
with the epoch, C(z) = R+1.924 [ωm (1+z)3 +ωr (1+z)4 ],

293
the time-independent input is the vacuum coefficient
R = ΩΛ C = 0.6042 ± 0.0027, and C is today’s value
of the early clock. The measurements are concordant:
C = 0.87784±0.00266 (BAO+θ∗ ) and C = 0.8772 (Planck
plik-lite), i.e. H0 = 67.52 ± 0.09 km/s/Mpc across five
concordant CMB-anchored channels. The value C = 1,
read as today’s early-clock value, gives H0 = 72.09 and
is excluded at ≈ 46σ by the CMB-anchored channels,
while bound-matter clocks read C = 1 (Postulate ZS.11),
and every closed formula proposed for C is a statement
about the present epoch. A further route, fixing the
whole shape of the spectral cutoff from a printed principle, fails as well: no printed principle fixes that shape,
and R depends on the internal (Kaluza–Klein) spectrum.
Status split: exponent 57 — theorem; coefficient C —
measured input. Rival exponent conventions remain excluded numerically, not by fiat: one unit of the exponent
moves H0 by a factor α1/2 ≃ 0.085, so no other integer is reachable and the measured C is an O(1) number,
not a disguised exponent shift. In coefficient space the
SH0ES-side booking C = 1.027 ± 0.029 (five-presentation
Λ theorem of the Extended Derivations; SH0ES’s 2022 H0
re-expressed) versus the acoustic C = 0.87784 ± 0.00266
is a ∼ 5σ(stat) split; under the two-clock postulate the
booking is a bound-matter reading, 0.9σ from C = 1, and
the absolute a0 scale (Euclid/DESI lensing RAR) tests
the postulate once baryonic masses are known to about
2% (App. ZS). Bookkeeping note: Lemma O.6 states the
ratio as a squared scale ratio while the Wheeler–DeWitt
ground-state theorem of the Extended Derivations states
an energy density; the two differ by the Friedmann factor
8π/3; the measured coefficient is R = ΩΛ C in density
form and C in clock form.
Axiom AP.7 (Scalar acceleration action dictionary).
The physical acceleration scale is extracted from the scalar
action through Eq. (AP12); the crossover a0 is the point
where the dimensionless acceleration invariant Ξ reaches
the stationary value selected by the S 3 microsector. It is
not inserted as a phenomenological fit parameter.

4.

The α57 hierarchy from topology and determinant
scaling
a.

Index count: kmax = 60

Lemma AP.8 (Hirzebruch–Riemann–Roch on RCP 2 ). Let
H ∈ H 2 (CP 2 , Z) be the hyperplane class with CP 2 H 2 =
1. For the holomorphic line bundle O(n),
χ(CP 2 , O(n)) =

(n + 1)(n + 2)
.
2

Proof. Hirzebruch–Riemann–Roch gives
Z
χ(CP 2 , O(n)) =
ch(O(n)) Td(T CP 2 ).
CP 2

(AP14)

(AP15)

For CP 2 ,
n2 2
H ,
2

(AP16)

3
Td(T CP 2 ) = 1 + H + H 2 .
2

(AP17)

ch(O(n)) = enH = 1 + nH +
while

The integral extracts the coefficient of H 2 :
3n
n2 + 3n + 2
(n + 1)(n + 2)
n2
+
+1=
=
. (AP18)
2
2
2
2
Theorem AP.9 (Finite microsector mode count). For
the DFD internal finite sector
E = O(9) ⊕ O⊕5

(AP19)

2

over CP , the total mode count is
kmax = 60.
Proof. Using Lemma AP.8,
(10)(11)
= 55.
χ(CP 2 , O(9)) =
2
Also χ(CP 2 , O) = 1. Therefore

(AP20)

(AP21)

χ(CP 2 , E) = 55 + 5χ(CP 2 , O) = 55 + 5 = 60. (AP22)

Axiom AP.10 (Generation kernel). The protected generation kernel of the finite DFD microsector has dimension
Ngen = 3.

(AP23)

These three zero modes are not included in the primed
determinant.
Corollary AP.11 (Nonzero determinant count). The
primed determinant contains exactly
Nnz = kmax − Ngen = 60 − 3 = 57

(AP24)

nonzero complex modes.

b.

Primed determinant scaling

Lemma AP.12 (One complex Gaussian mode). For
λ > 0,


Z
λ 2
πα
2
d z exp − |z| =
.
(AP25)
α
λ
C
Consequently, relative to the α = 1 normalization, one
nonzero complex mode contributes a factor α.
Proof. Writing z = x + iy gives
Z ∞
2
Z
2
2
πα
d2 z e−(λ/α)|z| =
e−(λ/α)x dx =
.
λ
C
−∞
(AP26)
At α = 1 the same integral is π/λ, so the ratio is α.

294
Theorem AP.13 (Primed determinant hierarchy). Let
the nonzero internal kinetic operator have eigenvalues
λi > 0 for i = 1, . . . , 57, and let the gauge-normalized
quadratic action be
Smicro =

57
X
λi

α
i=1

|zi |2 .

(AP27)

Then
Zα′
= α57 .
Z1′

(AP28)

det′ K
= α57 .
det′ (K/α)

(AP29)

Equivalently,

Proof. By Lemma AP.12, each nonzero complex mode
contributes α to the normalized partition ratio. There
are 57 such modes by Corollary AP.11. Hence
57
Y
Zα′
α = α57 .
=
Z1′
i=1

(AP30)

The determinant statement follows because
57
Y λi
′
′
det(K/α) =
= α−57 det K.
α
i=1

(AP31)

Thus det′ K/ det′ (K/α) = α57 .
c.

GℏH02
= (H0 tP )2 .
c5

By Theorem AP.13, the determinant ratio is α57 . Therefore Eq. (AP35) follows.
Remark AP.16 (What is theorem-grade, what is dictionary, and what is measured). The determinant scaling
Zα′ /Z1′ = α57 is a theorem once E = O(9) ⊕ O⊕5 and
Ngen = 3 are declared. The identification of this determinant ratio with (H0 tP )2 is a DFD spectral-action
dictionary axiom, and the O(1) coefficient C is a measured
input (Remark AP.6): the exponent is theorem-grade, the
coefficient is not.
5.

The action-derived MOND prefactor

The second parent result needed for the acceleration
invariant is
√
a0 = 2 α cH0 .
(AP37)
This section derives it from the scalar action and internal
gauge normalization.

[c] = LT −1 ,

Gauge-emergence proof of ka = 3/(8α)

α=

GℏH02 c−5 .

GℏH02
= C α57 ,
c5
C = 0.87784 ± 0.00266 (early clock, measured),
C = 1 (bound-matter clocks, Postulate ZS.11).

(AP35)

(AP39)

the magnetic fine-structure constant is
αM =

g2
1
=
.
4π
4α

(AP40)

2π
.
e

(AP41)

Proof. From eg = 2π,
g=
Therefore
1
αM =
4π

(AP34)

Theorem AP.15 (α57 master invariant). Under the finite
spectral-action clock dictionary, the DFD cosmological
hierarchy is

(AP38)

eg = 2π,

[H0 ] = T −1 ,

(AP33)
the mass exponent gives −a + b = 0, so b = a. The length
exponent gives 3a+2b+d = 0, hence 5a+d = 0. The time
exponent gives −2a − b − d − e = 0, hence −3a − d − e = 0.
With d = −5a, this gives e = 2a. Taking the primitive
choice a = 1 gives

e2
,
4π

and minimal Dirac quantization

(AP32)

Proof. Let Ga ℏb cd H0e be dimensionless. With dimensions
[ℏ] = M L2 T −1 ,

(AP36)

Lemma AP.17 (Magnetic dual coupling). In rationalized
units with

Lemma AP.14 (Uniqueness of the H0 dimensionless
group). The unique primitive dimensionless monomial
formed from G, ℏ, c, H0 is

[G] = L3 M −1 T −2 ,

GℏH02
Z′
= C α′ .
5
c
Z1

a.

The cosmological invariant

IH =

Proof. By Axiom AP.5,



2π
e

2
=

π
.
e2

(AP42)

Since e2 = 4πα,
αM =

π
1
=
.
4πα
4α

(AP43)

Axiom AP.18 (Internal gauge partition). The DFD
internal frame partition relevant to the scalar acceleration
channel has backbone and doorway dimensions
n3 = 3,

n2 = 2.

(AP44)

295
The frame-stiffness ratio entering the scalar backreaction
channel is n3 /n2 .
Theorem AP.19 (Acceleration-channel self-coupling).
The DFD scalar acceleration-channel coupling is
ka =

3
.
8α

(AP45)

Proof. By Axiom AP.18, the frame factor is
n3
3
= .
(AP46)
n2
2
By Lemma AP.17, the magnetic dual coupling is 1/(4α).
The acceleration-channel coupling is the product
n3
3 1
3
ka =
αM = ·
=
.
(AP47)
n2
2 4α
8α
Corollary AP.20 (Generation-normalized loop coefficient). If the same coupling is written in the generationnormalized form
ka = Ngen Cloop α−1 ,

(AP48)

with Ngen = 3, then
Cloop =

1
.
8

Thus
√

 3
2π − log(k + 2) + O((k + 2)−2 ),
2
(AP55)
so −∂ log ZS 3 /∂ log(k + 2) = 3/2 + O(k −2 ).
log ZS 3 = log

Definition AP.23 (Dimensionless acceleration invariant).
The scalar action uses the dimensionless invariant

2
|a|
.
(AP56)
Ξ = ka
cH0
Theorem AP.24 (S 3 stationary crossover). The local
topological potential
3
U (Ξ) = Ξ − log Ξ
(AP57)
2
selects the crossover value
3
Ξ∗ = .
(AP58)
2
Proof. Stationarity gives
dU
3
=1−
= 0.
dΞ
2Ξ
Solving gives Ξ = 3/2.

(AP59)

(AP49)
c.

Action-level derivation of the MOND prefactor

Proof. Set 3Cloop α−1 = 3/(8α). Cancel 3α−1 .
Remark AP.21 (The origin of the coefficient 1/8). The
coefficient 1/8 is not used as an unexplained heat-kernel
guess. It is the generation-normalized form of the independent product
n3 1
3 1
3
(AP50)
· = · = .
n2 4
2 4
8
This is stronger because every factor has a named origin.
b.

The S 3 stationarity theorem

Lemma AP.22 (S 3 scaling charge). For the SU (2)
Chern–Simons partition function on S 3 ,
r


2
π
ZS 3 (k) =
sin
,
(AP51)
k+2
k+2
the large-level scaling charge is
∂ log ZS 3
3
qS 3 = −
= + O(k −2 ).
(AP52)
∂ log(k + 2)
2

Theorem AP.25 (MOND prefactor from scalar action).
With ka = 3/(8α) and Ξ∗ = 3/2, the scalar acceleration
crossover is
√
a0 = 2 α cH0 .
(AP60)
Proof. At the crossover |a| = a0 , so Definition AP.23 and
Theorem AP.24 give

2
a0
3
(AP61)
ka
= .
cH0
2
Using Theorem AP.19,

2
3
a0
3
= .
(AP62)
8α cH0
2
Cancel 3 and solve:
2

a0
= 4α.
(AP63)
cH0
Taking the positive square root gives Eq. (AP60).

The topological scaling exponent is therefore 3/2.
Proof. For large k,


π
π
sin
=
+ O((k + 2)−3 ).
(AP53)
k+2
k+2
Hence
r


2
π
ZS 3 (k) =
+ O((k + 2)−3 )
k+2 k+2
(AP54)
√


−3/2
−2
= 2π (k + 2)
1 + O((k + 2) ) .

6.

The nonlinear kinetic action and galaxy law

The correct galaxy mechanism is not a direct source
term ka a2 /c2 . The acceleration scale a0 controls the
nonlinear kinetic branch of the scalar action.
Definition AP.26 (Static scalar kinetic energy). Let
Y =

|∇ψ|2
,
κ2α

κα =

2a0
.
c2

(AP64)

296
Define

In the deep branch µ(g/a0 ) = g/a0 , hence

Z
Z
c2
c4
κ2α F(Y ) d3 x −
ρψ d3 x. (AP65)
Eψ [ψ] =
32πG
2
The interpolation function obeys
2
F(Y ) ∼ Y (Y ≫ 1),
F(Y ) ∼ Y 3/2 (Y ≪ 1).
3
(AP66)

g2
GM
= gN = 2 .
a0
r
2
For circular motion g = v /r. Therefore
v4
GM
= a0 2 ,
r2
r

(AP77)

(AP78)

so v 4 = GM a0 .

Theorem AP.27 (Euler–Lagrange equation). The scalar
energy (AP65) yields
8πG
(AP67)
∇ · [F ′ (Y )∇ψ] = − 2 ρ.
c
Equivalently, in terms of g = (c2 /2)∇ψ,
   
|g|
∇· µ
(AP68)
g = −4πGρ ,
a0

Corollary AP.30 (Rotation curves from the kinetic
equation). If a term ka a2 /c2 is too small to act as a direct
galactic mass source, that does not invalidate the MOND
mechanism. In DFD, ka (a/cH0 )2 is the crossover-control
invariant Ξ in the scalar action, while galaxy rotation
curves arise from the nonlinear kinetic equation (AP68).

where

7.

µ(s) = F ′ (s2 ),

s=

|g|
.
a0

(AP69)

Proof. Vary Eq. (AP65). Since
2∇ψ · ∇δψ
,
(AP70)
δY =
κ2α
we have
Z
2∇ψ · ∇δψ 3
c4
κ2α F ′ (Y )
d x
δEψ =
32πG
κ2α
Z
c2
−
ρδψ d3 x
(AP71)
2
Z
Z
c4
c2
′
3
=
F (Y )∇ψ · ∇δψ d x −
ρδψ d3 x.
16πG
2
(AP72)
Integrating by parts and discarding boundary variations,

Z 
c2
c4
∇ · (F ′ (Y )∇ψ) − ρ δψ d3 x.
δEψ =
−
16πG
2
(AP73)
Stationarity for arbitrary δψ gives Eq. (AP67). Multiplying by c2 /2 inside the divergence converts ∇ψ to g and
gives Eq. (AP68).
Corollary AP.28 (Newtonian and deep branches). The
limits (AP66) imply
µ(s) → 1

(s ≫ 1),

µ(s) → s

(s ≪ 1).

(AP74)

′

2

Proof. If Y = s ≫ 1, F(Y ) ∼ Y , so F (Y ) → 1. If Y =
s2 ≪ 1, F(Y ) ∼ (2/3)Y 3/2 , so F ′ (Y ) ∼ Y 1/2 = s.
Theorem AP.29 (Baryonic Tully–Fisher relation from
the kinetic branch). For a spherical source in the deep
branch, Eq. (AP68) implies
v 4 = GM a0 .

Buckingham–π closure and the α58 acceleration
invariant

(AP75)

Proof. In spherical symmetry outside the source,
Eq. (AP68) integrates to
GM
µ(g/a0 )g = gN = 2 .
(AP76)
r

Lemma AP.31 (Two independent groups). For the dimensional set {a0 , G, ℏ, c, H0 }, there are exactly two independent dimensionless groups. A convenient basis is
Π1 =

GℏH02
,
c5

Π2 =

a20 Gℏ
.
c7

(AP79)

Proof. The dimension matrix in the (M, L, T ) basis is
a0 G ℏ c H0
M 0 −1 1 0 0
.
D=
L 1 3 2 1 0
T −2 −2 −1 −1 −1

(AP80)

The rank is 3. Since there are five dimensional variables,
Buckingham’s theorem gives 5−3 = 2 independent dimensionless groups. Direct substitution verifies both groups
in Eq. (AP79) are dimensionless. Independence follows
because Π1 contains H0 but not a0 , while Π2 contains a0
but not H0 .
Theorem AP.32 (Acceleration invariant). The DFD
acceleration invariant is
a20 Gℏ
= 4C α58 ,
c7

(AP81)

and hence
√
a0 = 2 C α29

r

c7
,
Gℏ

(AP82)

with C the clock coefficient, 1 on bound-matter clocks
(Postulate ZS.11) and measured on the early clock (Remark AP.6).
Proof. By Theorem AP.15,

By Theorem AP.25,

GℏH02
= C α57 .
c5

(AP83)

√
a0 = 2 α cH0 .

(AP84)

297
Squaring,
a20 = 4αc2 H02 .

(AP85)

Then
4αc2 H02 Gℏ
GℏH02
a20 Gℏ
=
=
4α
= 4α Cα57 = 4C α58 .
c7
c7
c5
(AP86)
Solving for a0 gives Eq. (AP82).
Remark AP.33 (Why the exponent is not fitted). The
exponent is
58 = 57 + 1 = (60 − 3) + 1.

(AP87)

The 57 is the primed determinant count. √
The additional
+1 is the single power of α from squaring 2 α. No galaxy
acceleration value enters this count.

a.

Using CODATA constants on bound-matter clocks,
where C = 1 (Postulate ZS.11), Eq. (AP82) evaluates
to
(AP88)

within the systematic error of the SPARC scale quoted below. On the early clock, H0early = 67.52 ± 0.09 km/s/Mpc
gives (1.1208 ± 0.0015) × 10−10 and the BAO+θ∗ channel alone 1.1212 × 10−10 . The theorem√remains falsifiable because the exponent and the 2 α prefactor
are fixed by laboratory constants and α, not by fitting
galaxy rotation curves. The bound-matter value sits
within the systematic error of the SPARC kinematic scale
(1.20 ± 0.02stat ± 0.24sys ) × 10−10 m s−2 ; the SH0ES-side
booking is C = 1.027 ± 0.029, 0.9σ from C = 1 (Remark AP.6). If precision lensing-RAR determinations
of the absolute a0 converge to the early-clock value on
distances computed with H0early beyond systematics, the
two-clock postulate fails; if they exclude both readings,
at least one parent input fails.

8.

Proof. A finite-radius optical horizon would require
cph (rH ) = 0, equivalently n(rH ) = ∞ or ψ(rH ) = +∞, at
some rH > 0. But Eq. (AP89) is finite for every r > 0. It
diverges only as r → 0. Hence there is no finite rH in the
isotropic exponential exterior. Without a finite horizon
and associated Euclidean regularity condition, no finiteradius tree-level Hawking temperature is defined.
Corollary AP.36 (Temperature-claim triage). Claims
such as
GR
GR
TH = e−1/2 TH
,
TH = TH
(1 + α/2),
(AP90)
GR
TH = TH (1 + δCasimir )
may not coexist as theorem-grade base temperatures. The
base temperature is Hawking’s, from the surface gravity
of DFD’s event horizon; each correction may appear only
as a correction to it, derived in a named model.

Numerical prediction and falsifiability

a0 = (1.19662 ± 0.00001) × 10−10 m s−2 ,

DFD’s exterior, with σ kept, is instead the Schwarzschild
metric in DFD’s coordinates (App. WS), with the event
horizon at the areal radius 2GM/c2 .

Black-hole sector: one base temperature

The base temperature rests on one branch rule.
Definition AP.34 (Isotropic exponential exterior). The
isotropic exponential exterior around a spherical mass
M , with the parent strain’s σ set to zero, is the optical
branch


2GM
2GM
n(r) = exp
,
ψ(r) = 2 .
(AP89)
c2 r
c r
The local coordinate phase speed is cph (r) = c/n(r).
Theorem AP.35 (No finite-radius horizon in the
isotropic exponential exterior). In the isotropic exponential exterior, there is no finite-radius optical horizon and
therefore no tree-level finite-radius Hawking temperature.

Remark AP.37 (Horizon temperature). DFD’s black holes,
with the parent strain’s σ kept, have a true horizon H
(App. WS), and its base temperature is computed from
its surface gravity, κH = c4 /4GM for a static hole:
ℏκH
H
TH
=
.
(AP91)
2πc kB
Local Unruh or Casimir corrections must be written as
corrections to that branch value, not as separate base
temperatures.

9.

Lensing and PPN branch discipline

Definition AP.38 (Lensing branches). The corpus distinguishes:
1. the PPN-closed physical metric branch, used for
solar-system parameter comparisons;
2. the pure optical-exponential ray branch, using Fermat propagation through n = eψ without all PPN
closure terms.
Theorem AP.39 (Lensing statement). The corpus may
state as a theorem that DFD reproduces the leading 1PN
light-deflection term only if that result is computed in the
declared physical branch. Any 2PN coefficient is theoremgrade only after being computed from the same branch
action/metric used for the PPN table.
Proof. By Axiom AP.3, coefficients computed in different
branches cannot be co-theorems of one corpus. A pure
optical-exponential 2PN coefficient and a PPN-closed
2PN coefficient answer different mathematical questions
unless an embedding theorem identifies the rays and affine
structures order by order. Therefore a 2PN anomaly can
remain as a model prediction in the pure optical branch,

298
while the PPN-closed branch may retain exact 2PN agreement only if it is derived from the same declared metric.
Keeping both as theorem-grade violates Axiom AP.3.
Corollary AP.40 (Lensing). DFD reproduces the leading
GR deflection in the declared physical branch. At second
order the coefficient is GR’s 15π/4 on DFD’s exterior
with the parent strain’s σ kept (§IV; App. WS).

10.

the scalar density field. A separate topological phaseinflation mechanism, if present, is a different branch with
a different action. By Axiom AP.3, success or failure
in one cannot be transferred to the other without an
embedding theorem.
Corollary AP.44 (Inflation). The DFD scalar ψ is not
the conventional slow-roll inflaton in the disfavored branch.
Any viable early-universe smoothing mechanism must be
stated as a separate branch with its own predictions.

Flavor/CKM sector: exact theorem boundary

Proposition AP.41 (Minimal real-kernel CP null). If
the minimal DFD flavor sector produces only real Yukawa
kernels and no additional complex orientation/discrete
shape-duality datum, then the CKM matrix can be chosen
real and the Jarlskog invariant vanishes:
J = 0.

(AP92)

Proof. If the up- and down-sector mass matrices are real,
their diagonalizing transformations may be chosen orthogonal in the absence of an intrinsic complex phase.
Then
VCKM = OuT Od

12.

PDE/well-posedness sector

The local-existence theorem is Theorem III.5.
Definition AP.45 (Critical Sobolev threshold). For a
first-order scalar evolution whose principal nonlinear structure contains one spatial derivative of ψ in three spatial
dimensions, the scaling-critical Sobolev index takes the
form
sc = 1 +

3
5
= .
2
2

(AP95)

(AP93)
13.

Theorem summary

is real orthogonal. The Jarlskog invariant is
∗
J = Im(Vij Vkl Vil∗ Vkj
)

(AP94)

for any distinct i, k and j, l. If all entries of V are real,
the product is real and its imaginary part is zero.
Theorem AP.42 (CKM claim). The minimal DFD corpus may claim theorem-grade magnitude relations only
to the extent that their projection/operator dictionary is
fixed without fitting. CP violation and nonzero J are
theorem-grade only in an explicitly extended DFD–SD
branch containing the required complex/discrete orientation datum.
Proof. The null result in Proposition AP.41 shows that
minimal real kernels cannot generate CKM CP violation.
Therefore any nonzero J requires an additional datum or
branch. If that datum is declared as DFD–SD, the CP
result belongs to that branch. Quoting it as a minimal
DFD theorem would contradict Proposition AP.41.

11.

Inflation and early-universe sector

Theorem AP.43 (No mixed inflation claim). The scalar
ψ sector and any separate topological/phase inflation
branch must be kept distinct. If ψ-inflation produces excluded or disfavored (ns , r), then the corpus may not simultaneously claim that ψ is the successful inflaton and
that inflation is unnecessary.
Proof. The statements answer different model questions.
A no-go theorem for ψ as a slow-roll inflaton constrains

The minimal statement is:
The finite DFD microsector has Spinc index
60 and a protected three-dimensional generation kernel. The primed determinant therefore
contains 57 nonzero complex modes. Gaugenormalized Gaussian integration gives one
factor of α per nonzero mode, so the finite
spectral-action clock dictionary gives
GℏH02
= C α57 ,
c5
C = 0.87784 ± 0.00266 (measured, early clock),
C = 1 (bound-matter clocks, Postulate ZS.11).
The scalar acceleration-channel coupling is
fixed by the internal frame ratio and magnetic
dual coupling,
n3
3 1
3
ka =
αM =
=
.
n2
2 4α
8α
The S 3 scaling potential U (Ξ) = Ξ −
(3/2) log Ξ selects Ξ∗ = 3/2 for
Ξ = ka (a/cH0 )2 .
Thus
√
a0 = 2 α cH0 .
Combining the two gives the H0 -free acceleration invariant
r
√
a20 Gℏ
c7
58
29
= 4C α ,
a0 = 2 C α
.
7
c
Gℏ

299
The galaxy law follows from the nonlinear
kinetic equation, not from a direct ka a2 /c2
source term.

14.

Dimension checks

Proposition AP.46 (Dimensionlessness of Π2 ).
Π2 =

a20 Gℏ
c7

(AP96)

is dimensionless.
Proof. Using [a0 ] = LT −2 , [G] = L3 M −1 T −2 , [ℏ] =
M L2 T −1 , and [c] = LT −1 ,
[Π2 ] = L2 T −4 · L3 M −1 T −2 · M L2 T −1 · L−7 T 7 = 1.
(AP97)
Proposition AP.47 (Relation to Planck acceleration).
 2
a0
Π2 =
.
(AP98)
aP
Proof. Since a2P = c7 /(Gℏ),
 2
a0
Gℏ
= a20 7 = Π2 .
aP
c

15.

(AP99)

Failure modes and falsifiers

The acceleration-invariant theorem fails if any one of
the following fails:
1. the finite microsector is not E = O(9) ⊕ O⊕5 or has
a different nonzero determinant count;
2. the generation kernel is not three-dimensional;
3. the clock dictionary (H0 tP )2 = C Zα′ /Z1′ is wrong
in its exponent content;
4. the gauge-emergence coefficient ka = 3/(8α) is not
the coefficient in the scalar acceleration action;
5. the S 3 stationarity potential is not the correct
action-level crossover selector;
6. the observed universal galaxy acceleration scale converges to a value inconsistent with Eq. (AP88) after
systematic errors shrink.

Appendix AQ: Executed Lattice-QCD Validation of
the DFD QCD Ledger (Program II)

DFD-QFT treats nonperturbative QCD as a generatorclosed sector: the Euclidean lattice path integral defines
the correlation functions and masses follow from exponential decay, C(t) ∼ e−M t . This appendix executes that
propagation at modest but real scale on commodity hardware, then states each claim at its actual status. The
governing discipline is: scale-free ratios are the primary
outputs; MeV values require an external scale such as
the Sommer parameter r0 ; and finite-volume, laptop-scale
calculations are consistency checks, not physical-point
determinations.

1.

Summary: what this appendix verifies

This appendix is a numerical bridge between the DFD
QCD ledger and standard nonperturbative lattice gauge
theory. It does not introduce a new lattice-QCD algorithm, and it does not claim a physical-point full-QCD
spectrum. Its purpose is narrower: take the DFD QCDsector ledger targets, run standard lattice machinery in
the corresponding pure-gauge and exploratory dynamical
sectors, and determine whether the resulting scale-free
lattice observables are consistent with the ledger. The
answer is yes, within the limits of the computation.
The primary pure-gauge check is the SU (3)
√ Wilsonaction string tension. The DFD ledger quotes σ DFD =
440 ± 25 MeV. The executed calculation does not treat
this MeV number as the fundamental object, because
converting a lattice observable to MeV requires
external
√
scale setting; the scale-free quantity is r0 σ. √
The largestvolume box-size check, 204 at β = 6.2, gives r0 σ = 1.178,
while the standard quenched reference region is about
1.18–1.19 [181, 182]. With the canonical Sommer convention r0 = 0.5 fm this corresponds to roughly 465 MeV,
at the upper edge of the DFD band; a phenomenological
larger-r0 convention quotes the same dimensionless output
closer to 440 MeV. The conclusion is therefore scale-free
consistency, not independent fixation of the MeV central
value.
The second pure-gauge check is the scalar glueball,
with DFD ledger target m0++ ≃ 1.69 GeV. The 164 ,
β = 6.0 run uses a zero-momentum A+
1 operator built
from smeared spatial plaquettes, with vacuum subtraction
and an arccosh effective-mass plateau. After conservative
error analysis, including autocorrelation inflation, the
defensible estimate is a m0++ = 0.81(8), i.e. m0++ ≃
1.71 ± √
0.17 GeV under r0 = 0.5 fm. The scale-free ratios
m0++ / σ ≃ 3.7 and r0 m0++ ≃ 4.3 sit in the standard
quenched range [183] and overlap the DFD ledger; because
the plateau is short and the effective sample is reduced
by autocorrelation, this is a successful consistency check,
not precision spectroscopy.
The third check is dynamical-fermion validation. A
from-scratch Nf = 2 Wilson HMC engine passes gamma-

300
matrix algebra, γ5 -Hermiticity, conjugate-gradient residual control, reversibility, the HMC identity ⟨e−∆H ⟩ ≃ 1,
and a direct finite-difference check of the fermion force.
A 64 κ-scan resolves the expected sea-quark screening
trend: as κ rises from 0.148 to 0.156, the dynamical plaquette rises monotonically from 0.5703 to 0.5808, well
above the matched quenched reference. This validates
the dynamical machinery at Program-II level; it is not a
physical-point ensemble.
2.

TABLE CXIII. Average plaquette validation. The 204 row is
the final β = 6.2 box-size check.
volume β measured ⟨P ⟩ reference deviation
164
164
164
204

5.8
6.0
6.2
6.2

0.56750(6)
0.59405(5)
0.61381(4)
0.61376(4)

0.5670
0.5937
0.6136
0.6136

+0.09%
+0.06%
+0.03%
+0.03%

Frozen DFD QCD ledger

The QCD-sector ledger targets considered here are
αs (MZ ) = 0.1187,
√

θ̄ = 0 (tree level),
(AQ1)

σ DFD = 440 ± 25 MeV,

mDFD
0++ = 1.69 ± 0.10 GeV,
(AQ2)

fπDFD = 92.0 ± 4.6 MeV.

(AQ3)

Only the pure-gauge string tension and scalar glueball are
evaluated as numerical outputs; the dynamical WilsonHMC results validate the execution engine and the seaquark screening response, not a physical-point fπ .
3.

Methods

a. Pure
P gauge. We use the SU (3) Wilson action
SW = β p (1− 13 Re Tr Up ). Configurations are generated
by multi-hit Metropolis updates on random SU (2) subgroups, interleaved with Cabibbo–Marinari microcanonical overrelaxation. The static potential uses APE-smeared
spatial links (α = 0.5, maximal-trace SU (3) projection).
Wilson loops W (R, T ) give


W (R, T )
V (R) = log
,
(AQ4)
W (R, T + 1) T ∈[2,4]
fit to the Cornell form V (R) = V0 + σR − e/R (e the
Coulomb coefficient). Statistical errors are jackknife over
measured configurations.
b. Dynamical fermions. We implement two degenerate flavours of unimproved Wilson fermions,
X

D = 1−κ
(1−γµ )Uµ (x)δx+µ̂ +(1+γµ )Uµ† (x−µ̂)δx−µ̂ ,
µ

(AQ5)
with pseudofermion action Spf = ϕ† (M † M )−1 ϕ and
M † = γ5 M γ5 . Molecular dynamics is reversible leapfrog
with Metropolis accept/reject; the fermion force uses a
loose conjugate-gradient tolerance and the Hamiltonian a
tight one.

FIG. 17. Measured plaquettes against the published SU (3)
Wilson curve.

values with standard SU (3) Wilson references; agreement
is at the sub-0.1% level.

5.

The static potential exhibits the textbook Coulombplus-linear (confining) form (Fig. 18); the slope is the
string tension in lattice units. The extraction
(Ta√
ble CXIV) is essentially exact at β = 6.0 (a σ = 0.217 vs.
literature 0.2189, a 0.8% difference) and carries a 1–5%
fit-window systematic at the coarse and fine ends.
The expected systematic is the 164 , β = 6.2 box: its
physical size is about 1.08 fm, so the fit window sits near
the onset of the linear regime and can
√ bias σ upward. The
4
larger 20
box
(≈
1.35
fm)
shifts
a
σ : 0.1623 → 0.1596
√
and r0 σ : 1.198 → 1.178, a 1.7%
downward
move that
√
straddles the literature value a σ ≃ 0.1610 and lands
in the standard quenched reference region (Figs. 19–20).
This is a directional finite-volume check, not a continuum
extrapolation.

6.
4.

Validation: average plaquette

The plaquette is UV-dominated and is the first unambiguous sanity check. Table CXIII compares measured

Static potential and string tension

Scale convention and the DFD string-tension
ledger

√
The MeV value of σ depends on the external physical
√
value assigned to r0 . The scale-free 204 result r0 σ =

301
√
TABLE
CXIV. String tension in lattice units and the scale-free r0 σ. The 204 , β = 6.2 row is the final box-size check; literature
√
a σ from EHK [184], Teper [185], and Bali–Schilling [186], with r0 /a from Necco–Sommer [181].
√
√
√
β volume
a σ
reference a σ r0 σ
note
5.8
5.9
6.0
6.1
6.2
6.2

164
164
164
164
164
204

0.329(10)
0.244(6)
0.217(3)
0.177(4)
0.1623(30)
0.1596(30)

0.313
∼ 0.26
0.2189
∼ 0.18
0.1610
0.1610

FIG. 18. Cornell-form static potentials reconstructed from the
fitted string tensions, using APE-smeared Wilson loops and
plateau fits.

1.206 coarse, high
1.093 interpolation
1.165 cleanest point
1.123 interpolation
1.198 small box, high
1.178
larger box

FIG. 20. Box-size test at β = 6.2: increasing
the box from 164
√
to 204 reduces the positive bias in r0 σ.

under canonical r0 = 0.5 fm scale setting, meeting the
DFD band 440 ± 25 MeV at its upper edge; a larger
phenomenological r0 convention quotes the same dimensionless output closer to 440 MeV [182]. The correct
conclusion is scale-free consistency, not independent fixation of the MeV value.

7.

Scalar 0++ glueball

The glueball operator is the zero-momentum cubicscalar (A+
1 ) sum of APE-smeared spatial plaquettes on
each timeslice, with vacuum subtraction. The arccosh
effective-mass plateau is short and the scalar channel is
autocorrelation-limited; the conservatively widened estimate is a m0++ = 0.81(8) (Fig. 21). Using r0 = 0.5 fm,
m0++ = 1.71(17) GeV,

4

FIG. 19. Scale-free string-tension trend. The 20 , β = 6.2
point moves the high 164 value down into the standard reference region.

1.178 gives
√
1.178
σ≃
(197.327 MeV fm) ≃ 465 MeV
0.5 fm

(AQ6)

m0++
√ = 3.7(4),
σ

r0 m0++ = 4.3(4),

(AQ7)
which overlaps the standard quenched reference [183]
and the DFD ledger value 1.69 ± 0.10 GeV. Being
quenched, single-spacing, single-volume, and based on
a short plateau, this is a consistency check, not precision
spectroscopy.

302
ical r0 = 0.5 fm; and (iv) “first-principles” is reserved for
dimensionless lattice ratios, not MeV values that require
external scale setting.

FIG. 21. 0++ glueball arccosh effective mass. The shaded band
is the autocorrelation-widened estimate a m0++ = 0.81(8).

TABLE CXV. Validation checks for the Nf = 2 Wilson-HMC
implementation.
check

result

{γµ , γν } = 2δµν and γ5
< 10−13
free-field Wilson Dirac operator
exact
γ5 -Hermiticity
4 × 10−13
CG residual
< 10−9
analytic vs. numerical fermion force 6.5 × 10−7 (run-recorded)
MD reversibility
10−12
−∆H
⟨e
⟩
≃ 0.997

8.

Executed Nf = 2 Wilson-HMC validation

The HMC implementation passes the core algorithmic
checks in Table CXV. The finite-difference force check is
the most important, since it directly tests the fermionforce gradient used in molecular dynamics.
The final dynamical validation is the 64 , β = 5.6 κ-scan
(Table CXVI). The matched quenched reference is ⟨P ⟩ =
0.52200. Turning on the Nf = 2 Wilson determinant
raises the plaquette well above quenched, monotonically
in κ (Fig. 22).
The sea-quark mass dependence is
resolved. It remains a validation scan, not a physicalpoint dynamical ensemble.

FIG. 22. Resolved dynamical plaquette κ-scan. The monotonic
increase with κ is the expected sea-quark screening response;
the heavy end lands in the expected SESAM/TXL WilsonHMC range [187, 188], while the light end retains finite-volume
excess.

10.

The pure-gauge string-tension and glueball sectors are
quenched. The dynamical sector is an HMC validation
and κ-scan, not physical-point QCD. No continuum or
infinite-volume extrapolation is claimed. What is established is that the DFD QCD-ledger targets meet executed lattice-QCD validation checks at the scale-free,
finite-volume consistency level: pure-gauge string tension,
scalar-glueball spectroscopy consistency, dynamical seaquark screening, and a finite-volume box-size test. The
runs use standard lattice inputs and the ledger’s 440 MeV
normalization is lattice-calibrated and its 1.69 GeV normalization is open (companion App. AH), so the MeV
agreement checks the machinery
√ and the scale convention;
√
the DFD ratio m0++ / σ = 4π (App. AT) is the one
DFD number tested, at ±11%.

11.
9.

Status boundary

Reproducibility

Claim discipline

The claims are stated as follows: (i) the glueball error
is autocorrelation-inflated to a m0++ = 0.81(8); (ii) the
HMC plaquette rise is an engine and sea-quark-screening
validation, not a precision determinant benchmark; (iii)
the 440 vs. 465 MeV comparison is a scale-convention
issue, with the lattice sitting ∼ 1σ above 440 at the canon-

python3 src/su3_fast.py

The companion code package (su3 fast.py,
wilson hmc.py, and the analysis scripts), included
as the supplement/ directory of this archive together
with the executed run outputs (results/), curated CSV
tables (data/), provenance logs, and the standalone
Program II paper, reproduces every number above;
representative invocations:

--L 16 --beta 6.0 --potential

# string tension

303
TABLE CXVI. 64 dynamical κ-scan at β = 5.6. Statistical errors are about 10−3 ; the matched quenched reference is
⟨P ⟩ = 0.52200.
κ

0.148 0.150 0.152 0.154 0.155 0.156

⟨P ⟩dyn 0.5703 0.5745 0.5776 0.5790 0.5801 0.5808

python3 src/su3_fast.py --L 20 --beta 6.2 --potential
# box-size check
python3 src/su3_fast.py --L 16 --beta 6.0 --glueball
# 0++ glueball
python3 src/wilson_hmc.py --test
# HMC validation checks
python3 src/wilson_hmc.py --L 6 --beta 5.6 --kappa 0.152 # dynamical kappa point
python3 src/analyze_potential.py ; python3 src/analyze_glueball.py

Appendix AR: A DFD-Native Yang–Mills Infrared
Gap from Loaded Spatial Frame Geometry

This appendix formulates and proves the Yang–Mills
infrared-gap statement that is native to Density Field
Dynamics. The target is not pure Yang–Mills theory on
empty flat four-dimensional Euclidean space; it is the
spatial gauge sector on a DFD-loaded frame
3

Σψ = (Ω ⊂ R , hψ ),

2αψ

hψ = e

δ,

(AR1)

where the scalar optical field ψ (the same field that drives
DFD optical gravity and galactic dynamics, cf. the main
text) enters through this stated frame (the physical metric’s spatial part is eψ δ, Eq. (12)). The main result is a
spectral statement: if the physical transverse Yang–Mills
fluctuation operator on (Ω, hψ ) has a strictly positive
Weitzenböck curvature floor, then its first nonzero eigenvalue is positive,
λ1 (LA0 ,ψ ) ≥ ΛDFD > 0,

mDFD-YM =

p

λ1 ≥

p

ΛDFD > 0.

(AR2)
For the DFD deep-field annulus the conformally flat spatial metric gives positive angular Ricci curvature; its
radial component vanishes. Any such floor is not the
hadronic QCD confinement scale and not a proof of the
Clay Millennium problem [189]. It is nevertheless structurally important: the same scalar-loading mechanism
that drives DFD optical gravity and galactic dynamics
also changes the infrared spectrum seen by spatial Yang–
Mills fluctuations.
1.

Purpose

The question is whether DFD generates a native Yang–
Mills infrared gap once Yang–Mills fields are placed on
the loaded DFD spatial frame rather than on an empty
mathematical background. It does, under an explicit and
checkable geometric condition: the loaded frame must
supply a positive lower bound for the physical transverse
fluctuation operator. In empty Euclidean space the Yang–
Mills magnetic Renergy is only the ordinary kinetic energy
EYM [A] = 2g12 R3 |FA |2 d3 x, and positivity of this stiffness alone does not imply a mass gap. Here gauge fields sit

on hψ = e2αψ δ, and the curvature of this loaded frame enters the Hodge–Weitzenböck decomposition for one-form
fluctuations.

2.

DFD-native arena

Let G be a compact simple gauge group with Lie algebra
g, and let Ω ⊂ R3 be a bounded spatial domain. The
DFD-loaded spatial frame is Σψ = (Ω, hψ ) with hψ,ij =
e2αψ(x) δij , where α is the fine-structure constant and ψ
the DFD optical scalar. A gauge field is a connection
one-form A ∈ Ω1 (Ω, g) with curvature FA = dA + A ∧ A,
and the spatial Yang–Mills energy on the loaded frame is
Z
1
EYM,ψ [A] = 2
|FA |2hψ dvolhψ .
(AR3)
2g Ω
Writing A = A0 + a about a reference background A0 , the
fluctuation a is restricted to the physical transverse sector
d∗A0 a = 0, with elliptic boundary conditions removing
pure-gauge and harmonic zero modes (without which no
positive first nonzero physical eigenvalue can be stated).
Definition AR.1 (DFD-native Yang–Mills infrared gap).
A DFD-native Yang–Mills infrared gap is a positive
lower bound for the first nonzero physical eigenvalue of
the spatial Yang–Mills fluctuation operator on (Ω, hψ ):
λ1 (LA0 ,ψ ) ≥ ΛDFD√> 0, with associated infrared mass
scale mDFD-YM = λ1 .
3.

Fluctuation operator

The Yang–Mills Hessian around A0 has the standard
covariant form LA0 ,ψ a = d∗A0 dA0 a + dA0 d∗A0 a + RA0 a,
where RA0 is the algebraic nonabelian curvature term
from linearization. On the physical transverse sector
d∗A0 a = 0,
⟨a, LA0 ,ψ a⟩ = ∥dA0 a∥2 + ⟨a, RA0 a⟩.

(AR4)

For one-forms on a Riemannian three-manifold, the
Hodge–Weitzenböck identity [130, 190] gives ∆Hodge a =

304
∇∗ ∇a+Richψ (a), equivalently at the quadratic-form level
Z
2
∗
2
2
∥da∥ + ∥d a∥ = ∥∇a∥ +
Richψ (a, a) dvolhψ (AR5)
Ω

up to boundary terms that vanish under the chosen elliptic
conditions. Thus the Ricci curvature of the DFD-loaded
spatial frame enters the physical Yang–Mills fluctuation
operator as a zeroth-order stiffness contribution.
4.

Main theorem

Assumption AR.2 (DFD physical-sector curvature
floor). There exists a constant ΛDFD > 0 such that every physical transverse fluctuation a obeys Richψ (a, a) +
RA0 (a, a) ≥ ΛDFD |a|2hψ .
Theorem AR.3 (DFD-loaded-frame Yang–Mills infrared
gap). Let (Ω, hψ ) be a bounded DFD-loaded spatial frame
with hψ = e2αψ δ, and let LA0 ,ψ be the Yang–Mills fluctuation operator on physical transverse one-form fluctuations satisfying d∗A0 a = 0, with boundary/gauge conditions
removing pure-gauge and harmonic zero modes. If Assumption AR.2 holds,√then λ1√(LA0 ,ψ ) ≥ ΛDFD > 0, and
therefore mDFD-YM = λ1 ≥ ΛDFD > 0.
Proof. For physical transverse fluctuations ⟨a, LA0 ,ψ a⟩ =
∥dA0 a∥2 + ⟨a, RA0 a⟩. The covariant Hodge–Weitzenböck
decomposition writes the kinetic part as a nonnegative
covariant-gradient contribution plus the loaded-frame
Ricci contribution; after physical-sector projection and
boundary-term removal,
Z


⟨a, LA0 ,ψ a⟩ ≥
Richψ (a, a) + RA0 (a, a) dvolhψ .
Ω

(AR6)
By Assumption AR.2 the integrand is ≥ ΛDFD |a|2hψ ,
so ⟨a, LA0 ,ψ a⟩ ≥ ΛDFD ∥a∥2 . By the Rayleigh–Ritz
characterization of the first nonzero physical eigenvalue,
2
λ1 = inf a⊥ker
√ L, a̸=0√⟨a, LA0 ,ψ a⟩/∥a∥ ≥ ΛDFD , hence
mDFD-YM = λ1 ≥ ΛDFD > 0.
5.

Deep-field DFD specialization

In the DFD deep-field branch the scalar profile in a
galactic annulus
√ is logarithmic, ψ(r) = ψ0 − B ln(r/r0 )
with B = c22 GM a0 , and the loaded spatial metric is
hij = e2αψ δij . For this conformally flat geometry the
main text records Ricθθ = Bα(2 − Bα), Ricrr = 0, so for
0 < αB < 2 the angular Ricci curvature is positive. The
angular curvature corresponds to a mass scale far below
the hadronic QCD scale and below the ∼ πℏc/R gap that
a bounded domain of size R has without it.
6.

Why this supports DFD

The same scalar field ψ appears in three places: (i)
optical gravity through the refractive index n = eψ ; (ii)

galactic dynamics through the deep-field scalar profile;
and (iii) gauge-sector infrared behaviour through the
loaded spatial frame hψ = e2αψ δ. Ordinary Yang–Mills
theory on an empty background has no such geometric
input; on the stated scalar-loaded frame the curvature
contributes to the gauge fluctuation spectrum. This is not
decisive empirical proof, but evidence of internal coherence
and structural compression: one scalar-loading mechanism
organizes optical gravity, galactic phenomenology, and a
curvature term in the gauge fluctuation spectrum.

7.

Scope and non-claims

The theorem deliberately does not make three stronger
claims (Table CXVII).

8.

Falsifiers and completion checks

The theorem is mathematical, but the DFD application
has concrete checks: (i) operator check — derive the exact
Hessian LA0 ,ψ for the DFD-loaded domain and verify
the sign of RA0 on the physical sector; (ii) projection
check — verify the boundary/gauge conditions remove
pure-gauge, radial, and harmonic zero modes in the deepfield annulus; (iii) spectrum check — numerically compute
λ1 for realistic DFD backgrounds; (iv) physics check —
confirm that any geometric floor lies far below QCD and
does not contaminate hadronic mass physics. The theorem
is complete as a conditional spectral theorem; the DFD
phenomenological specialization is testable by computing
the physical-sector operator on explicit backgrounds.

9.

Conclusion

DFD’s native Yang–Mills gap is a spatial-frame result:
gauge fields are placed not on empty R3 but on hψ =
e2αψ δ, and the Hodge–Weitzenböck identity contributes
the loaded-frame Ricci tensor into the physical fluctuation
operator. When the resulting physical-sector curvature
floor is positive, λ1 ≥ ΛDFD > 0 and hence mDFD-YM >
0. This is not a replacement for nonperturbative QCD
confinement (which DFD treats as the generator-closed
lattice sector of Apps. AQ and AS); it is a conditional
infrared-gap theorem on the loaded spatial frame.

305
TABLE CXVII. Status discipline for the DFD-native Yang–Mills infrared gap.
Statement

Status

DFD-loaded spatial Yang–Mills has a positive geometric infrared floor when Proved here (Thm. AR.3).
the physical-sector curvature floor is positive.
The deep-field DFD annulus supplies positive angular Ricci curvature for
DFD deep-field specialization.
0 < αB < 2.
The expected galactic scale is tiny, around 10−30 eV.
Curvature term only; the bounded
domain alone gives a larger gap.
This proves the Clay pure Yang–Mills mass gap.
Not claimed.
This analytically derives all hadron masses.

Not claimed.

Nonperturbative QCD remains computable from the Euclidean/lattice
path-integral generator (Apps. AQ, AS).

DFD-QFT generator-closed sector.

Appendix AS: Lattice Boundary Program: Staged
Execution and Provenance (Programs I–II)

The validated lattice results of Appendix AQ (“Program II”) were reached through a deliberately staged execution, recorded here for provenance and to document the
status discipline applied at each rung. The lineage is Program I (frozen inputs, smoke test, unquenched scaffold) →
Program I-A (first executed quenched 164 string-tension
runs) → Program II (consolidated 164 and 204 string tension, 0++ glueball, and validated Nf = 2 Wilson-HMC).
At no stage was a laptop-scale calculation presented as
continuum-extrapolated, physical-point QCD.

1.

Program I: frozen inputs, smoke test,
unquenched scaffold

Program I established the four-stage methodology that
the later rungs inherit. Stage 1 froze the DFD QCD
ledger (the same targets used throughout Apps.
AQ–
√
AS: αs (MZ ) = 0.1187, θ̄ = 0 at tree level, σ DFD =
DFD
440 ± 25 MeV, mDFD
=
0++ = 1.69 ± 0.10 GeV, fπ
92.0 ± 4.6 MeV). Stage 2 audited the analytic nonperturbative targets (string tension, lightest scalar glueball,
low-energy chiral scales) against the generator-closedsector statement. Stage 3 executed a small pure-gauge
SU (3) Wilson-action Metropolis smoke test on 44 lattices
at β = 5.4, 5.6, 5.8, measuring plaquettes, small Wilson
loops, an effective potential, and a crude string-slope
proxy. Stage 4 specified — without executing — the unquenched dynamical-fermion HMC bridge, enumerating
the Dirac-operator, pseudofermion, molecular-dynamics,
and accept/reject steps required for a production run.
The Stage-3 44 smoke test was explicitly a code-path
verification, not a physics result: at such a tiny volume
and with limited thermalization the plaquette values were
not at their infinite-volume benchmarks, and the “stringslope proxy” Veff (2)−Veff (1) was a placeholder, not a fitted
string tension. Program I’s conclusion was therefore that
the DFD QCD sector had been moved from “generator
exists” to “generator has an executable boundary-program
prototype” — with the production rung still to come. The

Stage-4 scaffold is the exact specification that was later
implemented and validated as the Nf = 2 Wilson-HMC
engine of Appendix AQ.

2.

Program I-A: first executed quenched 164 string
tension

Program I-A consists of executed, production-grade
quenched pure-gauge SU (3) runs on 164 lattices at β =
5.8, 6.0, 6.2 (a 44 smoke test is not used as physics). The
plaquette benchmark was reproduced to better than 0.1%
at all three couplings, and APE-smeared Wilson loops
with effective-potential plateaus and Cornell fits gave
√
σ a = (0.3292, 0.2171, 0.1623),
(AS1)
√
r0 σ = (1.206, 1.165, 1.198),
(AS2)
√
i.e. σ = 476±15, 460±7, and √
473±7 MeV at r0 = 0.5 fm,
for a best quenched estimate σ√= 470 ± 10 MeV. This
overlaps the DFD ledger target σ DFD = 440 ± 25 MeV
at its upper edge, with the residual assigned to quenching
and scale-setting convention.
These three 164 points are precisely the β = 5.8, 6.0, 6.2
4
16 rows subsequently consolidated in Table CXIV of
Appendix AQ; they are not re-tabulated here. Program II
then added the decisive refinements that Program I-A
flagged as “next rung”: the 204 box-size check at β = 6.2
(which confirmed the expected positive √
finite-volume bias
of the 164 fine-end point and moved r0 σ from 1.198 to
1.178), the autocorrelation re-analysis of the error budget,
the 0++ glueball estimate, and the executed Nf = 2
Wilson-HMC κ-scan.

3.

Status across the program

The staged lineage illustrates the discipline applied
throughout the DFD QCD sector: scale-free dimensionless ratios are the primary outputs; MeV values are quoted
only with an explicit external scale; finite-volume, singlespacing calculations are consistency checks rather than
physical-point determinations; and results that fail this
discipline (the 44 smoke test, the un-error-inflated glue-

306
ball) are not used. The consolidated, validated outcome
of this program is the content of Appendix AQ, and its analytic gauge-sector companion is the infrared-gap theorem
of Appendix AR.

Appendix AT: Cross-Sector Exact Identities,
Strong-Field Theorems, and New Predictions

This appendix collects results derived inside the DFD
axiom system that were not tabulated in the body.
Throughout we use the locked ledger values α√−1 =
137.036, MP = 1.220890 × 1019 GeV, v = MP α8 2π =
(5)
246.09 GeV, ΛDFD = MP α19/2 = 61.20 MeV, ΛMS =
√
4π ΛDFD = 216.95 MeV, MR = MP α3 = 4.744 ×
14
1012 GeV, and m3 = 14
= 50.16 meV (App. Z,
13 πMP α
X). Every exact identity below was verified symbolically
and to ≥ 20 digits in arbitrary-precision arithmetic.
1.

Exact cross-sector identities

Proposition AT.1 (Seesaw-triangle closure, (E)). Given
2
the stated rung yD
= Fν α of the Dirac neutrino Yukawa
(one discrete input), the priming factor fixes it to the
exact value

14
sin2 θW 
2
yD
=
α= 1+
α,
(AT1)
13
Ngen
with sin2 θW = 3/13 and Ngen = 3, and the seesaw then
fixes m3 . The neutrino absolute-scale exponent 14 follows
from the rung and is not a further postulate.
2
= 2m3 MR /v 2 .
Proof. The Type-I seesaw gives yD
2
Substituting 
 the 
 printed
yD
=

scales, 14 14+3−16
14
3
2 16
2 14
πM
α
M
α
/
M
α
2π
=
α
=
P
P
P
13
13
√
14
2π of v. The second
13 α, the π’s cancelling against the
3
1
sin2 θW = 1 + 13 · 13
form follows from 1 + Ngen
= 14
13 , so
the neutrino Yukawa and the weak mixing angle share
the denominator 13. Equivalently
√ the Dirac mass is the
seesaw
geometric
mean
m
=
m3 MR = 15.43 GeV
D
p
√
= 14α/13 v/ 2.

Proposition AT.2 (Four-sector scale lock, (E)). The
QCD, gravitational, electroweak, and neutrino scales satisfy
2π MP Λ2DFD = v 2 MR ,

(AT2)

exactly, with no constant other than the 2π inherited from
v.
Proof. v 2 MR = (MP2 α16 2π)(MP α3 ) = 2πMP3 α19 , while
2πMP Λ2DFD = 2πMP (MP α19/2 )2 = 2πMP3 α19 . The
equality encodes the exponent identity 2 · 19
2 = 16 + 3 =
19.
Proposition AT.3 (QCD–Majorana cube identity,
(E)).
√
With the microsector matching scale Λtop = α MP ,
ΛDFD MP3
,
Λtop
exactly, with prefactor unity.
MR3 =

(AT3)

Proof. ΛDFD MP3 /Λtop = (MP α19/2 )MP3 /(α1/2 MP ) =
1
MP3 α9 = (MP α3 )3 = MR3 , encoding 3·3 = 19
2 − 2 = 9.

307
Proposition AT.4 (QCD–Hubble bridge, (E)). The
Hubble energy carries the measured clock coefficient multiplying the cube of the confinement scale over the Planck
mass squared,
√ Λ3
c2 ,
(AT4)
ℏH0 = C DFD
MP2
a consequence of 57 = 6 · 19
2 = 3 · 19 and the printed invariant GℏH02 /c5 = Cα57 (exponent derived; C measured;
App. O). On bound-matter clocks, where C = 1 (Postulate ZS.11), the bridge holds with no measured factor,
ℏH0loc = Λ3DFD c2 /MP2 .
p
2 5
57
Proof. From
and t√
Gℏ/c5 one
P =
0 /c = Cα
√ GℏH
57/2
57/2
has
Cα
/tP , hence ℏH
=
p0 = C (ℏ/tP )α2
√ H0 =2 57/2
C MP c α
since ℏ/tP =
ℏc5 /G = MP c . On
the other side Λ3DFD /MP2 · c2 = (MP α19/2 )3 /MP2 ·
57
Equivalently
c2 = MP c2 α57/2 , using 3 · 19
2 = 2 .
the heaviest neutrino is the α-primed geometric mean
of the Hubble
rate and the Planck mass, m3 =
√
14
1
−1/4
−1/4
ℏH
M
(from 57
πC
0
P α
13
4 − 4 = 14).
Proposition AT.5 (Strange-quark mass as a rung frac(5)
tion, (E)). ms = 37 ΛMS exactly (a pure rational), and
more generally every charged-fermion mass can be written
(5)
in the closed form mf = (rational × 2-power) ΛMS αk .
√
√
(5)
Proof. ms = 67 α3/2 v/ 2 = 67 π MP α19/2 while ΛMS =
√
2 π MP α19/2 , so the ratio is 67 /2 = 37 ; α and π cancel identically. The same equal-exponent
mechanism
√
gives the pure-number ratios mτ /mc = 2, mt /mb = 42
(compared with the top pole mass over mb (mb ); at one
scale, Mt , the observed ratio is 59.7), me /md = 1/9,
mµ /ms = 7/6, with the down-type prefactor integers
7 = b0 and 36 = Nf2 reappearing verbatim from the QCD
running.
Proposition AT.6 (Thepthermal variable is the refractive scalar, (E)). Let τ ≡ gbar /aref be the dimensionless
variable of the occupation reading of ν, and take its reference acceleration to be DFD’s own derived surface gravity
κDFD = c4 /(4GM ) (App. AH of the companion volume,
the Hawking-temperature theorem) rather than the galactic
a0 . Then, for every mass,
p
2GM
rM ≡
GM/κDFD =
= rs ,
c2
(AT5)
p
rs
τ (r) ≡
gbar (r)/κDFD =
= ψ(r),
r
with ψ(r) = 2GM/(c2 r) the exterior solution (147) of
Sec. VI A, exact through second order in GM/(c2 r). In
particular τ = 1 at r = rs = 2GM/c2 for every mass.
p
p
Proof. p
GM/κDFD =
GM · 4GM/c4 = 2GM/c2 =
2
rs , and (GM/r )/κDFD = rM /r = rs /r, which is the
printed ψ(r). Setting r = rs gives τ = 1. All four residuals
(rM − rs , τ − rs /r, τ − ψ, τ (rs ) − 1) are identically zero
symbolically.

Remark AT.7 (What Prop. AT.6 buys). DFD’s statisticalmechanical variable and its optical variable are the same
function once the reference acceleration is the theory’s
own p
surface gravity. It therefore links the galactic sector
(τ = gbar /a0 ) to the refractive scalar (τ = ψ through
second order) by a change of reference scale only, and
it makes “τ = 1 at r = rs ” a theorem rather than a
coincidence. It introduces no new integer and no new
constant.
Proposition AT.8 √(Unruh–Gibbons–Hawking ratio,
(E)). With a0 = 2 α cH0 read on the same clock
(Thm. N.14, App. N), the Unruh temperature of the
MOND scale and the de Sitter Gibbons–Hawking temperature of the cosmic horizon stand in the exact ratio
√
TU (a0 )
ℏa0 /(2πckB )
a0
=
=
= 2 α = 0.1708490862,
TGH
ℏH0 /(2πkB )
cH0

(AT6)
with no numerical constant of any kind surviving: the
two 2π’s cancel identically. The DFD crossover scale is
therefore the
√ cosmic horizon scale reduced by exactly one
factor of 2 α, the same factor that appears in a0 itself.
Proof. √
Immediate substitution; the ratio is a0 /(cH0√
) and
a0 ≡ 2 α cH0 . The sympy residual
of
T
/T
−
2
α is
U
GH
√
identically zero. Numerically, 2 α = 0.170849086206 at
CODATA-2022 α−1 = 137.035999177 and 0.170849086264
at CODATA-2018 α−1 = 137.035999084 — a fractional
difference of 3.4 × 10−10 , so the identity is insensitive to
the CODATA release at any relevant precision.

2.

Strong-field and gravitational results

With the parent strain’s σ kept the strong-field exterior
is the Schwarzschild metric in DFD’s coordinates, and
Kerr’s for a spinning hole (App. WS; argued beyond first
order in the spin), so its innermost stable orbit, photon
ring, redshift and quasinormal modes are those of general
relativity: at zero spin the innermost stable circular orbit
sits p
at the areal radius 6GM/c2 with accretion efficiency
1 − 8/9 = 5.719%, and the photon sphere at 3GM/c2 .
This subsection collects the gravitational-wave memory
and the Cole–Hopf lapse family.
Proposition AT.9 (Tensor gravitational-wave memory
equals GR’s, (D)). The transverse-traceless sector is governed by the full nonlinear a2 Seeley–DeWitt Einstein–
Hilbert term (Sec. V A 4); the action Sh of Sec. II is
its quadratic part. DFD’s tensor-channel gravitationalwave memory therefore equals GR’s: the linear (ordinary)
memory sourced by the matter stress, plus the nonlinear
(Christodoulou) memory sourced by the gravitational-wave
stress-energy. The grade is Derived: the result rests on
reading the spectral a2 zero mode as the full Einstein–
Hilbert invariant, on the grounds given below.
Derivation. Four grounds fix the reading of the a2 zero
mode as the full Einstein–Hilbert invariant rather than its

308
linearized Hessian. (i) The corpus names the TT action
the a2 Seeley–DeWitt Einstein–Hilbert term (Sec. V A 4;
App. SM; Subsec. O 5). (ii) The zero-mode projection
keeps the cubic vertex: the constant internal zero mode
R
−1/2
−1/2
Y0 = VK
gives K Y03 = VK
̸= 0 (Rem. AT.10).
(iii) With the parent strain’s σ kept, the physical metric
near every compact source solves the nonlinear Einstein–
Hilbert equations in DFD’s coordinates (App. WS),
Schwarzschild’s outside a static source and Kerr’s around
a spinning hole (argued beyond first order in the spin),
solutions that no exactly quadratic action has, so the
TT sector there is nonlinear Einstein–Hilbert. (iv) The
linearized-Hessian alternative, on which Sh would be exactly quadratic and the Christodoulou memory would
vanish, has no printed mechanism: no corpus object removes the cubic and higher vertices of the a2 term. On
the full term ∂ 3 S/∂h3 ̸= 0, so the gravitational-wave
stress-energy sources hTT
ij at second order as in GR, and
the memory is GR’s flux integral.
Remark AT.10 (Channel scope). Prop. AT.9 concerns the
tensor channel: it rests on the cubic Einstein–Hilbert vertex of the TT sector, independent of whether the scalar
ψ radiates. A possible scalar-channel memory contribution is a separate question: for quasi-circular BBH
inspirals the ψ-sector sourcing is rigorous and negligible, while the general (e.g. head-on/burst) scalar case
is open pending the far-zone non-radiativeness analysis
of the ψ sector. Where the corpus elsewhere writes a
c-speed d’Alembertian for ψ, that form is the numerical
hyperbolic-relaxation of the elliptic AQUAL constraint,
not an independent claim that ψ radiates at c. The
spectral a2 zero mode carries the full Einstein–Hilbert
invariant (Sec. V A 4; Proposition WS.5), not only its
linearized Hessian, the “standard linearized action” of
Sec. II, on which ∂ 3 S/∂h3 ≡ 0 and the Christodoulou
memory would vanish. On the full invariant the constant
−1/2
internal zero mode Y0 = VK
is closed under products
R
−1/2
3
( K Y0 = VK
̸= 0) and the cubic vertex survives the
zero-mode projection: for hij = ϵ q(t) diag(1, 1, −2) the
kinetic density is 32 ϵ2 q̇ 2 + 3ϵ3 q q̇ 2 + O(ϵ4 ), and the nonlinear tensor memory of the standard flux form follows,
equal to GR’s. Separately, the local-relativity coefficient
A(ψ) = F e2ψ of the TT kinetic term (Rem. V.1) car2
ries the cubic vertex ψ(∂t hTT
ij ) , so GW energy sources
2
the elliptic ψ at O(h ) whether or not ψ radiates: the
scalar-channel contribution named above is a second-order
non-radiative response of the coupled (ψ, h) system, and
Gauss’s law fixes it. The outgoing shell of wave energy carries the monopole the source loses, so outside
the shell ψ is unchanged; after the shell passes a detector at distance r the response is the pure-trace term
δψ = −(2GEGW /c4 )(1/r − 1/ct), which changes both
arms of an interferometer and all nearby clocks together,
and the anisotropy of the shell leaves a tidal part suppressed by (L/r)2 . The scalar channel adds no memory
for any source.

Falsifier AT.11 (Gravitational-wave memory). For an
equal-mass nonspinning merger GR’s null memory is 27%
of the edge-on strain peak, and DFD’s tensor memory is
the same (Prop. AT.9). A memory measurement consistent with zero where GR predicts this 27% falsifies DFD;
a detection at the GR level does not distinguish DFD
from GR. LISA reaches single-event memory SNR 11–35
for 105 –106 M⊙ at z = 1.
Proposition AT.12 (Cole–Hopf lapse family, (E/T)).
The substitution χ = ekψ/2 linearises the acceleration
equation ∇· a + (k/c2 )a2 = −4πGρ (with a = (c2 /2)∇ψ)
to ∇2 χ = −(4πGk/c2 )ρ χ. Its point-mass exterior is the
one-parameter family nk (u) = (1 + ku)2/k , with β(k) =
1 + k/2, γ(k) = 1, and k → 0 recovering the exponential
e2u , DFD’s lapse-squared through second order in u. The
GR isotropic lapse-squared is the exact geometric mean
of the k = ± 12 members.
Proof. Writing a = (c2 /2)∇ψ turns the equation into
∇2 ψ+(k/2)|∇ψ|2 = −(8πG/c2 )ρ; with χ = ekψ/2 , ∇2 χ =
(k/2)χ[∇2 ψ + (k/2)|∇ψ|2 ], the stated linear equation.
The vacuum solution χ = 1 + kGM/(c2 r) gives nk =
(1+ku)2/k ; expanding the optical metric yields β = 1+k/2.
Lunar laser ranging (β − 1 = (−4.5 ± 5.6) × 10−5 ) pins
the vacuum member to k = (−0.9 ± 1.1) × 10−4 – gravity
sits at k = 0 to one part in 104 .

3.

Compact-star structure

Theorem AT.13 (Compact-star structure, (T)). Varying the matter action of a perfect fluid minimally coupled
to the physical metric (Sec. II, App. AN) with respect to ψ
uniquely fixes the relativistic source: in the µ → 1 regime
the static structure obeys


8πG
∂i (e−h )ij ∂j ψ = − 4 (ε + 3P ) eψ ,
c

dP
dψ
= 12 (ε + P )
,
dr
dr

(AT7)
with ε the energy density, P the pressure and hij the traceless part of the parent strain, whose scalar-longitudinal
piece σ obeys the traceless part of the Einstein equation.
The three equations are the Tolman–Oppenheimer–Volkoff
system in DFD’s coordinates (App. WS): for the reference
Γ = 2, K = 100 polytrope the maximum mass is Mmax =
1.637 M⊙ at central density ρc = 1.97 × 1015 g cm−3 , with
areal radius R(Mmax ) = 11.3 km, the general-relativistic
values.
Proof. The p
point-particle
Lagrangian
−mc2 e−ψ/2 1 − e2ψ v 2 /c2 of the physical metric
gives ∂L/∂ψ = 12 (ε + 3P )eψ on the static fluid (verified
symbolically, residual 0), which is the active (Tolman)
source; the lapse e−ψ/2 gives the equilibrium relation
by the standard static-fluid first integral, and the first
equation is the time-time component of the trace-reversed
Einstein equation for the static metric. With σ kept its
equation and the spatial identity of App. WS give general
relativity’s Hamiltonian constraint, so the system is the

309
TOV system; integrating it for the reference polytrope
gives the quoted values (computed, in DFD’s coordinates
and directly from the TOV equations, agreeing to
better than 10−8 ). The reduction ∇2 ψ = −(8πG/c2 )ρ
(Sec. IV) is recovered in the nonrelativistic limit P ≪ ε,
ψ ≪ 1.

1/(1 − e−1 ) = 1.582): Mdyn (< rM )/M = φ = 1.6180 . . .,
with phantom fraction 1/φ.

Reproducibility Note. The stellar-structure integrations reported here shoot the central scalar value ψc
so that the exterior solution satisfies asymptotic flatness,
ψ(∞) = 0 and n(∞) = 1. Holding ψc fixed instead computes a non-asymptotically-flat normalization and can
produce spurious runaway behavior. Radii quoted in this
subsection are areal (circumferential) radii, not raw coordinate radii. An independent integration of the TOV
equations confirms the quoted values: Mmax = 1.637 M⊙
at ρc ≃ 1.97 × 1015 g cm−3 , with areal radius 11.27 km.
With σ set to zero the first two equations alone give
1.474 M⊙ , and that truncation fails the Hamiltonian constraint inside the star (App. WS).

The Kuzmin-disk family solves the full nonlinear DFD
field equation
exactly, g = ν(|gN |/a0 ) gN with ν(y) =
p
1
4
1 + 4/y] and vflat
= GM a0 , providing an inte2 [1 +
grable thin-disk benchmark.

4.

Deep-sector conformal theorems

In the deep-field regime the energy functional is the
R1
R
c4
c2
3
cubic E = 16πGa
3 |∇ψ| − 2 ρψ.
⋆
Theorem AT.14 (Conformal structure and the exact
M –σ relation, (T)). The deep-field functional is invariant
under the full conformal group of R3 plus a ψ-shift (11
generators) and has identically traceless stress tensor. A
point
Q(m) =
√ mass3/2carries dilatation (anomaly)2 charge
√
2
Ga
m
,
the
two-body
force
is
F
=
(
Ga
/ℓ)
[(m1 +
0
0
3
3
3/2
3/2
3/2
m2 ) − m1 − m2 ] ∝ 1/ℓ, and any stationary isolated
deep system obeys the exact isotropic relation
4
4
σlos
=
GM a0 ,
(AT8)
81
together with the baryonic Tully–Fisher law vf4 = GM a0
with coefficient exactly unity.
Proof. Tracelessness σii = 0 off-shell and conservation
∂j σij = 0 on-shell are verified directly for the cubic Lagrangian; the dilatation current of the conformally invariant 3-Laplacian integrates
to the
P
P stated point charge.
The virial identity i xi · Fi =
) − Q(M ) for
i Q(m
√ i
a self-bound system gives 2T = 23 Ga0 M 3/2 , hence
4
the isotropic σlos
= (4/81)GM a0 ; and vf2 = dQ/dM =
√
GM a0 gives BTFR with unit coefficient. Dwarfspheroidal checks at the stellar masses of Tab. XIV: Leo I
σlos = 8.55 (obs ≈ 9.2), Fornax 12.17 (obs ≈ 11.7).
Corollary AT.15 (Golden-ratio
p phantom mass, (T)).
At the MOND radius rM =
GM/a0 (where gbar =
a0 ), the radial acceleration relation gobs = 12 [gbar +
p
2 + 4g
gbar
bar a0 ] forces, on the weak-response approximant, the enclosed dynamical mass to be the golden ratio
times the baryonic (the derived exponential response gives

At gbar = a0 the relation gives gobs = a0 (1 +
Proof.
√
5)/2 = φ a0 ; since Mdyn ∝ gobs and M ∝ gbar at fixed
rM , the ratio is φ identically.

5.

Strong-CP protection and the neutron electric
dipole

Theorem AT.16 (Determinant-phase protection for every value of the offset, (T)). Let Mf (ξ) = Sf + iξTf be
the up- or down-sector mass matrix of Postulate Y.10
(App. AO), with Sf real symmetric and Tf real antisymmetric. Then det Mf (ξ) is real for all real ξ (and
all orders in ξ), so arg det Mf (ξ) ∈ {0, π}, equal to 0
wherever det Mf > 0 (in particular on the component of
{det Mf ̸= 0} containing ξ = 0 when det Sf > 0); the
offset that supplies the weak CP phase never regenerates
a continuous strong-CP angle at tree level.
Proof. SfT = Sf and TfT = −Tf give Mf (ξ)T = Sf −
iξTf = Mf (ξ) (overbar = complex conjugate, since
Sf , Tf , ξ real). Hence det Mf = det MfT = det Mf =
det Mf , so det Mf is real and arg det Mf = 0 wherever
it is positive. This holds to all orders in ξ, beyond the
first-order (trace-form) statement of App. AO.
Corollary AT.17 (Neutron-EDM ledger and falsifier,
(P)). With θ̄ = 0 at tree level (θYM = 0 fixed by the
spectral action given that nothing is added to it for the
gauge fields; Theorem AT.16; App. L) and its loop-induced
value at the Standard-Model level, ≲ 10−16 (App. AO), the
DFD neutron EDM is the CKM long-distance background
scaled by JDFD /JSM = 0.94, giving dn ∼ (0.9–5.6) ×
10−32 e cm; the residual θ-channel is ≲ 2 × 10−32 e cm for
the upper-end induced θ̄ ∼ 10−16 (dn ≃ 2.4×10−16 θ̄ e cm),
comparable to the CKM background. DFD predicts null
at all foreseeable sensitivities, and any measured dn >
10−30 e cm falsifies the det-orthogonal architecture (margin
∼ 106 to the current bound). The same even-mappingtorus structure that closes the anomaly channel is silent
on the CP-even topological susceptibility χt , consistent
with a nonzero χt and the η ′ anomaly channel.
6.

New predictions

The following are predictions (not reformulations): each
is a DFD value confronted with data and carries a lookelsewhere note.

310
TABLE CXVIII. New DFD predictions; the As row is a calibrated amplitude that 32πα5 matches, not a prediction. “rung”
(5)
denotes the scheme scale ΛDFD = 61.20 MeV or ΛMS = 216.95 MeV (App. Z). The mt row evaluates with the v = 246.09 GeV
√
of this appendix; the muon-decay vev v = 246.22 GeV gives mt = (1 − α)v/ 2 = 172.83 GeV (0.85σ).
Observable

DFD form

Predicted

Measured

Dev.

3
Λ
p 5 MS

Fπ
130.17 MeV
130.2(8) MeV
0.03%
39/2 ΛMS 958.0 MeV
957.78 MeV
0.02%
η′
√
ω
782.2 MeV
782.66 MeV
0.06%
13 ΛMS
√
∆
32 ΛMS
1227 MeV
1232 MeV
0.4%
√
Ω−
59 ΛMS
1666 MeV
1672 MeV
0.4%
√
√
4π
3.545
3.529(97)
0.16σ
m0++ / σ
√
mt
(1 − α)v/ 2 172.74 GeV
172.60(27)
0.52σ
a0
2α29 c/tP
1.1966 e−10 1.20(2)stat (24)sys e−10 within σsys
As (calibrated) 32πα5 (match) 2.080 e−9 2.099(29)/2.105(30) e−9 0.6/0.8σ
sin2 θ13
3α
0.0219
0.02195(58)
0.1σ

a. Topological susceptibility, Λ-free and Nf -free on
the chiral-limit form (T). Two rows already p
in the table
′
above satisfy
an
exact
integer
identity:
η
=
39/2 ΛMS
√
and ω = 13 ΛMS with 39/2 = (3/2) · 13, hence the
dimensionless DFD statement
m2η′ = 32 m2ω .

(AT9)

Substituted into the Witten–Veneziano relation χt =
fπ2 m2η′ /(2Nf ) at Nf = 3, both Nf and ΛMS cancel identically, leaving
r
fπ mω
fπ2 m2ω
1/4
χt =
,
χt =
= 189.81 ± 0.58 MeV,
4
2
(AT10)
from PDG inputs alone, against the quenched lattice value
191(5) MeV: −0.24σ on this chiral-limit form; the full
Witten–Veneziano relation, with m2η − 2m2K added to m2η′ ,
gives 179.0 MeV, −2.4σ; both comparisons use the lattice
error alone, and the relation itself holds at leading order
in 1/Nc . Evaluating instead on the DFD rows themselves
gives 189.73 MeV (−0.25σ). The accuracy of the input
identity is (mη′ /mω )2 = 1.4975635 against 3/2, low by
0.162%.
The strong-CP discussion above states only that the
even-mapping-torus structure closing the anomaly channel is silent on the CP-even topological susceptibility.
Grade: equation channel derived (Nf verifiably absent
from the reduced expression); physics channel derivedconditional on the two catalog rows, with declared inputs
fπ and mω . The factor 3/2 grades selected, not forced,
until the 13 of the ω row is independently linked to the
13 of sin2 θW = 3/13; that identification is presently asserted, not derived. Look-elsewhere probability on the
unrestricted catalog is P = 0.262.
Falsifier.
Any continuum determination of χt
at quenched-equivalent normalization landing below
∼185 MeV falsifies the m2η′ = 32 m2ω row identity.
Consistency check only, not a derivation. Carrying the same chain to the pseudoscalar glueball, with

the Athenodorou–Teper continuum ratios and a lattice
1/4
m0++ /χt anchor imported as external inputs, gives
−+
m(0 ) = 2443.5 ± 74.8 MeV against the BESIII X(2370)
measurement 2395 ± 11 (stat) +26
−94 (syst) MeV, i.e. +0.61σ.
Because the prediction exceeds the measurement, the upward systematic is the live tail. This arm is not parameterfree and is not a DFD glueball-spectrum derivation.
b. Baryon decuplet (T/P). The J = 3/2 decuplet
follows the exact integer arithmetic progression m2 =
(32 + 9ns ) Λ2MS in the number of strange quarks ns :
∆, Σ∗ , Ξ∗ , Ω− at ns = 0, 1, 2, 3 all match within 0.4%.
The progression structure (intercept 32, slope 9), not the
individual hits, is the non-accidental content.
c. Baryon octet, incl. the nucleon (P). The J = 1/2
octet follows the companion progression
2
m2 = (19 + 9 ns ) ΛMS
,

(AT11)

with the same strange slope 9 as the decuplet and intercept 19 = h0 (O(1)) + h0 (O(2)) + h0 (O(3)) = 3 + 6 + 10,
the CP 2 line-bundle cohomology sum. The nucleon is
then a tower state:
√
√ √
MN = 19 Λ5 = 19 4π MP α19/2 = 946 MeV
(vs 939, +0.7%),
the integer 19 appearing in both the prefactor and the
exponent. The endpoints match — N (ns = 0) at +0.7%
and
√ Ξ (ns = 2) at +0.1% — while the ns = 1 value
28 Λ5 = 1148 MeV is the Λ–Σ average (within 0.5%
of (Λ + Σ)/2); the individual Λ(1116)–Σ(1193) splitting
requires the additional isospin/Gell-Mann–Okubo term
absent from the pure ns -linear tower, which is why the
spin-1/2 octet (+0.7%) is intrinsically less clean than the
spin-aligned decuplet (0.4%). Grade (P): the intercept
19 is the cohomology sum above; the closer numerical
fit (MN /Λ5 )2 = 18.73 ≈ 75/4 is not a forced DFD quantity and is deliberately not adopted. The decuplet–octet
intercept gap 32 − 19 = 13 tracks the measured ∆–N
mass-squared splitting (13.5 Λ25 ) to ∼ 4%.

311
d. Nucleon σ-term and the central nuclear coupling
c1 (conditional). The deep-central scalar-isoscalar pion–
nucleon coupling c1 —the one nuclear low-energy constant
not already fixed by the forced relativistic spin–orbit
mechanism, the saturation density ρ0 , or the ρ/ω and ∆
tower states—can be closed conditionally, by the same
mass-squared-tower logic that gives the intercept 19. It
rests on one rule: in DFD the light-quark scalar source
is not an independent chiral counterterm but the U (2)equivariant tangent-projector perturbation of the baryon
mass-squared tower. Granting that rule, the light scalar
2
spurion (⟨χ+ ⟩ = 4m2π for Nf = 2) can enter MN
only
through the projector onto the coset tangent T CP 2 of
CP 2 = SU (3)/U (2) (Schur / U (2)-equivariance), and
the index trace runs over its two complex directions,
dimC T CP 2 = 2, fixing the dimensionless local slope
2
TrT CP 2 (χ+ )
2 ⟨χ+ ⟩
∂MN
=
=
= 8
2
∂mπ
m2π
m2π

(8 = 2 × 4, tangent rank × scalar trace).
(AT12)
2
Since MN
= M02 −8 MN c1 m2π +· · · in chiral perturbation
theory, the forced slope 8 fixes
1
1
c1 = −
= −√
= −1.057 GeV−1 ,
MN
19 Λ5
9g 2 m3
whence σπN = −4c1 m2π − A 2π + · · · ≃ 56 MeV (the
64πfπ
leading −4c1 m2π = 4m2π /MN = 77 MeV minus the
forced 21 MeV chiral loop), inside the observed band
45–59 MeV. The central nuclear binding—and hence the
magic-number ordering at ρ0 —then closes with no free
coupling. Grade (conditional — strictly weaker than the
tower). Unlike the intercept 19 and the strange slope 9
(index/cohomology counts already carried by Eq. (AT11)),
the slope 8 is√not yet established in the corpus: the tower
gives MN = 19 Λ5 but does not derive the m2π -response
as a projector trace, and chiral symmetry explicitly permits the independent singlet counterterm c1 N̄ ⟨χ+ ⟩N .
The closure therefore requires a rule stronger than symmetry—“baryon mass-squared perturbations are internal
CP 2 ×S 3 projector traces, with no free IR counterterm”—
which the present corpus does not derive from a deeper
axiom. With that rule the nucleus is parameter-free at
leading chiral order; without it, c1 = −1/MN is a natural
value, not a theorem. This is a heavier insertion than the
baryogenesis Berry-holonomy channel (Extended Derivations, App. AH), which only names a clock inside the
fermion kernel DFD already uses — here a symmetryallowed counterterm must be actively forbidden. The
empirical slope (6.9–8.3, for σπN = 45–59) brackets 8, so
the integer alone is not the content; the content is the
tangent-projector rule that singles it out. The integer 8
is also a CP 2 cohomology dimension in its own right —
dim H 0 (CP 2 , T CP 2 ) = dim Aut(CP 2 ) = dim sl(3, C) =
8, the holomorphic tangent sections, the same kind of
Dolbeault index as the intercept 19 — which strengthens
that 8 is the right type of object. But the CP 2 structure

admits more than one route to 8 (the tangent-rank product 2 × 4 and the holomorphic-section count dim H 0 = 8
are distinct bookkeepings, 2 directions × 4m2π versus 8
sections × m2π ); the geometry does not uniquely select one
mechanism. That over-determination is precisely why the
forced content is the no-counterterm rule, not the integer.
e. Quark–lepton complementarity (P). The PMNS
reactor angle and the CKM Cabibbo angle are locked
3
PMNS
CKM
by sin2 θ13
= 31
sin θ12
, since sin2 θ13 = 3α (3 =
2
CKM
χ(CP )) and sin θ12
= 31α; 3/31 = 0.0968 vs data
0.0976 (0.8%).
f. CP-sector gauntlet (P). The exact DFD CKM
matrix (App. AO; its three integer counts and the Eulerprojected apex are selected, App. FK) yields, with no continuous parameter fitted: sin 2β = 0.719 (vs 0.708 ± 0.011,
+1.0σ); ϕs = −2.14◦ (vs −2.2 ± 0.9◦ , 0.1σ); Br(Bs →
µµ) = 3.40 × 10−9 (vs 3.34 ± 0.27, 0.2σ – where the
global-fit value sits 1.2σ high); and an εK deficit of 12–
17% (∼ −2σ), a falsifiable signature that lands on the
exclusive-|Vcb | side of the standing εK tension.
g. Look-elsewhere summary. Fπ = 35 ΛMS is the
unique expression in its window over both rungs,
pboth
functional forms, and denominators ≤ 10; η ′ = 39/2
is likewise the only structural candidate in its window;
As = 32πα5 is 1 of ∼ 225 grammar candidates within the
Planck 1σ band. The vector-meson and decuplet entries
sit on a denser integer grid; their evidential weight is in
the progression structure, scored separately.
h. As status: a calibrated amplitude and a numerical
match. We grade As = 32πα5 (Tab. CXVIII) as a numerical match to a calibrated amplitude: its power and its
coefficient come from the de Sitter dictionary at H⋆ , and
no DFD stage freezes perturbations there (Prop. FK.3).
In the de Sitter dictionary As = H⋆2 /(8π 2 εW M̄P2 ) with
the primordial (Wheeler–DeWitt) scale on the locked
Majorana rung, H⋆ ∝ MR = MP α3 (App. AT 1), and
the QED one-loop measure εW = α/(4π), the exponent
collapses exactly to α5 = (α3 )2 /α1 (sympy residual 0);
this power is rigid. The numeric coefficient 32π is not
derived from DFD geometry: it is fixed entirely by the
H⋆ –MR normalization, for which no first-principles argument (de Sitter horizon, S 3 /CP 2 internal volume, spectral
action, or Friedmann/Einstein coupling) exists in the corpus.
Planck-mass bookkeeping. DFD fixes MR = MP α3 =
4.74 × 1012 GeV using the non-reduced MP = 1.22 ×
1019 GeV, whereas the standard
amplitude formula carries
√
the reduced M̄P = MP / 8π = 2.44 × 1018 GeV. With
conventions made consistent √
the ratio that reproduces
the amplitude is√H⋆ /MR = 8π = 5.013 = MP /M̄P
(verified: H⋆ = 8π MR in the reduced-mass formula
gives As = 32πα5 = 2.080 × 10−9 , −0.82σ). The catalog’s
H⋆ = 8π MR writes the same number only because it
evaluates the reduced-mass formula with
√ the non-reduced
MP , so MP cancels; the “8π”
is
then
8π of reduced/non√
reduced conversion times 8π of an underived Hubble–
Majorana ratio.
√
Convention check. Writing H⋆ = 8π MR with the

312
non-reduced MP inside the reduced-mass formula gives
As = 4α5 = 8.28×10−11 (−67σ); that figure
√ is an artifact
of the mismatch, not evidence
against
8π. Under con√
sistent conventions H⋆ = 8π MR already lands in the
Planck band, and the relabelling to 8π double-counts the
Planck-mass
√ conversion. Either way the residual freedom
is a single 8π in H⋆ /MR that is not derived; we therefore present As as one calibrated amplitude, which
32πα5 matches numerically (App. FK).

7.

A zero-parameter amplitude–shape observable:
the (ν, u) relation

Every galactic test in this monograph so far has passed
through the acceleration scale a0 : the RAR (Sec. VII D),
the BTFR (Sec. VII C), the calibration table (Sec. VII E).
This subsection adds a test that does not. It ties the
mass discrepancy, an amplitude, to the rotation-curve
logarithmic slope, a shape, with no acceleration scale
anywhere in the statement, and it is therefore a test one
can run without knowing, fitting, or believing a0 .
a. The shape variable. Write ν ≡ gobs /gbar for the
local mass discrepancy and β ≡ d ln gbar /d ln R for the
baryonic log-slope, and define
d ln ν
u ≡ −2
.
(AT13)
d ln gbar
For extended mass the epicyclic frequency obeys κ2 /Ω2 =
3 + d ln gobs /d ln R = 3 + β (1 − u/2), which reduces to
the familiar point-mass form only at β = −2. With
s ≡ d ln Vobs /d ln R one has the exact estimator identity
u = 2 − 4 (s − 21 )/β, so u is measurable from a rotation
curve and a baryonic model alone.
Theorem AT.18 (Systematics immunity of the whole
derivative tower). Regard a galaxy as a curve in the plane
(X, Y ) = (ln gbar , ln ν). A distance error D → D(1 + ϵ)
and an inclination error Vobs → f Vobs each act on that
curve as a pure translation, (X, Y ) → (X +ln B, Y +ln A)
with A, B constant along the galaxy. Consequently every
derivative dn Y /dX n with n ≥ 1 — in particular u =
−2 dY /dX, its curvature, and every higher rung — is
exactly invariant, and the only motion is horizontal. The
displacement vectors in the (ln ν, u) plane are
 ∂ ln ν
∂u 
,
= (−1, 0),
∂ ln D ∂ ln D
(AT14)
 ∂ ln ν
∂u 
,
= (+2, 0),
∂ ln Vobs ∂ ln Vobs
both exactly, so distance and inclination are perfectly
degenerate with one another and collapse to a single horizontal nuisance parameter.
Proof. Under D → D(1 + ϵ) at fixed surface brightness,
R → R(1 + ϵ) and Mbar → Mbar (1 + ϵ)2 , so gbar =
GMbar /R2 is invariant (B = 1) while Vobs , a spectroscopic
2
quantity, is untouched and gobs = Vobs
/R → gobs /(1 + ϵ);
hence ν → ν/(1+ϵ), i.e. A = (1+ϵ)−1 and ∂ ln ν/∂ ln D =

−1. Under Vobs → f Vobs , gobs → f 2 gobs , gbar is untouched, so A = f 2 , B = 1 and ∂ ln ν/∂ ln Vobs = +2. In
both cases the transformation adds constants to X and
to Y ; adding a constant to a variable annihilates in every
derivative of order ≥ 1, so dn Y /dX n is unchanged for
all n ≥ 1. The rescaling R → cR is a reparametrisation
of the same curve and does not move its points in the
(X, Y ) plane. Verified symbolically to fourth order and
numerically to 10−12 on an explicit disc profile.
Remark AT.19 (What is not horizontal). A stellar massto-light error is not a translation, because the gas contribution does not scale with Υ⋆ ; it therefore has a vertical
component. Measured on SPARC the mass-to-light vector is (∂ ln ν, ∂u)/∂ ln Υ = (−0.562, +0.123) on average,
a stiffness of 4.6 : 1 in favour of the horizontal direction
(per-bin 1.4 : 1 to 25 : 1). Υ⋆ is the one systematic that
must be marginalised; distance and inclination need only
one shared horizontal nuisance.
Proposition AT.20 (Closed forms of u(ν), with τ or
x eliminated). For the saturating family µ(x) = x/(1 +
xn )1/n , x = gobs /a0 — whose n = 1 member is the
closure form µ(x) = x/(1 + x) of Thm. N.8 and Eq. (182)
— definition (AT13) collapses to
2(ν n − 1)
,
(AT15)
2ν n − 1
while for the exponentialp(single-mode occupation) response µ = 1 − e−τ , τ = gbar /a0 , it collapses to
ν
u(ν) = (ν − 1) ln
,
(AT16)
ν−1
via u = τ /(eτ − 1) and the exact inversion τ (ν) =
ln[ν/(ν − 1)]. Both families are monotone increasing, both
satisfy u(1+ ) = 0 and u(∞) = 1, and neither closed
form contains a0 : the acceleration scale cancels identically between the amplitude and the slope. For (AT16),
du/dν = ln[ν/(ν − 1)] − 1/ν > 0.
un (ν) =

Proof. Substitute y = µ x, ν = 1/µ into (AT13) and eliminate x (respectively τ ). All residuals vanish identically
in sympy; the limits are exact.
b. Confrontation with SPARC. Binning on gbar
(never on measured ν, which correlates bin assignment
with the Vobs noise that also enters s, and biases the
innermost bin by +0.47 in u), 8 equal-count bins over
2316 usable rows from 113 galaxies of the 149 that pass
the standard quality and inclination cuts, with a mockcalibrated bias correction and its systematic carried in
the error budget, the exponential response (AT16) gives
χ2 = 8.68 for 8 points, χ2 /dof = 1.08, p = 0.37, largest
single pull +2.26σ. Marginalising a global Υ⋆ under the
Schombert–McGaugh SPS prior (0.07 dex) and one shared
horizontal nuisance, the likelihood ratios against the same
data are
2 ∆ ln L = +0.43 (n=1),
+14.97 (n=2),
(AT17)
+30.89 (n=3),

313
i.e. the n = 2 “standard” and n = 3 members are excluded
by rotation-curve shape alone. The exclusion is robust:
+14.07 with Υ⋆ completely free, and in the raw (uncorrected, unmarginalised) statistic it stays in [+17.8, +40.4]
across all 22 pipeline variants of smoothing, bin count,
derivative scheme and minimum points per galaxy. The
raw statistic is however strongly Υ⋆ -dependent — at a
fixed Υdisk = 0.6–0.7 the ordering against n = 2 reverses
— which is precisely why the quoted ratios marginalise Υ⋆
rather than fixing it. The n = 1 closure form and the
exponential response, by contrast, are not separated by these data (+0.43). That is the limit of these
data, and it is not a defect of the estimator: an independent full-information analysis bounds every a0 -free shape
statistic — u, its curvature, or anything else built from
log-derivatives — at 2∆ ln L ≤ +3.99 on SPARC, because
77% of the available information sits in the amplitude,
not the shape. Separating n = 1 from the exponential
response requires either ∼ 8× more galaxies at the same
ν distribution, or the same data pushed to ν → 1.1 where
the gap between them doubles.
Remark AT.21 (Inverting the relation: a post-hoc consistency check). Flagged post-hoc; it was not pre-registered
and the test above stands without it. Read backwards,
the relation measures the stellar mass-to-light ratio: the
value of Υ[3.6] that minimises the shape χ2 is 0.475
with a statistical uncertainty of 0.024 dex under (AT16)
and 0.448 ± 0.023 dex under n = 1. The former is
−0.0223 dex from the independent stellar-population
value 0.50 ± 0.07 dex, i.e. −0.30σ combining the two
in quadrature.
Falsifier AT.22 (A mass-to-light ceiling, with no a0 , no
dark matter and no cosmology). DFD’s occupation reading of the boost carries ν ≥ 1 as a theorem. Raising
Υ[3.6] raises gbar and pushes the innermost (highest-gbar )
SPARC bin’s mean ν down, and that bin crosses the floor
ν = 1 at
Υ[3.6] = 0.5775 M⊙ /L⊙ = +0.063 dex above the SPS value,

(AT18)
holding Υbulge = 1.4 Υdisk at the SPARC fiducial ratio. At
the SPS value the bin sits at ν = 1.154 with a bootstrap
uncertainty of ≃ 0.06, i.e. ≃ 2.5σ clear of the floor. A
converged independent measurement of the 3.6 µm
disk stellar mass-to-light ratio above 0.58 M⊙ /L⊙
therefore refutes DFD using SPARC alone. The
falsifier is branch-independent (it uses only ν ≥ 1, not
the shape of µ) and involves no acceleration scale, no
dark matter and no cosmology. It is conditional on the
SPARC photometric and H i mass models as published; a
systematic revision of the gas masses moves the threshold.
Status AT.23 (Grades for this subsection). Theorem:
Prop. AT.20 (both closed forms, exact) and Thm. AT.18
(the two systematics vectors and the invariance of the
whole derivative tower). Validated: the SPARC confrontation of (AT16), χ2 /dof = 1.08, p = 0.37, and the
exclusion of the n ≥ 2 members at 2∆ ln L ≥ +14.07. Not
established: any separation between the n = 1 closure

form and the exponential response — SPARC provably
cannot deliver it (≤ +3.99 for any a0 -free shape statistic).
Post-hoc: Rem. AT.21. Branch bookkeeping: the
exponential response µ = 1 − e−τ used in (AT16) is the
derived law (App. BS, Thm. BS.9); µ(x) = x/(1 + x)
(Thm. N.8) is its weak-response form, and the Solar System selects the exponential branch (Cor. BS.14). Everything in this subsection that is graded Theorem or
Validated holds for both.

8.

Status and the remaining open lemma

Status AT.24. The identities of AT 1 are exact (machineverified), the results of AT 2–AT 5 carry the grades stated
with them, and none reuses a new integer; their role is
structural rigidity. The entries of AT 6 are predictions
at the stated grades. The α19/2 confinement exponent
that underlies several of these is reduced to a single open
lemma: writing 19
2 = (kmax − Ngen )/(2Ngen ) = 57/6, the
exponent is rigidity-selected (the unique catalog form in
the empirical window), but its derivation is contingent
on the cube law of Proposition AT.4, which is not independently proved here. We therefore present 19
2 as
rigidity-selected, not as derived.
Appendix AU: The Dark-Energy Sector from the α57
Clock

This appendix derives the cosmological vacuum-energy
deposit from the same finite-microsector nonzero-mode
count that fixes the Hubble scale. The central result is that
the deposit is a pure α-power in Planck units with no new
integer beyond the 57 = 60−3 of Appendix AP. The deposit
1/4
scale is ρdep = 2.547 meV, one α-rung from the heaviest
1/4

neutrino mass; the observed dark-energy scale is ρΛ,obs =
2.2455 meV, i.e. a measured ratio R = ρΛ,obs /ρdep =
0.6042 ± 0.0027 whose O(1) coefficient is underived — a
named open lemma (the vacuum-matching normalization;
it is the product of the measured clock coefficient C and
the dark-energy fraction ΩΛ ). The exponent is theoremgrade; the coefficient is measured. All exponent identities
verified to ≥ 40 digits in arbitrary-precision arithmetic.

1.

The dark-energy density as a pure α-power

The starting point is the master invariant proved in Appendix AP (Theorem AP.15) and re-used in Appendix AT

314
(Proposition AT.4),
GℏH02
= (H0 tP )2 = C α57 ,
c5
r

Gℏ
,
(AU1)
c5
C = 0.87784 ± 0.00266 (early clock, measured),
tP =

C = 1 (bound-matter clocks, Postulate ZS.11).
We use the Planck energy density ρP ≡ c7 /(ℏG2 ) =
EP /ℓ3P , the natural upper scale of any vacuum-energy
estimate.
Theorem AU.1 (Vacuum-energy deposit is α57 in Planck
units, (E)). The de Sitter vacuum-energy deposit associated with the DFD infrared clock is
 1/4
ρdep
3 57
1/4
3
=
α ,
α57/4 MP c2
ρdep = 8π
ρP
8π
(AU2)
where ρdep is the closure (critical) energy density
3c2 H02 /(8πG) evaluated at C = 1, the bound-matter clock
H0loc (Postulate ZS.11). This object is not ρΛ : labelling
it so conflates the deposit with the observed dark-energy
density, which is ρΛ,obs = R ρdep with the measured ratio
R = 0.6042±0.0027 (Eq. (AU5) below). The only constant
beyond the Friedmann 3/8π is the integer 57, which is the
primed-determinant mode count kmax − Ngen = 60 − 3 of
Theorem AP.11. No new integer enters.
Proof. The closure (critical) energy density of a spatially
flat universe is ρc = 3c2 H02 /(8πG) (the de Sitter value
when the clock H0 is dominated by the constant vacuum
term). Dividing by ρP = c7 /(ℏG2 ),
3 GℏH02
ρc
3c2 H02 /(8πG)
3
=
C α57 , (AU3)
=
=
7
2
ρP
c /(ℏG )
8π c5
8π
using Eq. (AU1). Setting the measured coefficient to its
bound-matter value C = 1 (Postulate ZS.11) defines
the exponent-theorem deposit, ρdep /ρP = (3/8π)α57 .
Taking the fourth root and using EP = MP c2 gives
1/4
ρdep = (3/8π)1/4 α57/4 MP c2 . Both equalities are exact;
the machine check reproduces ρdep /ρP = 1.89367×10−123
to 40 digits on both sides.
Remark AU.2 (The 10−123 “fine-tuning” is derived, not
tuned, given the matter and Planckian vacuum cancellations). The notorious statement “ρΛ /ρP ∼ 10−123 re3 57
quires 123-digit fine-tuning” is, in DFD, the output 8π
α
3 57
with log10 ( 8π α ) = −122.72. Nothing is tuned in the
exponent: it is the nonzero-mode count, and the fourth
root of the clock invariant (H0 tP )2 = C α57 is what darkenergy phenomenology measures — the residual O(1)
coefficient (R = 0.6042, a factor well inside one digit of
the 123) is measured, not tuned, given the cancellation
of the matter vacuum and of the spectral action’s Planckian vacuum term that Rem. AU.3 assumes. This is the
cosmological-constant analogue of the gauge hierarchy
being α8 in the Higgs sector (App. Z).

Remark AU.3 (Scope: microsector piece, not a full vacuum-energy sum rule). The α57 relation is the forced
microsector vacuum-energy piece together with the observed closure via the clock dictionary; its naturalness
statement is scoped to the 60-mode microsector. It does
not by itself bound or cancel the Standard-Model mattersector vacuum contribution (dominated by the top loop,
of order α32 MP4 in DFD units, formally far larger): DFD
forbids supersymmetry and derives no full-spectrum vacuum sum rule, so the matter piece is unaccounted — the
same open cosmological-constant problem faced by every quantum field theory coupled to gravity. Status: the
cosmological-constant exponent is the microsector relation
above and its coefficient R is measured; the matter-sector
cancellation problem remains open (and is universal, not
DFD-specific). The dark-energy relation assumes that the
matter piece cancels, as ΛCDM does through its fitted
constant, and that the Planckian vacuum term of the
spectral action (App. WS) cancels. Because the ψ source
is the cosmic-mean-subtracted density, a uniform vacuum
energy does not source ψ; it acts through the background
expansion law and, as Λ does, through the de Sitter term
on bound systems.
Numerically, with the locked ledger (α−1 = 137.036,
MP = 1.22089 × 1019 GeV),
1/4

ρdep = 2.547 meV

(deposit / closure scale at C = 1),
(AU4)
to be compared with the observed late-time dark-energy
1/4
scale ρΛ,obs = 2.2455 meV (Planck 2018 at H0early =
67.52 ± 0.09). The measured ratio is
ρΛ,obs
R≡
= 0.6042 ± 0.0027,
R1/4 = 0.8816.
ρdep
(AU5)
1/4
The ΩΛ factor alone (0.6881/4 = 0.911) leaves 2.32 meV,
not 2.2455 meV; the decomposition is R = ΩΛ C: the darkenergy fraction times the measured clock coefficient C =
0.87784 ± 0.00266, both measured. The deposit exponent
theorem stands; the O(1) coefficient R is underived — a
named open lemma (the vacuum-matching normalization
of Rem. AP.6).

2.

The Hubble–Planck seesaw and holographic form

Proposition AU.4 (Deposit scale as a Hubble–Planck
geometric mean, (E)). The deposit scale of Theorem AU.1
is the (3/8π)1/4 -dressed geometric mean of the Hubble
energy and the Planck energy,
 1/4 p
1/4
3
ρdep = 8π
ℏH0 MP c2
(AU6)
(with H0 = H0loc ) and, via the QCD–Hubble bridge
ℏH0 = Λ3DFD c2 /MP2 (Proposition AT.4), equivalently
1/4
3/2
ρdep = (3/8π)1/4 ΛDFD /(MP c2 )1/2 .
Proof. From Eq. (AU1) at C = 1, ℏH0 = MP c2 α57/2 ,

315
√
so ℏH0 · MP c2 = (MP c2 )2 α57/2 and ℏH0 MP c2 =
MP c2 α57/4 .
Multiplying by (3/8π)1/4 reproduces
Eq. (AU2). Substituting ℏH0 = Λ3DFD c2√
/MP2 gives the
1/4
ΛDFD form. Machine check: (3/8π)
ℏH0 MP c2 =
2.5468 meV, matching Eq. (AU4).
Remark AU.5 (Holographic dark energy, derived rather
than postulated). Equation (AU6) is precisely the Cohen–
Kaplan–Nelson holographic / infrared-cutoff relation ρΛ ∼
MP2 H02 that elsewhere is imposed as an ansatz. In DFD it
restates the clock relation: the IR cutoff is the α57 clock,
the MP2 H02 scaling follows from the de Sitter Friedmann
equation, and the prefactor 3/8π is fixed (not fitted). The
vacuum energy is thus a seesaw between the largest scale
(MP ) and the smallest curvature (H0 ), with the geometric
mean landing at the meV.
Proposition AU.6 (Deposit scale one α-rung from the
heaviest neutrino, (E)). With the printed neutrino mass
14
(App. X, App. AT), the exact ratio is
m3 = 14
13 πMP α
1/4

ρdep
m3 c2

=

13  3 1/4 1/4
α
= 0.05078,
14π 8π

Remark AU.8 (Non-coincidence). The e-fold count 140.29
is not equal to α−1 = 137.04 (they differ by 2.4%); we
explicitly decline to read ln(1/H0 tP ) ≈ α−1 as an identity.
The robust content is Eq. (AU8) as an exact restatement
of the α57 clock, and the Hubble time on the early clock,
1/H0early = 14.48 Gyr.
4.

One dark energy: the equation of state

(AU7)

1/4

so ρdep and m3 occupy adjacent rungs of the same α-tower
(57/4 = 14.25 versus 14), differing by one factor of α1/4
and the printed 14
13 π prefactor.
2 14
Proof. Divide Eq. (AU2) by m3 c2 = 14
13 πMP c α . The
57/4−14
1/4
MP cancels and α
= α
remains, giving
Eq. (AU7) (machine-verified to 40 digits).

The numerical near-coincidence flagged in the cosmol1/4
ogy literature (ρΛ ≈ mν ) is, in DFD, this α14 -tower
proximity. The lightest DFD neutrino, m1 = 2.34 meV
(dressed-closure spectrum, App. X), coincides with the
1/4
observed ρΛ,obs = 2.2455 meV to 4.2%; this is recorded
as a coincidence (both are α14 -class scales), not a derived
identity, because the m1 value carries its own branch
dependence.
3.

√
Proof. Equation (AU1) gives H0 tP = C α57/2 identi1
cally, so ln(1/H0 tP ) = 57
2 ln(1/α) − 2 ln C = 140.227 +
0.065 = 140.292 (machine check). The Hubble time on the
early clock is 1/H0early = 14.48 Gyr, the exponent content
of which is derived and the coefficient content measured;
on bound-matter clocks it is 1/H0loc = 13.56 Gyr, with
no coefficient. Neither is an age: the expansion age is
t0 ≃ 13.8 Gyr, 12.92 Gyr in laboratory units (App. ZS).
With the measured Ωm ≃ 0.31 the crossing is at zeq = 0.30
and the deceleration→acceleration onset at z = 0.64, both
O(1) today.5 No epoch-dependent tuning is invoked.

The “why now” coincidence

The second classical puzzle is the temporal coincidence
ρΛ ∼ ρmatter today. In ΛCDM the epoch at which the two
densities cross is set by the free constant Λ and appears
anthropically tuned.
Theorem AU.7 (The √
Hubble clock’s e-fold count, (T)).
The cosmic clock H0 = C α57/2 /tP is α-locked up to the
measured O(1) coefficient C (Eq. (AU1)). The logarithm
of the ratio of the Planck rate 1/tP to the Hubble rate is
1
57 1
1
ln
=
ln − ln C = 140.29,
(AU8)
H0 t P
2
α 2
a function of α and the nonzero-mode count alone up to
the 0.065 contributed by the measured C on the early clock;
on bound-matter clocks C = 1 (Postulate ZS.11) and the
count is 140.23.

The late-time distance
excess is not a ψ-screen ∆ψ(z) =


obs
matter
ln DL
/DL
(reconstructed value 0.274 at z = 1),
and the vacuum density of Theorem AU.1 and that screen
are not two faces of one optical field: the reconstructed
screen is the ΛCDM/matter-only distance ratio by construction, a homogeneous screen adds nothing beyond
the expansion (App. ZS), and the line-of-sight dressing
DL → DL e∆ψ is falsified (App. SL). DFD has one dark
energy: the constant vacuum term of Theorem AU.1.
This needs the four-dimensional three-form that the internal C3 leaves in its b0 = 1 piece, the part constant on
K (beside χ from b3 and one vector from b2 , App. AV),
to carry zero flux: that three-form propagates nothing,
but a nonzero four-form flux would add its own constant
vacuum energy.
Proposition AU.9 (Equation of state of the vacuum
term, (T)). The DFD dark energy is the constant vacuum
term ρΛ,obs = R ρdep of Theorem AU.1 alone. Its equation
of state is w = −1 identically, and DFD is degenerate with
ΛCDM on distance observables and on late-time linear
growth (Sec. XVI I).
Proof. A constant vacuum energy density has p = −ρ,
hence w = −1; this is the form of the term, not a separate
derivation. A CPL fit to the resulting distance ladder
returns (w0 , wa ) = (−1, 0).

5 The dark-energy appendices round the measured-clock matter

density Ωm = 0.312 (ΩΛ = 0.688) to 0.31 here; the conclusions
are insensitive to this spread. Across Ωm = 0.30–0.315 the
equality redshift is zeq = 0.33–0.30 and the acceleration onset
z = 0.67–0.63 (both O(1)), and w = −1 is exact for the constant
vacuum term. The Planck-2018 value Ωm = 0.315 is used in the
σ8 confrontation of App. AE.

316
Remark AU.10 (No dark-energy clustering, shared with
ΛCDM). A constant vacuum term carries no perturbations: no dark-energy sound speed and no late-time darkenergy clustering, exactly as for Λ in ΛCDM. This is not
a discriminator from ΛCDM but a shared falsifier: a confirmed evolving dark energy with or without clustering
(e.g. if the DESI DR2 dynamical-dark-energy hints are
confirmed) falsifies DFD’s vacuum term together with Λ.

5.

Look-elsewhere and status

a. Look-elsewhere. Scanning the grammar (prefactor
∈ {1, 3/8π, 1/8π, 3/4π, 1/4π, 21 , 2, π, 1/π}, exponent n/4
with integer n ∈ [50, 64]) for candidates landing in the
1/4
generous band ρΛ ∈ [1.8, 2.8] meV yields 3 hits out of
135 grammar candidates (n = 57 with prefactors 3/8π,
1/4π, 1/8π). The exponent n = 57 is not selected by the
data: it is forced to be the same 57 that appears in the
proven Hubble invariant (App. AP), and the Friedmann
prefactor 3/8π is forced by the closure-density definition.
The α-rung spacing is 1371/4 = 3.42 in energy, so no
neighbouring integer rung lands within a factor of the
observed scale. The result is therefore rigidity-selected
(zero new integers), not numerology.
Status AU.11. Theorem AU.1 (the α57 vacuum deposit),
Proposition AU.4 (the Hubble–Planck seesaw / holographic form), and Proposition AU.6 (adjacent-rung neutrino relation) are exact identities, machine-verified to
≥ 40 digits, reusing only the locked integer 57 = 60 − 3
and the Friedmann constant 3/8π. They inherit their
open coefficient from the clock dictionary (Axiom AP.5):
the identification (H0 tP )2 = C Zα′ /Z1′ is a spectral-action
dictionary statement whose exponent is theorem-grade
and whose O(1) coefficient C is a measured input (the
eight-route no-go proves it underivable from the printed
3 57
corpus) — so ρdep = 8π
α ρP is theorem-grade in its
exponent, and the observed ρΛ,obs = R ρdep carries the
measured R = 0.6042 ± 0.0027 (R = ΩΛ C), whose derivation is a named open lemma. Theorem AU.7 is a closed
consequence of the α-locked exponent. Proposition AU.9
(equation of state) restates the constancy of the vacuum
term; the no-clustering signature is a falsifier shared with
ΛCDM, not a discriminator. The empirical contact is the
derived 10−122.7 hierarchy landing within the measured
O(1) coefficient of the observed 2.2455 meV scale; its
evidential weight is governed by the look-elsewhere count
above.

Appendix AV: χ-Matter: the Weak-Sector Matter
Field and the True Definition of Dark Matter
1.

Overview, grade, and what this appendix claims

This appendix derives a cold, non-thermal pseudoscalar
matter field χ from the same internal geometry and
the same per-mode Gaussian determinant (Lemma O.5,
App. O) that fix the rest of the DFD α-tower. The
field is not postulated. It is the harmonic three-form on
the internal S 3 = SU (2) factor — a zero-mode of the
fixed compactification that the corpus’s rigidity classification (the parent-tensor rigidity theorem (App. AH))
does not list. With that mode included, its decay constant, its mass, and its relic abundance follow from
the tower and the stated inputs and premises named
below; the cold component is exactly what the CMB
acoustic kernel requires at the third peak. (Step 5b
(Thm AV.13) removes the free misalignment angle θi
— with no inflaton there is no chosen angle, and the amplitude is instead thePfinite SU (2)60 CS/WZW vacuum
2
expectation ⟨θ2 ⟩ =
j |S0j | C2 (j)/[k(k + 2)] = 0.073
— a variance in the period-1 (turns) flux coordinate
(Lemma AV.6) — giving, through the exact radiationera relic map, Ωχ h2 = 0.1193 (−0.60σ from Planck). A
factor-44 overshoot is predominantly a units artifact —
the classical-continuum measure π 2 /3 is the variance of a
period-2π angle, whereas the flux coordinate has period
1 — compounded by a continuum measure choice; see
Rem. AV.14.)
a. The de-“dark” thesis. DFD does not retire the
terms dark matter and dark energy; it supplies their true
definitions. “Dark” has only ever meant not knowing
what it is. DFD removes that ignorance on both counts:
the cold clustering component is the derived χ field of
this appendix, and the late-time acceleration is the α57
geometric vacuum energy (App. O), whose coefficient
is measured once (Rem. AP.6). We keep the historical
names — and compare to ΛCDM under them — precisely
so that the resolution is unmistakable: dark matter is the
χ particle; dark energy is the energy of the vacuum.
Remark AV.1 (Ledger-controlling status: derived carrier;
forced amplitude/stated conventions). This remark controls every later “theorem-grade” shorthand and aligns
it with the authoritative fitted-knob ledger. The carrier/operator
√ is derived. The numerical mass additionally
uses the 158 multiplicity of Sec. AV 5, one stated discrete input. The abundance is forced-amplitude/statedconvention: its operator is proven, while state preparation
and the relic-map normalization are named premises. The
chain is conditional on:
1. the two tower-wide conventions that already gate
the Higgs sector and α itself — the stated choices
of α’s exact construction (§X C) and the gauge normalization g 2 = α (App. O);
2. a zero photon coupling, E = 0 (Theorem AV.35),

317
so that the hot early plasma adds no thermal χ
(Sec. AV 7, channel (b));
3. the weak-block dimension dχ = 3 of Theorem AV.4
(this is exact Lie theory, see the proof);
4. Theorem AV.13 (Step 5b, Sec. AV 7) removes
the O(1) misalignment angle θi of Sec. AV 6,
and the State-Preparation Theorem AV.23 proves
it cannot be reintroduced: DFD has no inflaton, so no angle is chosen, and the amplitude is
the finite
PSU (2)60 CS/WZW vacuum expectation
⟨θ2 ⟩ = j |S0j |2 C2 (j)/[k(k + 2)] = 0.073, giving
Ωχ h2 = 0.1193 (−0.60σ from Planck). The operator (Casimir) is proven, the sector measure rests on
the asserted premise P4′ (Lemma AV.27), and the
normalization k(k + 2) follows from the Sugawara
level together with the half-period edge θmax = 12 —
the Z2 -folded edge of the period-1 flux coordinate
(Lemma AV.6) — that follows from χ’s derived Z2
orientation-oddness. So the amplitude is forced
and the abundance is forced-amplitude/statedconvention (its non-DFD bridge being the standard cosmological relic-redshift map, carried at its
exact radiation-era value, Rem. AV.12, shared with
every DM relic). A factor-44 overshoot is a unitsplus-continuum-measure artifact (Rem. AV.14).
We therefore assign Derived to the carrier and period
relation, motivated multiplicity to the printed mass
coefficient, and forced-amplitude/stated-convention
to the relic abundance (Step 5b) — not the loose “solved”
and not a fitted density. The residuals are named in
Rem. AV.14: the Z2 /Sugawara normalization convention
and the O(1) cosmological prefactor of the relic map.
2.

Step 1 — χ is the harmonic three-form outside
the rigidity classification

The corpus already establishes (the parent-tensor rigidity theorem, App. AH, companion Extended Derivations)
that the only metric zero-modes on K = CP 2 × S 3 are
the trace ψ and the TT graviton, and that the form-field
sector is constrained by b1 (K) = 0. That classification
addresses one-forms. It does not address the three-form,
and by Künneth the three-form is present:
X
b3 (CP 2 × S 3 ) =
bp (CP 2 ) bq (S 3 )
p+q=3
(AV1)
= b0 (CP 2 ) b3 (S 3 ) = 1 · 1 = 1.
Theorem AV.2 (Topological identity of χ). On K =
CP 2 × S 3 the space of harmonic three-forms is onedimensional, b3 (K) = 1. Its generator is the bi-invariant
Cartan three-form on S 3 ∼
= SU (2),


1
ω =
Tr (g −1 dg)∧3 ,
g ∈ SU (2),
(AV2)
2
24π
which coincides, up to standard normalization, with the
SU (2) Chern–Simons density. The DFD field χ is the

coefficient of this unique harmonic mode; its 4D reduction
on the S 3 three-cycle is a pseudoscalar in the sense that
it is odd under reversal of the S 3 orientation; DFD’s CP
leaves S 3 fixed and acts on C3 by pullback alone, so χ is
CP-even (Theorem AV.35).
Proof. S 3 is compact, connected, simply connected and
orientable, so by Hodge theory dim{harmonic p-forms} =
bp . For S 3 , (b0 , b1 , b2 , b3 ) = (1, 0, 0, 1), and b3 (K) =
1 follows from (AV1). On the group manifold SU (2)
the Maurer–Cartan form θ = g −1 dg is su(2)-valued and
satisfies the structure equation dθ = −θ ∧ θ. The biinvariant volume form is, up to scale, Tr(θ∧3 ): left/right
invariance follow from θ 7→ h−1 θh under g 7→ gh together
with cyclicity of the trace; closedness d Tr(θ∧3 ) = 0 is
immediate from the structure equation and cyclicity; coclosedness is automatic for a top-degree form on a closed
manifold. Hence ω is harmonic and generates H 3 (S 3 ).
Finally Tr(θ∧3 ) is the integrand of the SU (2) Chern–
Simons form CS = Tr(A dA + 23 A3 ) evaluated on the
pure-gauge configuration A = θ, establishing the stated
coincidence
and the normalization (the prefactor 1/24π 2
R
giving S 3 ω = 1). Reduction of an internal three-form
on the three-cycle S 3 yields a 4D scalar; the orientationodd character of ω makes it odd under reversal of the S 3
orientation, which DFD’s CP does not perform (App. L).
Remark AV.3 (The three-form sector and the parent-tensor theorem). The parent-tensor rigidity theorem
(App. AH) remains correct as stated: ψ and hTT are the
two components of the metric zero-mode, and χ is not a
metric mode — it is the three-form gauge field, a separate
sector, so it does not affect cT = c (the cT = c theorem,
App. AH). The antisymmetric-tensor sector is not empty:
b1 = 0, but b3 = 1. App. AH (Extended Derivations)
identifies the three-form with χ. χ is not a “60th internal
mode”: the index dim Hmicro = 60 counts chiral/gauge
modes of the Spinc Dirac operator, whereas χ is a bosonic
harmonic form of the same S 3 Chern–Simons structure
that already gives the closed form for α (App. AH, Extended Derivations; Berezin–Toeplitz at kmax = 60). It is
the dynamical zero-mode of a sector DFD already relies
on, not a new addition.

3.

Step 2 — the gauge-adjoint dimension of χ is
dχ = 3

Theorem AV.4 (Gauge-adjoint count for χ). The
Toeplitz (mode-quantized) subspace associated with the
Cartan three-form ω of SU (2) is the complexified algebra
of that SU (2), sl(2, C), of complex dimension dχ = 3, the
dimension of the weak algebra, which is why it is called the
weak block below. This SU (2) carries no quark or lepton:
it is not the weak SU (2) of the Standard Model fermions,
whose Chern–Simons coupling to χ would thermalize χ
(Theorem AV.35).

318
Proof. The Maurer–Cartan form θ in (AV2) is su(2)valued and the cubic Tr(θ∧3 ) saturates the adjoint trace
over that algebra. Under the corpus’s Toeplitz quantization the harmonic generator lifts to the operator content of the adjoint representation; complexification gives
su(2)C = sl(2, C), with dimC sl(2, C) = 3. This is the
exact parallel of the Higgs sector, where the corresponding construction on the colour factor yields sl(3, C) with
dimension 8. Thus dχ = 3 stands to the weak sector as
dH = dim su(3) = 8 stands to the colour sector.
The single integer carried forward is dχ = 3.
4.

Step 3 — decay constant from the per-mode
determinant

Lemma AV.5 (Lemma O.5 specialized to χ). The permode Gaussian determinant of a d-dimensional adjoint
Toeplitz block is α d with the uniform gauge normalization
g 2 = α (App. O, Lemma O.5). The λ-dependence cancels
R 2 −(λ/α)|ϕ|2  R 2 −λ|ϕ|2
exactly per mode:
d ϕe
d ϕe
= α.
Specializing to d = dχ = 3 fixes the χ kinetic scale
fχ = M̄P α3 = 2.435 × 1018 GeV × α3
≈ 9.46 × 1011 GeV,

(AV3)

with M̄P = (8πG)−1/2 the reduced Planck mass and α−1 =
137.036 (so α3 = 3.886 × 10−7 ). Which normalization
convention this scale carries — radian decay constant or
flux period — is settled by Lemma AV.6 below: fχ is the
period.
Proof. Each of the d Gaussian modes of the adjoint block
contributes one factor of α to the fluctuation determinant
(Lemma O.5); the determinant rescales the Planck-scale
internal stiffness to a physical decay constant f = M̄P α d .
For the weak block dχ = 3 (Theorem AV.4), giving (AV3).
The same lemma with dH = 8 returns M̄P α8 = 19.6 GeV,
the numerator of the χ see-saw; the Higgs vacuum value
follows from it only through the fitted prefactor of v =
4πΛ (Rem. AV.9). (AV3) is its d = 3 sibling, not an
independent input.
R
Lemma AV.6 (The flux period: S 3 CR3 has unit period, and (AV3) is that period). Let θ := S 3 C3 in units
of the normalized generator ω of (AV2). Then: (i) ω
3
3
is the integral generator
; Z): for a degree-n
R of1 H (S −1
3
map g : S → SU (2), S 3 24π2 Tr[(g dg)∧3 ] = n exactly
(verified symbolically for n = 1, 2, 3),
R and a large gauge
transformation of winding w shifts S 3 C3 by the integer
w, entering the level-k Chern–Simons phase as e2πikw .
Hence θ lives on R/Z: its period is 1 exactly — a
degree of a map, not a convention. (ii) Lemma AV.5 is
provably blind to the normalization of the coordinate it
integrates: rescaling ϕ → c ϕ in the per-mode ratio returns α identically, d(ratio)/dc = 0; a ratio of Gaussian
normalizations transmits whatever coordinate the geometry hands it and cannot fix one. (iii) Consequently the

coefficient (AV3) multiplies the period-1 coordinate of (i):
fχ = M̄P α3 is the period of the canonically normalized
flux field,
P ≡ fχ = M̄P α3 = 9.4635 × 1011 GeV,
(AV4)
P
= 1.506 × 1011 GeV,
2π
with frad the radian-normalized decay constant that enters axion-form formulas (see-saw mass, anomaly vertex).
The potential is the periodic parabola V = 12 m2χ P 2 ∥χ/P ∥2 ,
with ∥u∥ the distance from u to the nearest integer; its
curvature at each minimum equals that of 158 Λ4 [1 −
cos(2πχ/P )], with 158
√ the multiplicity stated in Theorem AV.8, so mχ = 158 Λ2 /frad . The photon coefficient
of a vertex gχγ = Eα/P is zero, E = 0, so χ has no
photon vertex (Theorem AV.35).
frad ≡

Remark AV.7 (Grade of the period identification, and the
structural supports). Part (i) and part (ii) are theoremgrade (exact symbolic residuals). Part (iii) is Derived:
it uses the corpus’s own stated accounting that the
Lemma O.5 determinant ledger carries no 2π (App. Z);
the one named residual is that Lemma AV.5 performs no
explicit Kaluza–Klein reduction — “Planck-scale internal
stiffness = M̄P ” is asserted, and the printed action cannot
supply it, because χ does not enter its Dirac operator,
so the stiffness is a stated input, which the measured
dark-matter density confirms to within 1% (computed) —
and an actual reduction returning 2π M̄P α3 in the turns
coordinate would restore the radian reading. Four independent structural facts support the period reading
and none supports the radian one — and an empirical
discriminator (e) selects it outright: (a) the relic amplitude QCS is a variance in turns2 — the exact identity
C2 (j)/[k(k + 2)] = (K/k) [u2j − (π/K)2 ]/(4π 2 ), uj = (2j +
1)π/K, exhibits the 1/(4π 2 ) radians2 →turns2 conversion,
and the pre-registered continuum limit limk→∞ QCS =
1
1
1
12 − 8π 2 = 0.0706682 has leading term 12 , the uniform
variance of a period-1 angle (a period-2π angle gives
π 2 /3); (b) the edge θmax R= 12 (AV13) is the Z2 -folded
half-period in turns; (c) S 3 ω3 = 1 is integer winding,
the same statement that derives NDW = 1; (d) the largegauge phase e2πikw of (i). The apparent counter-witness
δθ = H/(2πfχ ) is not evidence for the radian reading:
that 2π is the Gibbons–Hawking de Sitter factor, identical in both normalizations. (e) Empirical convention
discriminator. With the relic amplitude topologically
fixed (Step 5b: ⟨θ2 ⟩ = 0.073, no free angle), the radian
reading (frad = M̄P α3 , P = 2πfrad ) rescales the relic
by (m′ /m)1/2 (A′ /A) = (2π)−1/2 (2π)2 = (2π)3/2 = 15.75,
giving Ωχ h2 ≃ 1.88 — a 15.7× overshoot of the observed
0.12, excluded; the hybrid variant (radian scale with radian variance) is blocked by the turns-pinning facts (a)–(d)
and independently undershoots by ∼ 2.5×. Planck itself
selects the period reading within the no-free-angle chain.
This is a consistency discriminator on an already-scored
datum, not a new evidence row, and it is distinct from
the ∼9–44× continuum-measure artefact, which concerns

319
a different object. Conditional cross-link: M̄P cancels in
fχ /Λ = α−5 (both exponents derived-grade); via v = 4πΛ
(Rem. AV.9) this pins P √
from the measured v at 0.054%,
conditional on the fitted 2π prefactor of App. Z, whose
printed 0.054% best-fit gap this residual identically is.
Residual, sharpened: any KK reduction must consume
one dimensionful anchor (dimensional analysis on the
Toeplitz kinetic pattern, App. AH), and the reduction itself, which would fix the O(1) coefficient (1 vs 2π), cannot
be done in the printed action, because χ does not enter
its Dirac operator; the consumed-anchor status parallels
the consumed G of the Newton identity (App. ZU), with
the asymmetry that there the prefactor is proven unique,
while here the 1-vs-2π coefficient is a stated input.

5.

Step 4 — see-saw mass: χ is light, cold, and
non-thermal

Theorem AV.8 (χ mass). With the weak-block see-saw
scale Λ2 /frad — the radian decay constant frad = P/2π
of Lemma AV.6, since the periodic-parabola potential has
curvature 158 Λ4 (2π/P )2 — and the electroweak-scale numerator Λ = M̄P α8 (the same d = 8 scale that sets the
Higgs VEV),
√
√
Λ2
Λ2
mχ = 158
= 2π 158
frad
P
(AV5)
√
13
= 2π 158 M̄P α = 32.005 eV,
√
where 158 is the weak-block multiplicity prefactor, one
stated discrete input. It is read as 158 = 3(kmax −
dim M7 ) − 1 = 3 · 53 − 1, with the factor 3 the dimension
dχ = dim sl(2, C) of the weak block (Theorem AV.4) rather
than Ngen , which the O(9) block already contains, times
the 53 = kmax − dim M7 block, less a single zero-mode.
Status: no single counting rule gives 158. The corpus’s
uniform rule, all modes less the zero modes, which gives
57 = 60 − 3, gives 171 on the weak block, and a symmetry acting irreducibly on the three-dimensional factor and
trivially on the microsector modes makes every invariant count a multiple of 3, which 158 is not (computed).
The multiplicity is therefore a stated input, not a derivation; 171 would give mχ = 33.30 eV and Ωχ h2 = 0.1217
(+1.4σ) against 0.1193 (−0.6σ), a 2% difference that nextgeneration measurements of the cold-matter density can
separate. It is an integer count, not the loop factor 16π 2
it lies 0.05% from; the exact electroweak–dark scale link
is carried separately by v = 4πΛ (Rem. AV.9).
Proof. The see-saw form m ∝ Λ2 /f is the standard statement that a harmonic p-form axion acquires its mass
not from the Planck-scale moduli potential (which has
no size or shape modulus to fix on the truncated microsector, App. O) but from the non-perturbative SU (2)
effects that lift the three-form shift symmetry; the resulting scale is the geometric mean structure Λ2 /frad ,
with frad = P/2π the radian decay constant of the
period-P potential (Lemma AV.6). With Λ = M̄P α8

and P = M̄P α3 the net power is 16
√ − 3 = 13. Numerically α13 = 1.664 × 10−28 and 2π 158 = 78.98, giving
mχ = 78.98 × 2.435 × 1027 eV × 1.664 × 10−28 = 32.005 eV
(= 2π ×5.0937 eV). Because the finite-CS relic is produced
at rest (zero momentum; a variance relic, not a coherent
condensate, see Rem. AV.30), χ is cold regardless of this
low mass — unlike a 32 eV thermal relic, itsp
free-streaming
scale is set by the Jeans wavenumber kJ ∝ mχ H, which
lies far below all CMB and large-scale-structure scales.
The relic is therefore cold and non-thermal, clustering
as cold dark matter at recombination, and the hot early
plasma adds no thermal χ quanta, because χ has no photon coupling (E = 0, Theorem AV.35; Eq. (AV22)).
Remark AV.9 (Exact EW↔dark scale link: v = 4πΛ).
8
The χ see-saw
√ numerator Λ = M̄P α and the Higgs VEV
8
v = MP α 2π = 246.09 √
GeV (App. Z) sit on the same
d = 8 rung. With MP = 8π M̄P ,
√
√ √
v
= 8π 2π = 16π 2 = 4π
Λ

=⇒

v = 4π Λ

(exact).

(AV6)
This is an exact, type-correct relation between the √
electroweak and dark scales, conditional on the fitted 2π
prefactor of App.√Z: a ratio of two
√ continuum scales whose
4π is the fitted 2π times the 8π that relates MP and
M̄P . It is an exact algebraic consequence of the printed
scales, recorded here as one named relation: the Higgs
VEV and the χ see-saw numerator are one and the same
d = 8 scale up to the factor
4π. It is distinct from
√
the χ-mass multiplicity 158, a stated discrete count
that is not 4π: although 158 lies 0.05% above the 4D
loop factor 16π 2 = 157.91, that proximity is a numerical
near-coincidence, not an identity (a multiplicity is an integer; 16π 2 is transcendental). Consequently the tempting
“v 2 = 2 mχ P ” holds only to 0.03% and is not recorded
as exact — the exact EW↔dark content is carried by
v = 4πΛ alone.
Remark AV.10 (The key assumption, named). The one
physics input that is assumed rather than computed here
is that the three-form mode is lifted by the see-saw scale
Λ2 /frad (shift-symmetry protected, light) rather than
sitting at the Planck-scale moduli mass like the squashing
modulus. This is the standard expectation for a p-form
axion, and it is what distinguishes χ from the heavy
moduli; but it is the assumption on which the 32 eV value
rests, and it is the natural target of any attempt to refute
the χ identification.

6.

Step 5 — relic abundance by vacuum
misalignment

Theorem AV.11 (Relic density at a natural misalignment angle). For the derived (P, mχ ) = (9.46 ×
1011 GeV, 32.005 eV), standard vacuum misalignment
gives
√
Ωχ h2 ≃ 1.62 κrad 2π θi2 = 1.635 θi2 ,
(AV7)

320
√
with κrad 2π = 1.00923 the exact-onset/period-fix correction of Rem. AV.12, so that the observed cold-dark-matter
density ΩDM h2 = 0.12 is reproduced at the natural value
p
θi =
0.12/1.635 ≈ 0.27,
(AV8)
an O(1) angle requiring no fine-tuning.
Proof. The field begins to oscillate when 3H(Tosc ) =
1/2
mχ ; in the radiation era H = 1.66 g∗ T 2 /MP gives
4
Tosc ≈ 8.7 × 10 GeV (g∗ = 106.75). The comoving
number density nosc = 12 mχ θi2 P 2 redshifts as a−3 ; en3
tropy conservation (aosc /a0 )3 = g∗s,0 T03 /(g∗s,osc Tosc
) then
yields the present density ρχ,0 = mχ nosc (aosc /a0 )3 . Di−47
2
GeV4 gives Ωχ h2 =
viding by
√ ρcrit2/h = 8.10 × 10
1.62 κrad 2π θi (numerically verified; the coefficient
anatomy is Rem. AV.12). Solving for Ωχ h2 = 0.12 gives
θi = 0.271. The period and mass are fixed by Secs. AV 4–
AV 5; θi is the single free initial condition of this preinflation form—which Step 5b (Theorem AV.13) replaces:
with no inflaton there is no chosen θi , and the amplitude is instead the finite SU (2)60 CS-vacuum expectation
⟨θ2 ⟩ = 0.073. This step nonetheless supplies the O(1)
cosmological prefactor (the radiation-era H(T )/entropy
map from ⟨θ2 ⟩ to Ωχ h2 ), applied to the topological ⟨θ2 ⟩
rather than to a chosen angle.
Remark AV.12 (The relic-map coefficient, made exact).
The sudden-approximation relic map has the bare coefficient 1.62 at the mass 5.0937 eV. Two corrections,
both derived, make it exact: (i) exact radiation-era
onset. The exact RD misalignment equation χ′′ +
(3/2t)χ′ + m2 χ = 0 has the regular solution χ =
χi 21/4 Γ( 54 )(mt)−1/4 J1/4 (mt), whose comoving ρa3 is
smaller than the sudden-approximation value at 3H = m
by
Γ( 1 )2
κrad = √4 = 0.402625.
6 3π

(AV9)

(ii) the flux-period mass. With mχ = 2π × 5.0937 eV
(Lemma AV.6) and the onset amplitude unchanged
2
2
2
(frad
θrad
= P 2 θturns
identically),√only the m1/2 redshift
1/2 2 2
factor of Ω ∝ m f θ moves: × 2π = 2.50663. √
The net
factor on the sudden-approximation map is κrad
2π =
√
1.00923, i.e. the exact map is Ωχ h2 = 1.62 κrad 2π Q =
1.635 Q for any dimensionless quadratic amplitude Q. No
anharmonic correction is owed: the Sugawara energy is
exactly quadratic and the periodic parabola is strictly
harmonic on the fundamental domain. A cosine potential
of the same curvature is not DFD’s potential: it would
multiply the relic by an anharmonic factor of about 1.6
(Ωχ h2 = 0.187–0.192; computed). The residual spread
of the map over the choice of g∗s,0 is 0.1193–0.1202 (an
independent rebuild’s central value reaching 0.1200).

7. Step 5b — the relic amplitude is the finite
CS/WZW vacuum expectation (no free angle)

The single free input of Theorem AV.11 — the homogeneous angle θi — is not actually available to χ, and its
correct replacement is not a chosen number but a vacuum
expectation on a finite topological Hilbert space. Two
facts combine. First, DFD has no slow-roll inflaton, so
no homogeneous θi is selected and stretched across the
sky. Second, χ is not a classical continuous angle: it is
the compact harmonic C3 flux mode of the S 3 Chern–
Simons/WZW sector, whose physical states form the
finite integrable Hilbert space of SU (2)k at the locked
level k = 60 (dim = k + 1 = 61). The relic amplitude is
therefore the vacuum expectation of the quadratic flux operator on that finite space — a number fixed by topology,
with no free angle.
Theorem AV.13 (Finite SU (2)60 CS-vacuum relic amplitude (no chosen angle)). (No inflaton, no chosen angle.)
DFD has no slow-roll inflaton (Thm FK.2); the decay
constant fχ = M̄P α3 (Lemma
AV.5) and the de Sitter
√
dictionary’s rung H⋆ = 8π MR , MR = MP α3 , which is
not the expansion rate of any DFD
√ stage (Theorem SM.3),
are the same α3 rung (MP = 8π M̄P ), so
H⋆ =

√

8π MP α3 = 8π fχ = 2.38 × 1013 GeV,

H⋆
= 8π
fχ

(AV10)
(the two 8π are physically distinct — the reduced/nonreduced Planck conversion and the H⋆ /MR normalization
— so H⋆ /fχ = 8π is a genuine product, not a doublecounted Planck factor). With no inflaton no homogeneous
θi is selected; θi is not a knob.
(The amplitude is the finite-vacuum Casimir expectation.) As the compact S 3 CS/WZW flux mode quantized
as SU (2)k at k = 60 (K ≡ k + 2 = 62), χ’s dimensionless relic amplitude is the modular-weighted vacuum
expectation of the normalized quadratic Casimir:
√

k/2
X

C2 (j)
2
(2j + 1)π
, |S0j |2 =
sin2
,
k(k + 2)
K
K
j=0
X
C2 (j) = j(j + 1),
|S0j |2 = 1.

⟨θχ2 ⟩ =

|S0j |2

j

(AV11)
At k = 60,
⟨θχ2 ⟩ = 0.072957,

√
Ωχ h2 = 1.62 κrad 2π ⟨θχ2 ⟩ = 0.11928

(−0.60σ from Planck 0.1200 ± 0.0012; Rem. AV.12).
(AV12)
The amplitude (AV11) is a variance in the period-1
(turns) flux coordinate (Lemma AV.6); its own continuum
1
limit is 12
− 8π1 2 = 0.070668; the amplitude lies 3.2% above
it through the Sugawara factor K/k, while finiteness itself
lowers it by 0.09% at fixed (k + 2)2 . The classical value
π 2 /3 = 3.29 is the uniform variance of a period-2π angle
— the wrong unit for this coordinate, not its continuum
limit (Rem. AV.14).

321
Proof. Eq. (AV10) is arithmetic
√ on the locked rungs
(fχ , MR share α3 ; MP /M̄P = 8π). With no inflaton
there is no chosen homogeneous θi , so the amplitude is
a property of the vacuum, not an initial condition. Two
ingredients fix that vacuum value.
(i) Operator identity — the relic θ2 is the Casimir.
On the finite Hilbert space the flux sectors are the integrable representations j = 0, 12 , . . . , k2 . The unique SU (2)invariant quadratic operator is the Casimir C2 (j) =
j(j + 1) — diagonal on the irreps, equal to the group
Laplacian. The naive geodesic-angle-squared ψ 2 is not
admissible as the relic operator: it is basepoint-dependent
(not class-invariant), non-smooth at the S 3 cut locus, and
so state-dependent as to be meaningless (its expectation
ranges from ⟨ψ 2 ⟩ ≈ 5.3 on the flat group average down
to ≈ 0.01 on a localized state). The invariant, smooth
quadratic operator — hence the physical θ2 — is the
Casimir.
(ii) Normalization — k(k + 2) follows from two stated
conventions. The quantum-corrected quadratic flux energy is the Sugawara weight hj = C2 (j)/(k + 2), with
k + 2 = k + h∨ (h∨ = 2, the SU (2) dual Coxeter number). The dimensionless amplitude is this weight per bare
Chern–Simons
level k (the CS action is level-k, SCS =
R
k
2 3
2
Tr(A
dA
+
4π
3 A )), so θj = hj /k = C2 (j)/[k(k + 2)].
The denominator is fixed by an algebraic lock:
(k/2)(k/2 + 1)
1
C2 (k/2)
=
=
exactly, for all k,
k(k + 2)
k(k + 2)
4
(AV13)
placing the top sector at θmax = 12 — the Z2 -folded pseudoscalar fundamental-domain edge — independently of k.
Among {k 2 , (k+2)2 , k(k+1), (k+1)2 , k(k+2)} only k(k+2)
gives a k-independent edge (e.g. C2 (k/2)/k 2 = (k + 2)/4k
drifts); k 2 drops the Sugawara shift and (k + 2)2 doublecounts it. Summing (AV11) over the 61 states at k = 60
(⟨C2 ⟩ = 271.40, ⟨C2 ⟩/[k(k + 2)] = 271.40/3720) gives
⟨θχ2 ⟩ = 0.0729566, hence (AV12). The many-patch / fluctuation alternative δθ = H⋆ /(2πfχ ) = 4 (per e-fold; the
2π is Gibbons–Hawking, identical in both normalizations)
would wash the modular vacuum to the uniform measure
— at best the finite-space ceiling ⟨Q⟩∞ = 0.0853 — which
is the wrong state preparation for the finite mode and is
what (AV11) corrects. (The historical π 2 /3 washout value
was additionally a radian-units reading of the period-1
coordinate; Rem. AV.14.)
Remark AV.14 (Proven finite-CS operator; conditional
abundance closure; why there is no overshoot). What this
fixes. The classical continuum measure gives a factor-≈ 44
overshoot, Ωχ h2 ≃ 1.62 · π 2 /3 ≈ 5.3, and the 45× gap
between π 2 /3 and QCS factorizes exactly as
π 2 /3
= 45.0935
QCS
=
|39.4784
{z }

×

4π 2 (pure units)

|1.179220
{z }

continuum measure choice

× |1.000921
{z } × |0.967742
{z } ,
finiteness itself

Sugawara k/K

(AV14)

i.e. 96.5% of it (in log measure) is the radians2 →turns2
unit conversion of the period-1 flux coordinate
(Lemma AV.6), a further 1.179 is the uniform-vs-sin2
continuum measure choice, and the finite-Hilbert-space
effect itself is 0.09% and lowers QCS at fixed (k + 2)2 ; QCS
lies 3.2% above its continuum limit through the Sugawara
factor K/k, net of that 0.09%. The correct Hk measure
(AV11), through the exact relic map (Rem. AV.12), gives
Ωχ h2 = 0.11928 (−0.60σ from Planck). The operator
identity (the relic θ2 is the Casimir, not the non-invariant
ψ 2 ) and the k(k + 2) algebraic lock (AV13) are forced in
the proof, so the amplitude ⟨θ2 ⟩ = 0.073 is a topological
deliverable, not a fit. There is no θi knob (no inflaton).
What is proven and what is stated. The operator identity — the relic θ2 is the Casimir, not ψ 2 — is proven: ψ 2
is provably non-invariant (its Haar-character expansion
has nonzero coefficients for every j ≥ 12 , so it mixes irreps
and is no legal operator on Hk ) and is phenomenologically excluded (it overshoots by 35–150×). The remaining ingredients rest on already-derived properties of χ,
standard cosmology and√two stated conventions: (a) the
relic prefactor 1.62 κrad 2π = 1.635 in (AV12) is a separate cosmological bridge — it carries the radiation-era
H(T )/entropy content of the relic map (AV7) (at the
exact onset, Rem. AV.12), not pure topology; the puretopology deliverable is ⟨θ2 ⟩ = 0.073, and the closure is
(topological amplitude)×(this O(1) relic bridge), not a single self-contained derivation. (This prefactor is operatoragnostic: it is the redshift packaging of ρχ = 12 m2χ fχ2 ⟨Qχ ⟩
at the forced onset 3H(Tosc ) = mχ , identical for any dimensionless quadratic amplitude Qχ — so applying it to
the flux Casimir Qχ = C2 /[k(k + 2)] is consistent, not
a mixing of the flux and holonomy-angle pictures; the
angle ψ 2 is the conjugate-dual description, not a second
admissible operator on the same mode.) (b) the normalization k(k + 2) follows from two stated conventions:
k + 2 = k + h∨ is the standard Sugawara shift, the bare
level is k, and the edge θmax = 12 — which the algebraic
lock (AV13) shows is the unique k-independent edge —
is fixed by χ’s derived Z2 (χ → −χ) orientation-oddness,
the same Z2 that yields NDW = 1 (Rem. AV.16). The
edge rests on this derived property; that the top sector
sits on it is the first stated convention. (c) K = k +2 = 62
is the Sugawara shift (k = 60 derived + h∨ = 2), the
second stated
— normalization does not even
Pconvention
K−1
bear on it ( n=1 sin2 (πn/K) = K/2 for every integer
K).
We therefore grade the topological operator and amplitude as proven/forced, while the physical relic abundance
remains forced-amplitude/stated-convention in the
corpus ledger: ⟨θ2 ⟩ = 0.073 is forced — measure conditional on the asserted premise P4′ (Lemma AV.27), proven
operator, normalization from two stated conventions —
and the abundance Ωχ h2 = 0.11928 (−0.60σ from Planck)
follows through the standard cosmological
√ relic-redshift
map (the operator-agnostic 1.62 κrad 2π, the identical H(T )/entropy factor that every dark-matter relic —
ΛCDM’s included — carries, exact at onset, Rem. AV.12).

322
The single non-DFD input is thus standard cosmology, not
a DFD convention. (One optional refinement, not a gap in
the theorem: deriving that standard relic-redshift map natively from the impedance/Friedmann branch, App. AU,
would make the chain end-to-end DFD.)
Remark AV.15 (The thermal-onset overshoot analysis
does not apply). Treating χ as a classical misalignment
relic with a temperature-dependent (instanton) mass turning on at Λ = M̄P α8 ≈ 20 GeV makes the π 2 /3 estimate worse (Ωχ h2 ∼ 103 –104 ) rather than better. That
treatment does not apply (Theorem AV.13): the relic
amplitude is the topological finite-vacuum expectation
(AV11), set by the fixed S 3 CS/WZW spectral curvature — a geometric, temperature-independent scale —
not by a patch-averaged classical angle evolved through
a thermal oscillation onset. There is consequently no
Tosc /g⋆ /Kibble-patch dependence in the abundance; the
only residual cosmological input is the O(1) relic-map prefactor of caveat (a) in Rem. AV.14. Caveat: the ∼ 103 –104
“onset makes it worse” number holds within a thermal misalignment picture; it does not apply to the finite-vacuum
derivation, which carries no such onset.
Remark AV.16 (Two consequences that are not free: an
isocurvature pass and a string/wall network). (i) Zero
CDM isocurvature, for the opposite reason. The χ amplitude is the topological finite-CS-vacuum expectation
(Thm AV.13), and DFD has no inflaton (Thm FK.2), so
there is no inflaton-seeded super-horizon mode to seed
isocurvature; χ predicts essentially zero CDM isocurvature and passes βiso < 0.038 — by the absence of inflation,
not by a tuned-small inflationary fluctuation. If the optical engine inflates the optical metric at a low scale
(App. Q 6), the super-horizon variance it leaves enters the
density quadratically, ∆2iso ≃ 4 ln(k/k0 ) (Pθ /QCS )2 with
Pθ = rAs /(8α6 ) (Eq. (SM6); k0 today’s horizon scale),
and βiso < 0.038 holds for r ≲ 10−10 (estimate), so the
isocurvature pass and the B-mode null are one prediction.
The local sector label adds white-noise isocurvature of
at most ∆2iso ≃ 8 × 10−43 at k = 0.05 Mpc−1 (computed;
Rem. AV.30).
(ii) Domain walls (NDW = 1), but no global strings.
Once the potential of Lemma AV.6 switches on, its minima
define domain walls. The potential
has a single minimum
R
per period (the unit winding S 3 ω3 = 1 makes the period
one turn), so NDW = 1. This is the safe case: with a single
vacuum per period the walls are unstable and collapse
before they can dominate (NDW > 1 would over-close the
Universe with stable walls). The parameter-free NDW = 1
remains a sharp, falsifiable structural prediction of the
S 3 origin of χ. It does not depend on the photon vertex
of Theorem AV.35.
We do not claim a global cosmic-string network. A
global string requires a spontaneously-broken U (1) with a
winding complex order parameter whose modulus |Φ| → 0
at the string core. DFD’s χ is the real coefficient of the
single harmonic
three-form (Theorem AV.2) whose norR
malization S 3 ω3 = 1 is a topological constant fixed by
a per-mode determinant (Lemma AV.5), not a complex

amplitude that can relax to zero — there is no string
core, hence no global string. (The string-tension estimate
Gµ = (fχ /M̄P )2 = α6 imports a vacuum-expectationvalue picture χ does not realize and is not a DFD prediction; the wall self-destruction above does not rely on
string-bounding.)

8.

Step 5c — the topological flux-density form

Step 5b derived the relic amplitude as a finite-vacuum
expectation. We restate it in the form that makes the
abundance and the clustering manifestly independent variables, and then answer the standard objections. Everything turns on one identification: χ is neither a classical
raw-angle misalignment condensate nor a later-produced
gas with an arbitrary particle number — it is a local
topological flux-density field whose internal fiber is the
finite SU (2)k CS/WZW Hilbert space.
Theorem AV.17 (Topological flux-density relic). The
local χ Hamiltonian density is
Hχ (x) = 12 m2χ fχ2 QCS [1 + δχ (x)] + O(vχ2 ),
E
D C
(AV15)
2
,
QCS =
3
k(k + 2) S CS vac
where QCS is the fixed internal quadratic flux-density
coefficient (AV11) and δχ (x) is the spacetime density perturbation of the cold flux fluid. Hence the mean abundance at onset, ρ̄χ (aosc ) = 12 m2χ fχ2 QCS , is fixed by the
internal finite-CS expectation (it redshifts as ρ̄χ ∝ a−3
thereafter, Eq. (AV26)), while clustering is carried by
ρχ (x) = ρ̄χ [1 + δχ (x)]. There is no independent background displacement θi and no independent production
number nχ : those knobs exist only for a classical rawangle condensate or a later-produced gas, neither of which
is the DFD χ sector.
Proof. The S 3 CS/WZW compactification supplies a finite internal Hilbert space of flux sectors j = 0, 12 , . . . , k2 .
The unique quadratic SU (2)-invariant is the Casimir
C2 (j) = j(j + 1); the quantum-corrected quadratic flux
energy is the Sugawara weight hj = C2 (j)/(k + 2); and
since the level-k action has quadratic stiffness k, the
squared dimensionless flux-density amplitude is Qj =
hj /k = C2 (j)/[k(k + 2)] (Step 5b). Its finite-vacuum
expectation
the modular measure pj = |S0j |2 is
P over
2
QCS = j |S0j | C2 (j)/[k(k + 2)] = 0.0729566 at k = 60
(AV12). With n = 2j + 1 and K = k + 2 the sum admits
the closed form
QCS =



(K − 1)K(2K − 1)
K
K
1
K2
−
+
−
,
2
2kK 2
12
4
2
4 sin (π/K)

(AV16)
which evaluates to 0.0729565815 at k = 60, K = 62 (agreeing with the direct sum to all printed digits) — so QCS
is an exact analytic number, not a numerical truncation. The spacetime dependence enters through δχ (x),
not through a new background value of QCS ; so the same

323
sector carries a fixed abundance coefficient and still clusters.
9.

Parity degeneracy of the CS measure, and the
vector count

Theorem AV.18 (Exact parity halving of the modular
measure). For every even K = k + 2 ≥ 6, the modular
measure pj = |S0j |2 of SU(2)k splits exactly 1/2 : 1/2
between the integer-spin and half-integer-spin classes:
X
X
|S0j |2 =
|S0j |2 = 12 .
(AV17)
j∈Z

j∈Z+ 21

The Casimir-weighted asymmetry is not zero but is quartically suppressed,
h
i
Qhalf-int j
6π 4
δ(K) = 1 −
1 + O(K −2 ) ,
=
2
4
Qint j
(2π − 3) K
(AV18)
with limiting constant 6π 4 /(2π 2 −3) = 34.91530293402780
and δ(62) = +2.369 × 10−6 , the integer-j class being the
larger.
Proof. By the character sum: |S0j |2 = (2/K) sin2 (πn/K)
P
2
with n = 2j + 1, and
=
n odd sin (πn/K)
P
2
sin
(πn/K)
=
K/4
for
even
K
by
the
stann even
dard Gauss sum. Residuals verified below 10−51 at
K = 6, 8, 12, 22, 62, 102, 1002. The Casimir weighting
breaks the exact halving only at O(K −4 ), giving the
stated closed form; the constant is confirmed by C(K) =
35.1162 (K=42) → 35.0073 (62) → 34.94927 (102) →
34.915337 (3202).
Remark AV.19 (Two mandatory statements of proof).
The halving must be proved by the character sum above
and never by the map (2j + 1) 7→ K − (2j + 1): for
even K that map preserves parity, and at k = 60 the
two classes hold 31 and 30 members respectively, so no
bijection between them exists. The constant 34.9153 . . .
is an asymptotic limit with the closed form given, not a
fixed numerical constant.
Theorem AV.20 (Massless vector count from the internal S 3 ). The S 3 coexact 1-form sector at tachyon-free
mass parameter (µ ≤ 2) delivers exactly N ∈ {0, 3} massless four-dimensional vectors. The achievable counts over
all µ are {n(n + 2)} = {3, 8, 15, 24, . . .}, so N = 6 is
kinematically unreachable, and N = 3 holds if and only if
µ = 2 — a value which is simultaneously the dual Coxeter
number h∨ (SU(2)), the adjoint Casimir C2 (adj SU(2)),
and λmin |⋆d| on S 3 . Independently, b2 (CP 2 × S 3 ) = 1
gives exactly one four-dimensional U (1) from C3 and zero
Kalb–Ramond axions.
Remark AV.21 (Right-handed W mass and convention).
The associated M (WR ) = 2.95 × 1018 GeV using the full
Planck mass (5.88 × 1017 GeV in the reduced convention),
giving (mW /MWR )2 = 7.4 × 10−34 . The conclusion is
convention-independent: 30–32 orders of magnitude below
the experimental bound either way.

Remark AV.22 (What is not claimed here). The gaugeboson budget 8 + 3 + 1 = 12 is not booked. Iso(S 3 ) =
SO(4) supplies six metric Kaluza–Klein graviphotons that
the argument above does not dispose of, so the printed
budget remains 18. Closing this requires showing that
DFD’s action carries no dynamical internal metric over
K = CP 2 × S 3 , or otherwise disposing of the six graviphotons separately. This is recorded as open.
a. Objections, answered. The finite-CS flux-density
identification disposes of the standard objections:
O1. “0.073 was tuned.”:
No:
it is the finite-vacuum expectation
P
2
|S
|
0j C2 (j)/[k(k + 2)] at the already-fixed
j
level k = 60. Level, measure, operator, and
normalization are all fixed before the abundance is
evaluated.
O2. “The WZW object is C2 /(k + 2), not C2 /[k(k + 2)].”:
Correct, and C2 /(k +2) = hj is the Sugawara energy;
the relic map needs the squared amplitude, which is
energy over stiffness, hj /k = C2 /[k(k+2)]. The extra
1/k is the level-k stiffness conversion — k 2 would
drop the quantum shift, (k + 2)2 would double-count
it.
O3. “A CS vacuum expectation is vacuum energy, so
w = −1.”:
No: a local vacuum-energy term is volume-extensive
(ρ =const, w = −1). QCS is not a volumeextensive local density; it is a finite topological statepreparation measure over flux sectors. The massive
flux-density content has fixed comoving energy, so
ρχ ∝ a−3 , w = 0.
O4. “A single global flux number cannot cluster.”:
DFD uses no smooth global number: (AV15) fixes
the local coefficient QCS while δχ (x) carries clustering. Abundance and perturbations are different
variables.
O5. “Inflation freezes a real-space amplitude into a random displacement.”:
That applies to an ordinary local spectator scalar
with a continuous raw coordinate; restoring the
random-angle branch needs an added DFD-native
map (finite CS flux sector)→(continuous raw angle
uniformly randomized on [−π, π]). An appended external inflationary stage is not harmless: it kicks the
flux coordinate by H/(2πfχ ) per e-fold, and the relic
tolerates it only below H ≃ 4×1010 GeV (channel (a)
below; Eq. (Q13)); any stage that left a detectable
tensor signal fails that test (App. SM, Cor. SM.4).
O6. “A gas of quanta has a free production number.”:
χ is a pre-existing compact flux-density mode, not
a later-produced gas: QCS in (AV15) is the fixed
internal quadratic coefficient of the local fluid, not
the energy of one quantum needing an occupation
number; δχ (x) controls clustering, not the mean
abundance.

324
O7. “1.62 belongs to a misalignment angle, so pairing it
with a flux
√ Casimir is inconsistent.”:
1.62 κrad 2π (Rem. AV.12) is the operator-agnostic
cosmological relic-redshift factor for a unit dimensionless quadratic amplitude in the massive χ mode
(ρχ = 12 m2χ fχ2 Q at 3H(Tosc ) = mχ , with fχ = P the
period and Q in turns2 — matching normalizations,
Lemma AV.6): Q = θi2 in the raw-angle branch,
Q = QCS in the flux-density branch. Applying the
identical H(T )/entropy map (the one every darkmatter relic, ΛCDM included, carries) to the forced
QCS is self-consistent.
b. State preparation: χ cannot be Kibble-randomized.
The would-be random-angle branch ⟨θ2 ⟩ = π 2 /3 is not
a competing DFD calculation; it is a statement about a
different object.
Theorem AV.23 (Finite-CS state-preparation). The
DFD-native initial state of χ is the finite SU (2)60
CS/WZW vacuum measure pj = |S0j |2 (given the asserted
premise P4′ of Lemma AV.27), not a Kibble-randomized
continuous-angle ensemble. P
Consequently the relic coefficient is fixed, Qχ = QCS = j |S0j |2 C2 (j)/[k(k + 2)] =
0.0729566, with no free initial angle θi and no reopening
of the (π 2 /3) overshoot.
Proof. The Kibble value ⟨θ2 ⟩ = π 2 /3 is the output of
a specific classicalization mechanism, not a neutral default. It requires, jointly: (i) a local continuous order
parameter θ(x) with a meltable radial partner |Φ|; (ii)
a high-temperature symmetry-restored phase in which
|Φ| → 0; (iii) a later symmetry-breaking transition; and
(iv) causally disconnected domains that reselect independent classical
whence θi ∼ Uniform[−π, π]
R πangles,
1
2
and ⟨θi2 ⟩ = 2π
=
π 2 /3. DFD’s χ has none
θ
dθ
−π
of (i)–(iv). It is the coefficient of the unique harmonic
three-form ω3 on S 3 R= SU (2) (Theorem AV.2, b3 = 1),
whose normalization S 3 ω3 = 1 is a topological invariant
fixed by the per-mode determinant, not a VEV; there is
no complex amplitude |Φ| that can relax to zero, hence
no string core (Rem. AV.16). Its physical state space
is therefore not S 1 with a uniform classical-angle measure but the finite CS/WZW Hilbert space HSU (2)60
of 61 integrable sectors j = 0, 12 , . . . , 30. The statepreparation measure on this space is the no-boundary
3
closed-S
Chern–Simons Born weight pj = |S0j |2 , S0j =
p
P
2/(k + 2) sin[(2j + 1)π/(k + 2)], j |S0j |2 = 1 — derived (not asserted) by S 3 surgery in the two paragraphs
following this proof, which also exclude the only wrongvalued rival (the linear-dj Cardy weight). Pairing it with
the unique quadratic invariant Qj = C2 (j)/[k(k + 2)]
(Theorem AV.17) gives QCS = 0.0729566. To convert this
into the π 2 /3 ensemble one must add a non-DFD map
(finite CS flux sector)→(continuous raw angle)→(Kibble
domains); DFD’s fixed S 3 geometry supplies no such
map. The free angle is removed not by naming but by
quantizing the correct object.

c. Brick 1 — the measure is the closed-S 3 surgery
vacuum, not a name. The weight pj = |S0j |2 is not
posited; it is the Born amplitude of the closed-S 3 noboundary Chern–Simons state, computed by standard
Witten–Reshetikhin–Turaev surgery and wired to the
S 3 partition function this manuscript already uses elsewhere. Present the closed internal S 3 = SU (2) as two
solid tori glued by the modular generator S (genus-one
Heegaard split). A solid torus with empty core (no
Wilson line threaded) prepares the no-boundary Hartle–
Hawking state |0⟩ in the boundary-torus Hilbert space
k/2
HT 2 = span{|j⟩}j=0 ; this is DFD’s own cosmogenic
Wheeler–DeWitt no-boundary state (the DFD primordial no-boundary/Wheeler–DeWitt selecting condition on
ψ, App. Q, Rem. Q.11), and there is no native inflaton
(Thm. FK.2) nor sustained de Sitter stage (Thm. SM.2)
to re-prepare it. Gluing the second solid torus by S gives
q
π
2
ZS 3 = ⟨0|S|0⟩ = S00 = k+2
sin
,
(AV19)
k+2
identically the S 3 Chern–Simons partition function used
for the α57 closure (App. AP, Lem. AP.22). InP
the gluedtorus basis the no-boundary state is |0⟩ =
j S0j |j⟩,
and because S is unitary the Born weights are selfnormalized
with no ad hoc division, pj = |S0j |2 with
P
†
j pj = (SS )00 = 1. The measure is thus derived as
the closed-three-manifold no-boundary amplitude built
from the very ZS 3 = S00 that fixes the α-closure; only its
j-resolved row is read off here.
d. Brick 2 — the linear-dj (Cardy) weight is excluded
by the absence of a boundary. The only measure that
would reproduce the high value Q = 0.0767 (Ωχ h2 =
0.1254 under the exact map, Rem. AV.12) is the linear
P
quantum-dimension weight pCardy
= dj / j ′ dj ′ , dj =
j
S0j /S00 . This is not a competing vacuum on the closed S 3 :
it is the diagonal of the Cardy trivial-brane boundary state,
whose annulus partition function produces a weight linear
in dj only on a manifold with boundary. The decisive
algebra is that the genuine (squared, true-probability)
quantum-dimension measure collapses exactly back onto
the no-boundary value,
 S 2
X
d2j
1
0j
2
2
2
=
S
=
|S
|
,
D
=
d2j = 2 ,
0j
00
D2
S00
S
00
j
(AV20)
so the only quantum-dimension weight that can live on
a closed manifold simply reproduces pj = |S0j |2 ; only
the un-squared, linear dj deviates, and a linear, nonamplitude-squared weight is an open-string/boundary
trace, not a closed-manifold quantum probability. The χ
relic is the coefficient of the unique harmonic three-form
on the closed internal S 3 = SU (2) (b3 = 1, ∂S 3 = ∅;
Thm. AV.2); it carries no brane, defect, horizon, or Wilson line (changing j costs at least the Sugawara gap,
2.95 × 1016 GeV; Rem. AV.29). With no internal boundary there is no annulus on which a g-function can be
defined, so pCardy
∝ dj has no manifold to live on; a
j
horizon or observable-universe boundary lives in exter-

325
nal spacetime, not on the internal S 3 worldvolume, and
cannot license it. To force the linear-dj value one must
append a non-DFD boundary the fixed S 3 geometry does
not contain — the exact mirror of the π 2 /3 Kibble nogo. The closed-S 3 no-boundary vacuum, equal to the
Plancherel row d2j /D2 , is therefore the unique admissible measure. (Conditionality: this selection holds given
DFD’s no-boundary Wheeler–DeWitt condition (App. Q,
Rem. Q.11) — a central DFD postulate shared with all of
its cosmology, not an independently derived fact. Granting
it, the linear dj is excluded, and with the asserted premise
P4′ (Lemma AV.27) |S0j |2 is forced.)
e. Brick 3 — conditional uniqueness of the abundance
measure (all rivals dispatched).

Remark AV.26 (Anchoring dichotomy — the status of
P4). P4 is operator-selected, not derived. The written
abundance operator Q̂ has 61 distinct eigenvalues (C2 (j)
strictly increasing), whereas the preparation-core observable is deterministic on |0⟩ with eigenvalue C2 (0) = 0: the
written operator therefore provably reads the glued core,
and the only rival reading predicts zero dark matter. But
S 3 is simply connected — both Heegaard Rcycles bound
discs — and no written decomposition of S 3 ω3 selects
one cycle over the other, so P4 remains the single asserted
premise of this theorem. Deriving it (a Heegaard decomposition of the χ flux jump) would make the uniqueness
unconditional given P1–P3.

Theorem AV.24 (Conditional uniqueness of the sector measure). Assume (P1) the closed-S 3 no-boundary
CS state prepared by empty-core WRT surgery (Brick 1
above); (P2) the RT/TQFT axioms for SU (2)60 with the
two-solid-tori Heegaard preparation, including the Sugawara identity hj = C2 (j)/(k + 2); (P3) the Born weighting of sector amplitudes (derived elsewhere in DFD from
additivity and no-signaling); and (P4, cycle-anchoring,
asserted by construction) that the physical χ winding —
the flux jump costing the 2π/α instanton — is carried by
the glued-torus core cycle of the Heegaard splitting. Then
the sector measure is uniquely pj = |S0j |2 , the abundance
P
operator is uniquely Q̂ = j [C2 (j)/k(k + 2)]Pj .

Lemma AV.27 (Exhaustion: sector-changing processes
are glued-core processes). On the 61-dimensional torus
Hilbert space of SU (2)60 , the Wilson-loop algebras of the
two Heegaard core cycles jointly generate the full operator
algebra End(C61 ): the preparation-core loops act diagonally with 61 distinct Verlinde eigenvalues Sqj /S0j (for
q = 12 they are the distinct values 2 cos(π(2j + 1)/62)),
and the spin- 12 glued-core loop acts as the fusion shift
connecting all sectors; the joint commutant is therefore
scalar (Verlinde 1988; Witten 1989 §4; Elitzur–Moore–
Schwimmer–Seiberg 1989). Consequently any operator
that changes the sector label j necessarily contains gluedcore insertions — preparation-core insertions are exactly
j-inert (diagonal). Cycle-anchoring thus follows automatically from the single sharpened premise:

Proof sketch. State uniqueness: every genus-1 gluing presentation of S 3 is of the form ±T a ST b ; since T is diagonalunitary, the Born weights |⟨j|T a ST b |0⟩|2 = |S0j |2 are
presentation-independent, and by Waldhausen’s theorem
(the genus-1 Heegaard splitting of S 3 is unique up to
isotopy) the no-boundary state is unique up to phase. Orbit completeness: the admissible cut states form exactly
the stabilizer orbit {T m ST n |0⟩}, all yielding the identical
measure; every other candidate is excluded structurally
— other S-columns require Wilson-line insertions (violating no-boundary emptiness), and longer SL(2, Z) words
present lens spaces, not S 3 . Fusion stationarity: the Verm
linde fusion kernel Ta (j → m) = Naj
dm /(da dj ) is exactly
stochastic and leaves pj = |S0j |2 stationary for every a
(detailed balance), so fusion processes preserve the same
measure.
Remark AV.25 (Rival table). Each rival weight fails a
named premise: linear-dj (Cardy) — no boundary to live
on (Brick 2); knotted-cut weights pj ∝ |S0j |2 |Jj (K)|2 —
excluded by the stated two-solid-tori preparation (P2);
thermal e−βEj — no Hamiltonian evolution on the closed
preparation; djp for p ̸= 2 — not an amplitude-squared
state weight (and p = 1 is Brick 2); the meridian-dual
reading δj0 — the Born weight of the same state in the
preparation basis, corresponding to an abundance operator that appears nowhere in DFD and predicts Ωχ h2 = 0
identically, excluded by any nonzero dark-matter observation.

P4′ : the 2π/α winding instanton is a
sector-changing (fusion) process.
P4′ is the part of P4 that remains asserted; it is invoked (not derived) only for cycle-anchoring (Theorem AV.24). It cannot refer to the unit large gauge
winding of Lemma AV.6(i), which multiplies every Chern–
Simons state by e2πikw = 1 and so changes no sector: P4′
asserts that a physical instanton of action 2π/α changes
j through glued-core insertions, which this lemma permits
but does not show, and no protection of the sector label
rests on it (Rem. AV.29). A standing obstruction to any
first-order derivation of P4′ is the symplectic factorization
δχ ∧ δj ≡ 0; the live route is second-order (the instanton’s
on-shell action evaluated between neighboring flux vacua).
Note the construction here is exactly symmetric under
S-conjugation, so this lemma supplies consistency and
exhaustion, not evidence for P4′ over its dual.
Two dynamical channels confirm the flux state is nondecoherable, not only the default at t = 0:
(a) No inflaton epoch (decoupling). Channel (iv) needs
many super-horizon e-folds for a stochastic angle to
random-walk to π 2 /3. DFD has no inflaton (Thm. FK.2)
and no sustained de Sitter stage (Thm. SM.2): there is
no epoch in which the walk runs. The rung-locked scale
H⋆ = 8πfχ gives H⋆ /(2πfχ ) = 4 exactly, so even a single e-fold of a hypothetical de Sitter stage would inject
2
σinf
∼ 16 — not small (this is the Inflation-Dictionary

326
Exclusion Theorem SM.3: a slow-roll reading of As , r
forces Hinf = 8πfχ , δθ = 4 per e-fold on a period-1 coordinate, hence immediate washout of the modular vacuum
to the uniform ceiling Ωχ h2 = 0.139, excluded by Planck
— so the slow-roll branch and the relic cannot coexist).
The relic survives because that stage does not exist, not
because its kick would be small:
 H 2
⋆
2
σinf
≃ Neff
= 16 Neff
2πfχ


if an inflaton epoch existed; Neff = 0 in DFD .
(AV21)
(b) The finite Hilbert space caps the cold relic at every
temperature. The relic is robust because a large overshoot is unreachable from the finite CS Hilbert space
at any temperature, independent of the gap magnitude. Thermalizing the 61-state space can at most
drive it to the infinite-T
P uniform measure pj = 1/61,
1
for which ⟨Q⟩∞ = 61
j C2 (j)/[k(k + 2)] = 0.0853, i.e.
√
2
Ωχ h ≤ 1.62 κrad 2π × 0.0853 = 0.139 — a 17% ceiling. (The “continuum catastrophe” π 2 /3 fails twice: it
is a radian-units reading of the period-1 flux coordinate,
Lemma AV.6, and in any case not in the spectrum of a
finite-dimensional flux Hilbert space.) Below that ceiling
the finite-CS vacuum is itself protected by the Sugawara
gap: every change of the sector label costs at least the
first flux excitation, Egap = h1/2 M̄P = 2.95 × 1016 GeV
(h1/2 = C2 ( 12 )/(k + 2) = 0.0121), far above H⋆ and
above every temperature of the radiation era, and the
modular measure is stationary under the fusion kernel
(Rem. AV.29). The protection is this gap, not a small
interaction rate. The light 4D quanta of χ are a separate question. The√plasma at the leptogenesis temperature TL = MP α3 / 2π = 1.89 × 1012 GeV that DFD’s
baryon asymmetry requires (App. AH) would produce
them through any photon or gauge coupling of χ, and
even the smallest nonzero coupling that the flux period allows would produce a fifth of their equilibrium abundance
or more and leave a hot relic, Ωχ,th h2 ≃ 0.005–0.023 on
top of the cold 0.1193, which the CMB reads as extra
cold dark matter and Planck excludes (computed). χ
has no such coupling: its photon coefficient is E = 0
(Theorem AV.35), so the plasma adds no thermal χ.
The general expression is therefore
√
2
Ωχ h2 = 1.62 κrad 2π [ QCS + σinf
] + Ωχ,th h2 ,
2
σinf
= 0 (no inflaton epoch),

QCS = 0.0729566,

Ωχ,th h2 = 0 (E = 0),
(AV22)
so the cold relic is Ωχ h = 1.62 κrad 2π QCS = 0.11928,
and the thermal term, the hot relic of the plasma, vanishes because χ has no photon coupling. The Kibblerandomized branch is not the DFD default — it is an
added external state-preparation assumption that DFD’s
fixed S 3 geometry does not realise.
Residual and sharpest falsifier. Because the finite
Hilbert space caps the cold relic at 0.139 at any tem2

√

perature (channel (b)), a large overshoot of the cold relic
cannot be reopened by thermalizing the sector label. The
only way to recover a continuum random-angle value is
to append a continuous degree of freedom outside the CS
Hilbert space — a light radial partner of χ at fχ that a
super-scale bath could excite — which DFD’s fixed S 3
geometry forbids (no complex amplitude relaxing to zero,
Rem. AV.16); a fully independent no-go on such a partner
is the one place a determined adversary can still push.
The single sharpest falsifier is therefore external: should
any future DFD-native epoch ever supply sustained quasi–
de Sitter e-folds above H ≃ 4 × 1010 GeV (Eq. (Q13)),
channel (a) reopens — so survival is conditional on Theorem FK.2 remaining airtight and on the optical engine
staying below that scale. Such an epoch is observable: it
freezes tensor modes with Pt = 8α6 Pθ (Eq. (SM6)), so a
primordial B-mode detection at r ≳ 10−7 would mark it.
Lemma AV.28 (Preservation through the pre-BBN optical phase). Let the pre-BBN optical screen flow ψ0 (t)
(App. Q 6) act during the primordial era, and let Pj be
the projector onto the integrable CS sector j. The screen
ψ0 is a gauge-singlet spacetime field acting on propagation (ceff = ce−ψ0 ), whereas the j label the integrable
sectors of the internal SU (2)60 CS/WZW Hilbert space.
Every transition j → j ′ costs at least the Sugawara gap
Egap = h1/2 M̄P = 2.95 × 1016 GeV, so a matrix element between sectors oscillates at least 9.2 × 1023 times
faster than the χ mass oscillation and far faster than the
screen varies. The coupling the screen exerts is therefore
block-diagonal in j, up to terms suppressed by that ratio,
X
Pj ′ Hψχ Pj ≃ 0 (j ′ ̸= j),
Hcoupl (t) =
hj (t) Pj ,
j

(AV23)
so [Htot (t), Pj ] ≃ 0 and


d
Tr ρχ Pj ≃ 0 =⇒
dt

pj (t) = pj (ti ) = |S0j |2 .

(AV24)
The optical phase can stretch and freeze spacetime perturbations but cannot move probability between integrable
representations: it prepares spacetime coherence, it does
not thermalize the internal finite-CS label.
Remark AV.29 (Scope of the preservation lemma). Blockdiagonality (AV23) is not assumed; it rests on two protections of the sector label. The Sugawara gap: a transition
j → j ′ costs at least Egap = 2.95×1016 GeV, so a flow that
is slow on that scale drives none, and a thermal process
at the leptogenesis temperature supplies it with weight
10−6760 (computed). Fusion stationarity: the modular
measure |S0j |2 is stationary under every Verlinde fusion
kernel (proof of Theorem AV.24), so any fusion process
that does occur leaves pj unchanged. The winding instanton is not a protection: a unit winding of the flux acts on
the Chern–Simons state as e2πikw = 1 (Lemma AV.6(i))
and changes no sector. The Kibble-randomized branch
returns only if a non-DFD operator that mixes sectors at
rates comparable to Egap is added by hand (converting

327
the finite SU (2)60 sector into a continuous Kibble angle)
or if Htrue exceeds the erasure bound (Q13). Together
with the modular-vacuum preparation (Theorem AV.23),
this gives: the finite-CS abundance is not assumed; it is
prepared by the closed S 3 modular vacuum and preserved
through the pre-BBN optical phase by the Sugawara gap
and fusion stationarity.

Ωχ h2 = 0.1193 (fractional error below 10−24 ), and the
domain-to-domain spread adds white-noise isocurvature
of at most ∆2iso ≃ 8 × 10−43 at k = 0.05 Mpc−1 (computed). A global label would instead make the observable
universe one draw from pj , with a ±65% spread about
0.1193, and Ωχ h2 would be the value of a single rung
(j = 16 gives 0.1195) rather than an average.

Remark AV.30 (χ is a comoving-variance relic: not a
condensate,
not a constant variance). The sector state
P
ρχ = j |S0j |2 Pj is diagonal, and χ is a local field that
acts within each sector, where the state is even under
the orientation symmetry χ → −χ; so the one-point
function vanishes, ⟨χ⟩ = Tr(ρχ χ) = 0. This is not a
problem — it is the correct nature of the relic. The
conserved CS quantity is the dimensionless comoving
quadratic action, the expectation of the diagonal operator
b = P [C2 (j)/k(k + 2)] Pj ,
Q
j
X
C2 (j)
b =
⟨Q⟩
|S0j |2
= QCS = 0.0729566
k(k + 2)
(AV25)
j

Net.
With χ taken as DFD defines it — the
finite S 3 CS/WZW topological flux-density relic —
the abundance closes numerically at the ledger grade
forced-amplitude/stated-convention:√ mχ = 32.005 eV,
QCS = 0.0729566, Ωχ h2 = 1.62 κrad 2π QCS = 0.11928
(−0.60σ from Planck). The would-be 44× overshoot,
(π 2 /3)/QCS ≃ 45, was the cost of reading the period-1
flux coordinate in radians and quantizing the wrong object (a continuous random angle) (AV14), not a physical
catastrophe.

(conserved: pj fixed),
which fixes the physical field variance at oscillation onset aosc to ⟨χ2 ⟩phys (aosc ) = fχ2 QCS . It is not a constant local variance: a constant ⟨χ2 ⟩ would give ρ ≃
1 2
2
2 mχ ⟨χ ⟩ =const, i.e. w = −1. Rather, as for any massive
scalar Fock/mixed state, the comoving occupation ⟨N ⟩ is
conserved while the physical variance and energy dilute
with physical volume:
 a 3
osc
⟨χ2 ⟩phys (a) = fχ2 QCS
,
a
 a 3
osc
ρχ (a) = 12 m2χ fχ2 QCS
∝ a−3 ,
wχ = 0,
a
(AV26)
clustering through δχ (x). Thus χ is a finite-CS comovingvariance relic: ⟨χ⟩ = 0, ⟨χ2 ⟩phys ̸= 0 and redshifting as
cold matter — exactly as a cold Fock state has zero field
expectation but a real, diluting mass density. QCS therefore enters the abundance as a conserved comoving second
b not a classical displacement θ2 ; the relic
moment ⟨Q⟩,
i
√
2
map Ωχ h = 1.62 κrad 2π ⟨ b
Q⟩ = 1.635 QCS = 0.11928
is a variance map, not an axion-misalignment formula.
The coherent value π 2 /3 belongs to the other object — a
classical compact scalar with ⟨χ⟩ = θi ̸= 0, read in radians
— and does not apply here. DFD dark matter is therefore
neither a coherent misalignment condensate nor a constant vacuum variance: it is finite topological quadratic
energy whose conserved comoving action redshifts as cold
matter.
The sector label is local. χ is a local four-dimensional
field (Theorem AV.17), so its sector is set independently
in causally separate patches when the relic forms. The
domains are small: no larger than the Hubble patch at oscillation onset, 2 × 1010 m comoving today (∼ 10−26 M⊙
of χ), and about 1 km if the label is set at the leptogenesis temperature. On every observable scale the
realized abundance is therefore the domain average,

10.

Step 6 — the third peak, with no postulated
CDM

Theorem AV.31 (CMB third-peak height from χ). A
pressureless component of cold density Ωχ h2 = 0.12, clustering at recombination z ≃ 1100, reproduces the Planck
acoustic-peak height pattern [H1 , H2 , H3 ] ∝ [1, 0.45, 0.44]
(H3 /H2 ≈ 1.0). Since χ is cold, non-thermal, and pressureless on all CMB scales (Theorem AV.8), it enters the
photon-baryon perturbation kernel identically to a colddark-matter component of the same Ωh2 , and supplies
exactly this height.
Proof (CAMB-validated). A fixed-Ωb Boltzmann computation (CAMB) gives, for a baryon-only universe (Ωc h2 →
0), peaks at ℓ ≃ 294/771/1212 with [H1 , H2 , H3 ] ∝
[1, 0.39, 0.205] (H3 /H2 = 0.53); restoring a cold pressureless component Ωc h2 = 0.12 moves the peaks to
ℓ ≃ 220/536/813 and lifts the ratio to [1, 0.45, 0.44]
(H3 /H2 ≈ 1.0), matching Planck. The third-peak height
is set by matter–radiation equality and the driving of the
cold well, not by Silk damping (full Silk removal reaches
only H3 /H1 = 0.38 < 0.44). Because the χ field is dynamically identical to a cold pressureless fluid of the same Ωh2
at recombination, the CAMB result at the derived abundance is the χ result. Run at the derived ωχ = 0.11928
(Thm AV.13) on the measured clock (H0early = 67.52,
which fixes the peak positions ℓ ≃ 220/536/813), the
pipeline gives the height pattern H3 /H2 = 0.977 (0.0σ),
H3 /H1 = 0.444 (+0.5σ), H2 /H1 = 0.454 (+0.5σ) against
the Planck 2018 bandpowers; on the same clock the Planck
2018 high-ℓ (plik-lite) likelihood places the derived abundance at −0.98σ (App. ZS).
Remark AV.32 (Why no other DFD device delivers this).
Every non-χ route to the third peak is closed: deepMOND gravity enhancement is wrong-signed per mode
(it lifts H1 above H3 ); the nonlinear AQUAL harmonic
injection is negligible and lands in troughs; a line-of-sight

328
ψ-screen cannot move the peak locations because the
sound-horizon rod and angular-diameter distance transform together (App. ZS); the Jacobian lift is therefore
excluded as a global CMB mechanism; sound-speed structure cs (ψ) is time-only and cancels in height ratios; and
a single forced echo at the sound-horizon delay cannot
selectively boost the third peak without filling the troughs.
The cold clustering charge is irreducible — and χ is the
derived field that carries it.

11.

Step 6b — stability: χ has no photon coupling

Remark AV.33 (χ stability: E = 0 under DFD’s CP). χ
is stable because it has no photon coupling: its photon
coefficient is E = 0 (Theorem AV.35), so it has no twophoton decay and its lifetime is infinite. The selection
rule is DFD’s CP, which acts as complex conjugation
on CP 2 and as the identity on S 3 (App. L). Under it
χ is even and F F̃ is odd, so χF F̃ is odd and absent.
The operator χF F̃ is even only under a parity that also
reverses the orientation of S 3 (ω3 → −ω3 , χ → −χ); that
reflection is not DFD’s CP, and it cannot be made one:
the CP map of App. L must preserve the orientation
of CP 2 × S 3 , and CP 2 has no orientation-reversing selfmap
to compensate a reflection of S 3 . The reduction
R
ω = 1 ̸= 0 does let an orientation-odd internal threeS3 3
form descend to a nonzero 4D operator, so the absence
is a CP selection rule, not a cohomological one, and the
exact period independently excludes any small nonzero
coefficient (Theorem AV.35, route (b)).
Remark AV.34 (DFD’s CP forbids an axial χ coupling;
DFD’s baryogenesis does not use one). Because χ is even
under DFD’s CP (Rem. AV.33), the derivative coupling
∂µ χ J µ to any fermion current is CP-odd and absent, like
χF F̃ , so spontaneous baryogenesis through χ is closed.
DFD’s baryon asymmetry does not use it. Its driver is
the determinant-line connection A5 = Q d ln A, Q ∝ α, of
the internal axial Berry holonomy (App. AH, Extended
Derivations), whose temporal holonomy is the chemical
potential µB−L = A50 = QH; it contains no χ. A χsourced chemical potential would fall short by about 27
orders of magnitude, because the relic χ is frozen at TL
and starts to oscillate only at Tosc ≃ 8.7 × 104 GeV
(computed). The driver’s α4 power is derived, and so
is its bias current, B−L: a current with an axial quark
component would be QCD-anomalous and, coupled to
the never-ending A5 , would move θ̄ away from zero. Its
sign is fixed by an adopted orientation axiom. With the
stated carrier, which gives each generation a full B−L
turn per flux step, the mechanism gives ηB of about (2–
3)×10−9 , three to five times Planck’s value, and agreement
would need one winding of the determinant line per efold, where the stated carrier gives three (App. AH); the
baryon asymmetry is carried as one calibrated input, ωb
(App. FK).

12.

Step 7 — no photon coupling: E = 0

Theorem AV.35 (χ has no photon coupling). The photon coefficient of χ vanishes, E = 0: χ has no vertex
L ⊃ − 41 gχγ χ F F̃ (gχγ = Eα/P = 0) and no linear coupling to any Standard Model field. It neither decays nor
converts to photons, and it is stable. Any dark-matter
photon line at 16.0 eV (77.5 nm, two-photon decay) or
axion-like signal at the χ mass, 32.005 eV (vacuum wavelength 38.74 nm, conversion), at any strength, falsifies
χ.
Proof. Three routes give E =P
0. (a) Fermion loops. The
loop contribution is E = 2 f Xf df qf2 , with Xf the
charge of fermion f under the shift χ → χ + P . χ is a
separate sector, not a 60th internal mode (Rem. AV.3);
it carries no Peccei–Quinn charge (the BSM-forbidden
theorem of App. AH); and the saturated index 60 admits
no further heavy fermion. So Xf = 0 for all 48 Weyl
fields and the loop part vanishes. The one alternative
reading, χ as the Chern–Simons number of an SU (2) that
the Standard Model fermions see (the S 3 spectral flow
of Thm. F.14), gives in every weighting of the fermions
other than a hypercharge gauge rotation either gauge
couplings of order ten or a coupling of order 30–90 to the
right-handed-neutrino mass phase, and each of these thermalizes χ at the leptogenesis temperature (computed); so
the SU (2) whose Chern–Simons number is χ carries no
quark or lepton (Theorem AV.4). (b) Discrete coefficients
and the hot plasma. The period χ → χ+P is a gauge identification (Lemma AV.6), so on the Standard Model group
with its Z6 quotient (App. F) the anomaly coefficients are
discrete: E is an even integer if χ does not couple to gluons and |E| ≥ 2/3 if it does. The unit value E = 1 is not
allowed, and the photon Chern–Simons level E/2 on the
domain walls of Rem. AV.16 must likewise be an integer.
At the leptogenesis temperature TL = 1.89 × 1012 GeV
every smallest nonzero coupling allowed by a Z6 , Z3 or
Z2 quotient produces χ quanta at Γ/H = 0.26–965 and
leaves a hot relic Ωχ,th h2 = 0.005–0.023, 1.8–8.5 times
Planck’s limit 0.0027 (computed; the rates are estimates
good to about 20%). A group with no quotient would need
colour-singlet charges of e/6 or e/3, which DFD excludes.
Only E = 0 survives. (c) CP. DFD’s CP is complex
conjugation on CP 2 combined with the identity on S 3
(App. L, Cor. L.2); a reflection of S 3 is excluded, since the
map must preserve the orientation of CP 2 × S 3 for the
mapping torus to be oriented, and CP 2 has no orientationreversing self-map to compensate it. With that CP acting
on the real form C3 by pullback alone, with no sign of its
own, the Chern–Simons number χ, its integral over S 3 , is
therefore CP-even, and every shift-symmetric linear coupling (χF F̃ , χW W̃ , χGG̃, and ∂µ χ J µ for any fermion
current) is CP-odd. DFD’s CP violation sits in the CKM
phases and in the adopted orientation of the baryogenesis
transport (Axiom E-transport′ , Ext. Deriv. App. AH),
and χ has no linear coupling to any Standard Model field
for loops to dress (no shift charges, no mixing with ψ,

329
whose internal modes are forms of another degree, and an
even potential), so E is its tree-level value, zero. Routes
(b) and (c) are independent: E = 0 follows from the exact
period with DFD’s CP, given that χ’s period is an integral
period of C3 (Lemma AV.6 (i)), or from the exact period
with the Z6 quotient and a plasma that reaches TL , as
DFD’s baryogenesis requires. The step lattice, the CP
parity and the operator table are theorems; the couplings
and relic densities are computed.
Remark AV.36 (Experimental status: gravitational only).
With no photon coupling χ predicts a null result in every
haloscope, light-shining-through-walls, direct-detection
and indirect-detection experiment, and it produces no
conversion line, no decay photons and no dark-ages ionization floor. A photon line at 16.0 eV (77.5 nm) or
32.005 eV (38.74 nm) from dark matter, or a confirmed
halo-recoil or haloscope signal in any current-reach channel, at any strength, falsifies χ. The complementary,
mass-independent signatures are gravitational/clustering:
the third-peak scaffolding (Sec. AV 10) and the rotationcurve scatter (Rem. AV.38).
13.

Why χ is the true definition of dark matter, not
adopted CDM

Remark AV.37 (χ versus a parameter-fit CDM component). Four properties separate χ from cold dark matter
grafted onto the model:
1. Forced by topology, not postulated. χ is the
unique harmonic three-form on S 3 = SU (2) (Theorem AV.2); its existence is a theorem of the fixed
internal geometry, in a Chern–Simons sector DFD
already uses for the closed form of α.
2. Does not break absolute time. χ is an ordinary a−3 matter field propagating on the fixed ψ
background; it does not modify the absolute-time
foliation or any background dynamics that anchor
DFD.
3. Mass and period from the determinant
ledger,
√
with the stated multiplicity 158 and the
3
stated stiffness. P = fχ = M̄P α√
(Lemma AV.5,
Lemma AV.6) and mχ = 2π 158 M̄P α13 =
32.005 eV (Theorem AV.8) come from the determinant ledger, with the mass tied to the same α8 scale
as the Higgs.
4. Relic abundance — derived, conditional on
the flux-period identification, the stated
count 158 and premise P4′ (the Finite SU (2)60
CS Vacuum Relic). With no inflaton there is no
chosen angle (Theorem AV.13); the amplitude is
the finite SU (2)60 CS/WZW vacuum
expectation of
P
the quadratic Casimir, ⟨θ2 ⟩ = j |S0j |2 C2 (j)/[k(k+
2)] = 0.073, giving Ωχ h2 = 0.11928 (−0.60σ from
Planck 0.1200±0.0012). The operator is proven and

the normalization k(k + 2) rests on two stated conventions (θmax = 12 from χ’s derived Z2 , K = k + 2
from Sugawara), given which the amplitude is forced;
the only non-DFD input is the standard cosmological relic-redshift map, exact at onset (Rem. AV.12).
A factor-44 overshoot is a units-plus-continuummeasure artifact (AV14).
χ is the weak-sector see-saw sibling of the Higgs in the
same α-tower: a derived, cold, non-thermal a−3 field
that supplies the clustering component the CMB kernel
requires — and so it is the named definition of cold dark
matter rather than an external “dark” unknown.
a. How χ and the µ-law coexist in galaxies: an open
question. DFD’s gravity has two channels (App. GR):
an unscreened rest-mass channel, sourced by all rest mass,
and the optical/ψ µ-AQUAL response (the exponential response of App.
√ BS, weak-response form µ(x) = x/(1 + x))
with a0 = 2 α cH0 (the MOND-scale theorem, App. AH),
whose polarization the baryons source. Every species
couples minimally to the single optical metric ds̃2 =
−c2 e−ψ dt2 + eψ δij dxi dxj of App. AN, Eq. (AN1) (the
opposite-sign exponents are essential — the timelike part
g̃00 = −c2 e−ψ gives the matter acceleration a = (c2 /2)∇ψ
and PPN γ = 1, while light reads both g̃00 , g̃ij ), so in
the rest channel the weak equivalence principle holds for
neutral χ exactly as for the neutral H i gas that drives
DFD’s best dwarf-galaxy fits; inside bound systems χ
gives it up where it falls short of the µ-law’s demand
(below).
As printed, the two channels add: χ contributes its
Newtonian pull on top of the baryons’ µ-law field. The
same printed physics makes χ form ΛCDM-like halos
around every galaxy (χ is cold and pressureless with
Qχ = 1, and every small-scale modifier acts some twenty
decades in mass below galaxy scales; computed). Added
to the µ-law, such halos count the missing mass twice:
on 153 SPARC galaxies they shift the radial-acceleration
relation by −0.20 dex (−0.27 dex if χ’s mass also feeds
the µ-law) against a tolerance of 0.03 dex, excluded at 3.4–
4.6σ on the mean offset once a global stellar mass-to-light
systematic is included, and at more than 3σ in about
half the galaxies individually even with each galaxy’s
mass-to-light ratio fitted (computed). Galaxy by galaxy,
the ΛCDM-like χ halo matches the Newtonian missing
mass (median ratio 0.93 at 2.2Rd and 1.01 at the last
measured radius; computed), so Newtonian gravity of the
baryons plus such a halo reproduces the µ-law relation on
average, and a χ halo at the phantom’s surface density
a0 /2πG ≃ 137 M⊙ pc−2 (Donato et al. 2009; Milgrom
2009) stands in for the phantom. That reading holds only
if the two channels do not add.
The leading candidate for how they compose is the
floor composition: the baryons set a floor under gravity,
their µ-law field; χ’s Newtonian pull counts toward that
floor; and the optical medium supplies only the shortfall,
exactly so in spherical symmetry, where




g = max ν gN,b /a0 gN,b , gN,b + gN,χ ,
(AV27)

330
with ν(y) the µ-law boost, µ(νy) νy = y. It cannot come
from an action of the printed type: in any local action
with a Poisson rest channel the equation of the polarization cannot contain the rest channel’s field, so the medium
cannot see χ and every such action gives the additive law
(theorem; checked on a 34-term ansatz). It does follow
from one energy of the two Newtonian fields, Eq. (AV28)
below, the only local energy that gives the baryons the
floor law and leaves χ Newtonian where it overfills the demand (theorem, spherical symmetry). It follows uniquely,
too, from total-field equilibrium, the same input stated for
the medium: inside a bound system the medium settles
at the minimum of its µ-law energy measured on the total
field and, in spherical symmetry, never screens the rest
channel. It states for the medium what the energy of
Eq. (AV28) states for the baryons, that the total field
counts; in the Bose reading of App. BS the mode counts
the total field, with occupation n = g/gN,b − 1 held at
n ≥ gN,χ /gN,b . Never screening, the χ-free limit and
equilibrium on the total field give exactly Eq. (AV27)
(theorem, checked on four energies to 7 × 10−8 dex); with
an energy of the medium’s own field alone the unique
result is the additive law. The floor law replaces the
printed additive equation; it is not derived. In spherical
symmetry χ that stays below the floor adds no pull of
its own, and only χ in excess of the baryons’ phantom
shows in a rotation curve, as a one-sided lift above the
relation. The tight radial-acceleration relation then stays
the µ-law’s signature on the baryons, with no halo needed
for the rotation curves. Clusters, where χ is denser than
the phantom, are then Newtonian plus χ and carry the
ΛCDM dark budget (App. I). For the static field of the
medium one three-dimensional form is the field-level projection of the µ-law field onto the curl-free fields whose
component along the rest-channel field is at least that
field, which exists and is unique; the density-level version
is excluded, because the confinement makes each bound
system’s induced charge sum to zero and that version
would then leave no phantom at all (theorem). The working three-dimensional form, for static and moving matter,
is the energy of Eq. (AV28), evaluated mode by mode,
which gives the same force as the projection in spherical
symmetry; near a disc the two differ where χ fills the
demand, and the Sun’s vertical force chooses the energy
(below). The floor composition passes SPARC for halos no
heavier than the baryon-closure mass (one galaxy in 120
shifted by more than 0.03 dex); for halos of low concentration, 0.22 dex below the mean, the median shift is zero
but a tenth of the galaxies, nearly all massive discs, are
shifted by more than 0.03 dex; it fails for contracted halos,
uncontracted halos at the Moster stellar-to-halo masses
miss the shape of the residuals, and the feedback profiles
of Di Cintio et al. (2014) at the abundance-matching
masses fail in the massive discs (computed; the halo families are compared below). Near a disc the vertical force
separates the static field from the adopted energy. With
the measured local stars and gas (47.1 ± 3.4 M⊙ pc−2 in
all and 43.8 ± 3.4 within 1.1 kpc of the plane; McKee et

al. 2015) in a Milky Way whose µ-law rotation speed at
the Sun is the observed 229 km/s, the µ-law’s own field
pulls toward the plane with 2πG × (84 ± 4) M⊙ pc−2 at
1.1 kpc, because its phantom follows the disc (computed).
The measured values are 68 ± 4 (Bovy & Rix 2013), 74 ± 6
(Holmberg & Flynn 2004) and 63.6 (Bienaymé et al. 2024)
at 1.1 kpc, and 67±6 and 68.5±1.0 at 1.0 kpc (Zhang et al.
2013; Bienaymé et al. 2014), together 68.8 ± 1.9 at 1.1 kpc
(with an error of 4 assumed for Bienaymé et al. 2024 and a
systematic of 3 added to Bienaymé et al. 2014), 3.1–3.2σ
below that value (computed). The static projection keeps
this pull for every χ halo, because its bound acts only
along the rest-channel field, which at the Sun is nearly
radial. Under the energy of Eq. (AV28) the baryons feel
the Newtonian field of the baryons and χ wherever χ fills
the demand; with χ at the density that the rotation curve
beyond the Sun gives, 0.0079 ± 0.0008 M⊙ pc−3 (Eilers
et al. 2019), the vertical force is 2πG × (66–69) M⊙ pc−2
under the filling rule below with χ at the rotation-curve
density, within 0.7σ of the measurements for a round
χ halo, the spread coming from the disc model (computed), and environmental confinement changes nothing
here, since the Galaxy’s own field stays boosted. This
needs χ close to the demand at the Sun: a local χ density of at least about 0.0068 M⊙ pc−3 for a round χ halo,
or 0.0076–0.0082 for one flattened to an axis ratio of
0.8 (computed). The lensing of isolated blue galaxies
in the Milky Way’s mass range caps a round NFW χ
halo at about 0.0055–0.0066 at the Sun, depending on
how the lens sample is modelled (computed): below the
round halo’s threshold, so for a round halo the two do
not meet, and below the rotation-curve density. At the
Sun DFD therefore behaves like standard dark matter,
so this test separates the adopted energy from the static
field rather than DFD from ΛCDM. Its early-type minus
late-type lensing contrast is +0.076 dex (+0.138 inside
0.3 h−1
70 Mpc), against +0.275 ± 0.065 (split by colour)
and +0.194 ± 0.040 (split by Sérsic index) measured by
Brouwer et al. (2021): 2.9–3.1σ low overall and, against
the +0.206 ± 0.072 and +0.162 ± 0.049 measured there,
0.5–0.9σ low inside 0.3 h−1
70 Mpc (computed). The contrast
turns on the early-type masses: with early-type stellar
masses raised by the factor of up to 1.7 that Mistele et al.
(2024) find, the floor gives +0.143 (+0.170 inside), 1.3–
2.0σ low overall and within 0.5σ inside, and Mistele et al.
(2024) measure no split, where the floor gives +0.085 dex
(computed).
Three properties cannot all hold: action and reaction,
χ’s equivalence principle, and χ’s exemption from the
µ-law inside bound systems; the first two together force a
universal active mass, so χ would be boosted too (Bondi
1957). DFD keeps action and reaction and the exemption,
and gives up χ’s equivalence principle only where χ falls
short of the µ-law’s demand. Matter moves under one
energy of the rest channel’s Newtonian fields, gb of the

331
baryons and gχ of χ,
Z
1
d3 x K,
W =−
4πG
Z |gb |


2
1
p(u) − c + du,
K = 2 |gb + gχ | +
0


p(u) = ν(u/a0 ) − 1 u,

(AV28)

where p is the µ-law’s demand, the phantom field that
a baryonic field u calls for, and c is the part of χ’s field
that counts toward it. The baryons are accelerated by
the curl-free part of ∂K/∂gb and χ by that of ∂K/∂gχ .
W does not depend on the origin and orientation of the
coordinates, so momentum and angular momentum are
conserved exactly (Noether), and a χ clump and a galaxy
pull each other equally; W depends on time only through
a0 , so energy is conserved at fixed a0 . In spherical symmetry (c = gN,χ ) the baryons feel exactly Eq. (AV27),
and χ feels gN,χ plus the part of the baryons’ field it
fills, |{u ≤ gN,b : p(u) ≤ c}|: the Newtonian field of all
matter wherever χ fills the demand, and less where it does
not. K is the only local function of (gN,b , gN,χ ) that gives
the baryons the floor law and leaves χ Newtonian where
it overfills the demand (theorem). The demand peaks
at 0.648 a0 , at a baryonic field of 2.54 a0 , and p/gN,b is
10−330 at Saturn (computed).
χ’s departure from the baryons’ fall, δ = 1 − Aχ /Ab , is
zero wherever χ fills the demand. Around a galaxy with
6×1010 M⊙ of baryons this holds out to about 208 kpc for
a halo of 2×1012 M⊙ and 65 kpc for 1.2×1012 , while a halo
of 1012 falls short of the demand everywhere (δ = 0.12 at
60 kpc); beyond, δ ≃ 0.14 at 300 kpc and 0.25 at 400 kpc
for the heaviest, and at Fornax’s 90 and 147 kpc δ = 0.08
and 0.23 for 1.2 × 1012 M⊙ (computed). The Sagittarius
streams allow |δ| ≲ 0.10 (Kesden & Kamionkowski 2006)
and Fornax δ ≲ 0.17–0.19 (Zamlung & Nusser 2025),
which need χ at 0.91–0.93 of the demand or more at
18–60 kpc and 0.83 or more at 90–147 kpc (computed).
Where χ falls short, a satellite’s stars are pulled ahead
of its χ halo and its leading tidal arm is enhanced, the
opposite sign to a dark-matter fifth force. Cluster cores
and the Bullet cluster are filled, δ = 0 (Harvey et al. 2015
measure dark-matter offsets of 5.8 ± 8.2 kpc). The threedimensional filling rule is stated, not derived: χ counts
toward the demand by what it adds to the size of the
total rest-channel field, c = max(|gb + gχ | − |gb |, 0), which
is gN,χ in spherical symmetry. It counts the total field
by its size, as the medium’s mode does under total-field
equilibrium, and other rules that give the floor law in
spherical symmetry pass the present tests as well. Under it
a 1010 M⊙ clump of χ and a 5 × 1010 M⊙ galaxy without
χ, 100 kpc apart, repel each other at about 0.3 times
their Newtonian attraction (0.2–0.4 across the sizes of
the two), and with the galaxy’s own χ halo in place
they attract at the Newtonian value where that halo
fills the demand at the clump and up to about 4% more
weakly beyond it (computed). The forces jump where
χ stops raising the size of the total field, |gb + gχ | =

|gb |, by |gχ | for the baryons and by |gb | for χ; they
stay bounded and the conservation laws hold (computed).
The demand is evaluated on the scale-additive depth of
App. NL, mode by mode, so a host’s χ fills only the host’s
demand, and a subsystem that environmental confinement
keeps Newtonian (Axiom NL.14) makes no demand of
its own. This mode clause, not the filling rule, keeps the
planets’ orbits: read on the total field at every point,
the Sun’s demand would give them a tidal quadrupole
Q2 ≃ 3 × 10−25 s−2 , about 170σ from the DE440 value
and built up between about 102 and 104 AU from the
Sun, while mode by mode the Sun’s demand reads only
the Sun’s own field, which is spherical about the Sun,
and Q2 = 0 (computed). The mode-by-mode form has no
constructed action (App. NL), so it has no rule of motion
of its own, and the conservation laws above are those of
the energy read on the total field; where two systems’
fields blend, as near the Large Magellanic Cloud, which
system’s demand a χ halo fills is open. The energy keeps
Qχ = 1 on linear scales because unbound perturbations
do not polarize the medium (Cor. GR.11); without that
confinement its linear boost at z = 0 on R8 would be 8.7
for the baryons and 0.84 for χ (computed).
The alternatives fail the data (computed on published
values; the kicks and the assembly are estimates). One
field, with χ boosted inside bound systems like the
baryons, over-predicts the masses of the sixteen clusters
of App. I by a factor 1.43 (3.6σ), needs groups nearly
empty of χ (0.06–0.11 of the cosmic share), fails the
lensing of isolated galaxies (≳ 14σ) and misses SPARC
by −0.27 dex (≥ 4.6σ). A rule in which every species
moves in the total ψ gives up action and reaction: a
pair drifts at 44.7 km/s per Gyr, a χ-poor companion
falling in receives 78–95 km/s against at most 29 km/s
allowed by the cold local Hubble flow (Makarov et al.
2025), and unequal χ fractions in the Local Group are
excluded at ≥ 4.9σ. A simple three-field action that exempts χ wherever the phantom acts is excluded by the
Sagittarius and Fornax streams (δ = 0.37–0.72 against
about 0.10). The energy predicts no drift of the Local
Group’s matter barycentre against the cold local Hubble
flow, where the total-ψ rule gives 15–20 km/s along the
Milky Way–M31 line with ΛCDM-like halos; the measured
value is 6.9 ± 3.8 km/s towards M31, the component along
that line of the (24 ± 4) km/s motion towards Sculptor
found by Karachentsev et al. (2009), with the error of
the barycentre’s position included, 1.8σ from zero. A
measurement to 5 km/s or better, with the tidal motions
modelled, decides, and a drift of 15 km/s or more falsifies
the energy.
Under the energy of Eq. (AV28) χ feels the Newtonian
field of all matter wherever it fills the demand and is blind
to the unfilled part of the baryons’ floor elsewhere, so
the µ-law’s a0 acts on it only through
√ the part of that
floor it fills: the decline a0 (z) = 2 α cH(z) does not
expand χ halos but contracts them (computed), and the
runaway infall that a one-dimensional assembly model
gives when χ feels the whole floor (halos 8–120 times

332
heavier, turnaround radius 4.4 rather than 0.9 Mpc for
1012 M⊙ ) does not occur. χ assembles as in ΛCDM (estimate). Inside the optical galaxy such a halo matches the
Newtonian missing mass on average (above), and the floor
law keeps the radial-acceleration relation only if the inner
halo is not contracted: on 120 SPARC galaxies, against a
tolerance of 0.03 dex, halos at the abundance-matching
masses with the feedback profiles of Di Cintio et al. (2014)
shift the relation by +0.004 to +0.02 dex in the median,
with a third to a half of the galaxies above the tolerance
(depending on the halo-mass convention), most of them
discs with more than 1010 M⊙ of baryons, while the dwarfs
pass; uncontracted NFW halos give +0.007 dex with 29%
above it and a one-sided tail that the data do not show;
adiabatically contracted halos give +0.076 dex with 90%
above it (computed). Fitted with the stellar mass-to-light
ratio, distance and inclination free within their errors,
and with χ the cosmic dark share of the virial mass (the
convention of the smallest shift above), the feedback profiles fit the rotation curves about as well as the µ-law
alone in all but a few of the galaxies whose halos lie below
1012 M⊙ , while above that mass every ΛCDM-like halo
tried at the abundance-matching masses fits a quarter or
more of the discs worse than the µ-law alone (computed).
A χ halo contracts under the disc’s Newtonian pull, so
the rotation curves need cored inner halos. Slow disc
formation under the energy contracts χ as Newtonian
gravity does at z = 0 (computed on 120 SPARC galaxies),
and in spherical simulations of dwarf and low-surfacebrightness galaxies the bursty feedback from stars that
cores dark-matter halos under Newtonian gravity stalls
under the energy with χ near or above today’s demand,
so it does not make those cores there (computed); what
cores the χ halos, and in the massive discs how heavy
they are, is open. The fraction of χ inside galaxies is the
ΛCDM one at estimate grade: about the µ-law’s phantom
inside the optical disc, less where the halo is cored, and
the abundance-matching halo in total, except in discs
above about 1011 M⊙ of baryons, where that halo fails
both SPARC and the lensing of isolated late-type galaxies
and lighter halos pass both (computed). The cosmological
role of χ is unaffected: it supplies the cosmological cold
dark matter, namely the CMB third-peak height, cluster
mass (App. I), and large-scale structure.
What cosmology does depend on is where the µ-law
acts, and the Rest-Mass Channel fixes it: the polarization
is confined to bound low-acceleration systems, and the net
induced mass of each bound system is zero (App. GR, Remark GR.12). In three-dimensional structure-formation
runs with the floor composition, which resolve groups
and filaments but not the insides of galaxies, the polarization then carries 0.0003–0.002 of the mean matter density
and raises σ8 by at most 0.5% and the lensing power
at k = 0.1–0.4 h/Mpc by at most 7%; a phantom reaching beyond the bound regions would carry 0.47–1.25 of
the mean matter density and multiply the lensing power
2.9–6.4 times (computed). Beyond the bound radius lensing follows χ and the baryons, as in ΛCDM, so the flat

weak-lensing profile of isolated galaxies is not kept there
(computed); spectroscopically isolated lens samples decide
it (App. GR).
At recombination the mean density satisfies g/a0 ≫ 1
(µ → 1, Newtonian); but the small linear perturbation
contrast that drives the acoustic peaks (δ ∼ 10−5 ) selfsources a field at x = gN /a0 ∼ 10−6 — deep-MOND, by
Lemma GR.3 of App. GR. The literal single-W µ-law thus
gives χ a deep-MOND running Q that over-grows linear
structure (excluded by f σ8 at > 7σ); the data decisively
prefer Q = 1, and this is a proven obstruction of the
single-operator action (App. GR). DFD therefore adopts
the Rest-Mass Channel postulate, a rest-mass (µ = 1)
gravitational channel distinct from the gradient-energy
MOND screen. χ feeds only the rest channel because the
medium is polarized through photon–fermion vertices and
χ has no charged constituents (a conditional derivation;
App. GR, Remark GR.9); what is postulated is the rest
channel itself. It is the minimal fix: with the energy of
Eq. (AV28) action and reaction hold, and χ falls like the
baryons wherever it fills the µ-law’s demand (App. GR).
Given this postulate the Q = Geff /G = 1 cold-CDM
clustering invoked for the third peak and the CMB-lensing
pass holds; the cost is one added gravitational axiom, and
a falsifiable cluster-vs-galaxy signature follows (its clusterside amplitude follows from the derived abundance, at
estimate grade; App. GR). Interlock (consistency check,
both directions): Qχ = 1 and the no-CDM third-peak
result stand or fall together — the running-Q branch
that might sharpen S8 would instead destroy the thirdpeak height (and overgrow structure), so neither is traded
for the other. DFD’s distinguishing result stands
√ either
way: it derives the acceleration scale a0 = 2 α cH0 ,
and with it the universal surface density a0 /2πG of the
phantom, from α and the S 3 topology, explaining the
famous a0 ≃ cH0 coincidence that ΛCDM cannot.
Remark AV.38 (Falsifiable√prediction: the RAR intrinsic
scatter). Because a0 = 2 α cH0 is a fixed derived constant (not a per-halo property), DFD forces the galaxyto-galaxy a0 -drift channel of the RAR scatter to exactly
zero. Conditionally on the DLSC axiom (App. ZT), a
domain-local environment modulation reopens this channel in signed form, ∆a0 /a0 = −(f /3) δenv , contributing
0.015–0.027 dex at RD = 10–20 h−1 Mpc — inside the
intrinsic budget below, and a candidate explanation for
part of it (a0 anti-correlates with environment density).
That channel is the dominant term in ΛCDM’s budget
— the a0 -universality that ΛCDM treats as a fine-tuning
puzzle is here a structural consequence. The residuals are
the halo-shape floor under the single µ-AQUAL law (the
in-plane geometry channel, ∼ 0.013–0.025 dex computed)
and a one-sided χ term, zero for χ halos thinner than
the baryons’ phantom under the floor composition above
and up to about 0.2–0.35 dex in massive, concentrated
ones (computed); for the feedback profiles of Di Cintio et
al. (2014) it exceeds 0.03 dex in a third to a half of the
galaxies, most of them massive discs (computed above).

333
For χ halos below the phantom DFD predicts
intrinsic
σRAR

≲ 0.037 dex,

(AV29)

the halo-shape floor alone giving 0.013–0.025 dex and
the upper value applying if the domain-local channel
above operates (estimate; the χ term adds nothing while
χ halos stay thinner than the phantom), against the
ΛCDM expectation ∼ 0.06–0.08 dex. This is a falsifiable
bet: DFD predicts that the intrinsic scatter lies below
the tight (Lelli–Li, ∼ 0.057 dex) end, as the joint SPARC
inference finds (0.034±0.001 (stat)±0.001 (sys) dex under
the SPARC error model [169]), and that Stone et al.
(2019)’s 0.11 dex is observational systematics (stellaronly, gas-missing gbar and an under-propagated error
budget). Falsifier: if the true intrinsic RAR scatter
exceeds ∼ 0.06 dex (e.g. Stone’s 0.11 dex confirmed at
the intrinsic level) and is not the one-sided upward lift
that χ halos denser than the phantom produce in massive,
concentrated galaxies, DFD is falsified; under the floor
composition χ can only lift galaxies above the relation.
Grade: Derived/Pending — the a0 -channel-zero is forced;
the halo-shape floor is part first-principles, part literatureestimated (medium confidence), and is hardened by a
full SPARC-matched µ-AQUAL population synthesis; the
prediction holds if the ΛCDM-like χ halos of the adopted
energy are cored, as the rotation curves require (above).

Appendix AW: The Electroweak Precision Program
from sin2 θW = 3/13

Theorem Z.1 fixes the bare weak mixing angle to the
pure rational sin2 θW = 3/13 from the gauge partition
(3, 2, 1), the hypercharge trace Tr(Y 2 ) = 10/3, and the
stiffness ratio κ2 /κ1 = 2. The Higgs quartic is the conjectured λH = 1/8 (App. Z 3), used below only for the
Higgs rows. This appendix shows that the angle reproduces the entire Z/W precision sector at sub-percent
accuracy, with the residuals carrying the single universal
sign of the known top/Higgs radiative
uplift. Throughout,
√
c2 ≡ cos2 θW = 10/13, v = ( 2 GF )−1/2 = 246.22 GeV
(muon-decay vev), and we write s2 ≡ 3/13.

1.

Exact electroweak identities (E)

Proposition AW.1 (Mass-ratio identities, (E)). The
combinations entering the Z, W , and Higgs masses are
exact rationals, reusing only 3/13 and, for the Higgs, the
conjectured λH = 1/8:
3
2λH s2 = 52
,

15
2λH s2 c2 = 338
,

30
s2 c2 = 169
.
(AW1)

Consequently the tree mass ratios are
m2H
3/52
2 = πα ,
MW

m2H
15/338
=
,
MZ2
πα

(AW2)

and the famous
electroweak tree relation s2 c2 =
√
πα(MZ )/( 2 GF MZ2 ) is satisfied by 30/169 to 0.84%.
3
3
Proof. 2 · 81 · 13
= 52
; multiplying by c2 = 10
13 gives
15
3
10
30
2 2
2
;
s
c
=
·
=
.
With
m
=
2λ
v 2 and
H
H
338
13
13
169
2
2 2
2
the tree MW = παv /s (from e = 4πα, MW = gv/2,
2
g 2 = e2 /s2 ) the ratio is m2H /MW
= 2λH s2 /(πα) =
(3/52)/(πα) = 2.5165 (computed).
√ The electroweak
tree relation evaluates to πα(MZ )/( 2GF MZ2 ) = 0.17901
against 30/169 = 0.17751, a −0.84% check (the residual is
the standard ∆r). These identities reuse no new integers;
their empirical content lives in α, GF , MZ .

2.

The Z-pole suite from sin2 θW = 3/13 (P)

We evaluate the on-shell partial widths in the improvedBorn form


GF M 3
2
2
Γf = Ncf √ Z ρ gV,f
+ gA,f
Rf ,
6 2π
(AW3)
gA,f = T3f , gV,f = T3f − 2Qf s2 ,
with s2 = 3/13, the QCD/QED radiators Rf (Rq =
1 + αs /π + 1.41(αs /π)2 , Rℓ = 1 + 34 α/π), and ρ the
custodial parameter. The tree prediction sets ρ = 1; the
entire suite then follows from the single number 3/13.
Theorem AW.2 (Tree Z-pole from one input, (P)). With
ρ = 1 and sin2 θW = 3/13 the Z-pole observables are

334
Observable DFD (3/13) Measured
Γℓ [MeV]
Γinv [MeV]
Γhad [MeV]
ΓZ [MeV]
Rℓ
0
σhad
[nb]

83.58
497.6
1741.7
2490.1
20.840
41.44

Dev.

83.985(86) −0.49%
499.0(15) −0.27%
1744.4(20) −0.15%
2495.5(23) −0.22%
20.767(25) +0.35%
41.480(33) −0.10%

All six are reproduced to better than 0.5% absolute by
the single rational 3/13. The four partial widths share
the same negative sign: this is the signature of the omitted positive custodial uplift ρ > 1, not of independent
mismatches.
Theorem AW.3 (One-loop custodial uplift, in-framework, (T/P)). Restoring the standard custodial parameter
ρ = 1 + ∆ρeff with the effective leptonic uplift computed
in-framework,
∆ρeff =
+0.00834
| {z }
top √
loop, QCD-corrected:
mt =(1−α)v/ 2=172.74 GeV, αs =0.1187

− 0.0031
| {z }

(AW4)

= +0.0052,

SM bosonic remainder
at mH =125.1 GeV

together with the measured effective angle s2eff = 0.231522
in gV , gives χ2 /6 = 3.0–3.6 in the same improved-Born
formula (αs = 0.1179–0.1187 in Rq ), against 6.7 at tree
level. The derived +0.0052 reproduces, to 10−4 , the effective value +0.0053 preferred
√ by the electroweak fit. The
inputs mt = (1 − α)v/ 2, mH (the loop-shifted v/2,
Thm. AW.12), and DFD’s derived αs (MZ ) = 0.1187
(App. Z) are all ledger quantities (the top relation holds
for the pole mass, the quartic 1/8 only as an MS value at
a matching scale near 182 GeV); the top-loop piece is the
same computation as Thm. AW.9 (which gives +0.00836
at its muon-decay-anchored mt = 172.83 GeV with PDG
αs — a sub-permille spread); only the standard loop functions are imported. The uplift therefore carries the same
loop-derived grade as the MW convergence theorem.
Remark AW.4 (Residual of the loop uplift). One residual is booked explicitly. The −0.0031 bosonic remainder is the standard Standard-Model one-loop value at
mH = 125 GeV, cited rather than recomputed diagramby-diagram here. The Higgs leading-log piece is separately
verified small (−3×10−5 ), so the remainder is gauge-boson
self-energy dominated and insensitive to the Higgs mass.
Rederiving it in DFD’s framework is the same one-loop
computation at DFD’s inputs, since DFD’s electroweak
sector is the Standard-Model gauge theory, and moving
mH from the measured value to the tree value v/2 changes
it by only 2×10−5 (computed), a tenth of the ρ-parameter
precision.
Remark AW.5 (Effective leptonic mixing angle). The
DFD tree angle sin2 θW = 3/13 = 0.230769 is the bare
mixing angle. Against the measured effective leptonic

angle s2eff = 0.231522(29) it sits −0.33% low. Part of
the deficit is the electroweak form factor κZ = s2eff /s2bare
generated by the top quark; DFD supplies the bare angle,
the loop supplies κZ . The residual, which includes the
radiative relation of the bare angle to MZ , is not computed
here: undelivered, not contradicted.
3.

The W mass and the scale-ambiguity theorem (T)

The W mass carries an internal scale ambiguity. Because 3/13 is a bare relation, it may be anchored two
ways, and the two anchors bracket MW .
Theorem AW.6 (MW double reading, (T)). Anchoring
sin2 θW = 3/13 to the two natural reference scales gives
q
(I)
MW = MZ 10
13 = 79.977 GeV,
{z
}
|
MZ -tied, tree

(II)
MW = v2

|

q

(AW5)

4πα(MZ )
= 80.313 GeV .
3/13

{z

v-tied, running α

}

These bracket MW ∈ [79.98, 80.31] GeV. The PDG world
average MW = 80.369(13) lies 56 MeV above the upper
edge of the bracket (anchor II); the CDF II value 80.434(9)
lies 120 MeV above the entire bracket.
2
Proof. Anchor I uses the on-shell
p definition sin θW = 1 −
2
2
MW /MZ , giving MW = MZ 10/13 exactly. Anchor II
uses MW = gv/2 with g 2 = e2 / sin2 θW = 4πα(MZ )/s2
and the muon-decay v; with α−1 (MZ ) = 127.951 this
evaluates to 80.313 GeV (verified to 40 digits). The
(II)
(I)
gap MW − MW = 0.336 GeV is exactly the runningα/scheme difference between the two definitions of the
bare angle.

Corollary AW.7 (DFD and the MW controversy, (P)).
The tree bracket lies 120 MeV below the CDF II value
and 56 MeV below the LEP/ATLAS/LHCb world average.
With the leading custodial ∆ρ the anchors converge at
80.31–80.33 GeV (Thm. AW.9); the comparison of DFD’s
MW with the world average and with the CDF value is
open.
The two preceding statements left ∆ρ (the custodial
uplift that closes anchor I) as an undelivered StandardModel loop. Theorem AW.9 removes that gap at leading
order: the leading custodial uplift is computed from the
DFD-derived top mass, so with it the two anchors converge
rather than merely bracket.
Remark AW.8 (Scheme comparison of the bare angle).
The pure rational sin2 θW = 3/13 = 0.230769 is the bare
mixing angle; its radiative relation to the angles measured
at MZ is not computed here. Numerically
2
3
13 − ŝMS
2,lept
3
13 − seff
2
3
13 − son-shell

= 0.225%,
= 0.325%,
= 3.39%,

(AW6)

335
against ŝ2MS (MZ ) = 0.23129, s2,lept
= 0.231522(29), and
eff
2
s2on-shell = 1 − MW
/MZ2 = 0.22320. The on-shell value is
15× farther from 3/13 than the MS value. Moreover 3/13
is the unique irreducible p/q with q ≤ 24 within 0.3% of
ŝ2MS (machine-verified scan). Of the +0.325% residual

to s2,lept
, the MS-to-effective form factor supplies about
eff
0.1%.
Proof. Direct evaluation with ŝ2MS (MZ ) = 0.23129 (PDG

running value) and s2,lept
= 0.231522; the on-shell value
eff
uses MW = 80.369. A search over all irreducible p/q,
2 ≤ q ≤ 24, returns the single hit 3/13 inside the 0.3%
MS window.
Theorem AW.9 (MW convergence from the DFD top
mass, (T/P)). With 3/13 inserted at MZ (Remark AW.8),
the custodial uplift of anchor I is not an external input:
it is the one-loop ∆ρ √evaluated at the DFD top-mass
relation mt = (1−α)v/ 2, taken at this appendix’s muondecay vev (mt = 172.83 GeV, consistent with the GF
entering ∆ρ below; the derived v = 246.09 GeV gives
mt = 172.74 GeV, App. AT),
i
2

3GF m2 h
∆ρDFD = √ t 1 − 32 1 + π3 απs = +0.00836,
8 2 π2
(AW7)
√
with no free quantity (GF = 1/ 2v 2 , and the PDG input
αs = 0.1179 used here for the ∆ρ precision computation —
DFD’s own derived value is αs (MZ ) = 0.1187, App. Z, a
sub-σ difference with negligible [<MeV] impact on MW ).
The two DFD anchors then agree at this order:
q p
(I)
MW = MZ 10
1 + ∆ρDFD = 80.310,
13
(AW8)
q
(II)
α̂(MZ )
MW = v2 4π3/13
= 80.313,
2
agreeing to 2.9 MeV.
√ The full Sirlin−1relation MW (1 −
2
2
MW /MZ ) = πα/( 2GF ) (1 − ∆r)
with the DFD
inputs ∆α = 1 − α/α̂(MZ ) = 0.0663 (ledger-α running) and ∆r = ∆α − (c2 /s2 )∆ρDFD = 0.0384 gives
MW = 80.328 GeV. All three routes land at MW = 80.31–
80.33 GeV, keeping only the leading custodial ∆ρ (anchor II only the running of α). Run with the measured MS
angle 0.23129, the same routes land 26–134 MeV below
the Standard Model’s complete value (80.353–80.357 GeV
in current global fits).
√
Proof. mt = (1 − α)v/ 2 is the top-mass row of App. AT;
substituting
into the leading custodial ∆ρ with GF =
√
1/( 2v 2 ) and the standard QCD reduction factor at αs =
0.1179 gives +0.00836, verified to 40 digits. The uplifted
anchor I and the running-α anchor II were each evaluated
to 40 digits. The Sirlin route uses ∆α from the same
ledger α run between 0 and MZ , no other input.

Remark AW.10 (Anchor convergence). Theorem AW.9
computes ∆ρ from the DFD top mass: the same
√ integer
ledger that fixes 3/13 and mt = (1 − α)v/ 2 brings
anchor I to within 2.9 MeV of anchor II at leading order.

4.

The ρ parameter (T)

Theorem AW.11 (Tree custodial ρ = 1, (T)). The
DFD weak sector has an exact tree-level custodial relation
ρ = 1: the gauge partition assigns the SU(2)L stiffness κ2
2
uniformly across the isodoublet, so MW
= MZ2 cos2 θW at
tree level with unit coefficient.

5.

The Higgs mass from the conjectured quartic

Theorem AW.12 (Higgs mass from λH = 1/8 (conjecture)). With λH = 1/8 the tree Higgs mass is
p
v
mtree
= 123.11 GeV,
(AW9)
H = v 2λH =
2
−1.61% below the measured 125.13 GeV. Imposed at
the top-quark mass, where the Standard-Model quartic
is matched, λH = 1/8 gives mH = 124.63 ± 0.15 GeV
under two-loop running (computed): the radiative shift
from this tree value, which uses the muon-decay v, is
+1.2%, and the result lies 0.40% (2.7σ) below the measured 125.13±0.11 GeV (PDG 2026). The running quartic
equals 1/8 at 182 ± 4 GeV; the matching scale is not derived. With the quartic fixed at a matching scale the Higgs
sector is the Standard Model’s at the resulting mass, so the
trilinear self-coupling is the Standard-Model one, κλ = 1;
the ratio (1/8)/0.12914 = 0.968 measures the tree-to-pole
mass shift, not a coupling modifier.

6.

Look-elsewhere and status

a. Look-elsewhere. sin2 θW = 3/13 is the unique irreducible fraction p/q with q ≤ 24 lying within 0.3% of the
measured 0.23129 (no competitor exists in that window);
it is moreover not fitted: with the stated trace weights of
Thm. Z.1 it follows from Tr(Y 2 ) = 10/3 and κ2 /κ1 = 2.
The conjectured quartic 1/8 = 1/(dim X + 1) lies 0.8%
below the running quartic at the top-quark mass and
3.2% below its tree value at the measured 125.13 GeV;
it is the unique 1/n within 2% of the running value at
mt (the 1/n grid spacing near n = 8 is ∼ 8%). The Z/W
precision sector is reproduced from the one integer input
13, and the Higgs mass to 0.4% from the conjectured 8.
b. Status. The identities of AW 1 are exact and reuse
only 3/13 and, for the Higgs rows, the conjectured λH =
1/8. The Z-pole suite (AW 2) is a parameter-free tree
prediction accurate to < 0.5% across six observables. The
MW double reading (AW 3) is a theorem; its comparison
with the standard average and with the CDF value is
open, since the routes are truncated (Thm. AW.9). The
undelivered ingredients are the radiative relation of the
bare angle to MZ and the electroweak form factor κZ
(with the universal ρ uplift), Standard-Model loops rather
than DFD postulates; we present these as loop-completed,
not derived, and the tree values stand as the parameterfree DFD content.

336
Status AW.13. The weak-mixing input 3/13 and the
0
Higgs input 1/8 reproduce ΓZ , Γℓ , Γinv , Γhad , Rℓ , σhad
,
2 lept
sin θeff , ρ, MW , and mH at sub-percent accuracy. The
tree residuals carry the single universal sign of the omitted
custodial uplift. With the leading custodial ∆ρ DFD’s
routes give MW ≈ 80.31–80.33 GeV; the comparison with
the world average and the CDF II value is open. The
Higgs quartic, conjectured to be 1/8 at a matching scale
that is not derived, counts as one continuous number
(App. FK); nothing else is fitted.

Appendix AX: The Rotating-Source Exterior

The static exterior g̃ = diag(−c2 e−ψ , eψ δij ) with ψ =
2u, u ≡ GM/(c2 r), is the leading-order form of the
Schwarzschild metric in DFD’s coordinates, the full static
exterior when the parent strain’s σ is kept (App. WS).
This appendix adds rotation by solving the gravitomagnetic (shift-vector) sector of Sec. IV D for an angularmomentum source at first order in the spin, carries the
shift to O(JM ) with the Einstein–Hilbert coupling, and
extracts the strong-field observables for spinning black
holes.

1.

The stationary axisymmetric exterior

Theorem AX.1 (Rotating exterior at first order in the
spin, (T)). In the high-acceleration regime µ → 1, to first
order in J and to leading order in GM/c2 r in the shift,
the DFD scalar and gravitomagnetic fields obey separate
linear Poisson equations (Sec. IV D, App. AN): ∇2 ψ =
−(8πG/c2 )ρ and ∇2 Ni = −16πG ji⊥ . For a stationary
source of mass M and angular momentum J = J ẑ the
vacuum exterior at that order is


ds̃2 = −c2 e−ψ dt2 + eψ dr2 + r2 dθ2 + r2 sin2 θ dϕ2
−

4GJ
sin2 θ dt dϕ,
c2 r

(AX1)
with ψ = 2GM/(c2 r) unchanged from the static case.
Beyond this order the Einstein–Hilbert term couples the
two sectors (Proposition WS.5): the gravitomagnetic field
energy sources ψ at O(J 2 ),

2G2 J 2 
δψ =
1 + 2P2 (cos θ)
(AX2)
3c6 r4
plus interior-dependent terms, and ψ corrects the shift at
O(JM ),
h
 G2 M 2 i
2GJ
GM
gtϕ = − 2 sin2 θ 1 − 2 + O
, (AX3)
c r
c r
c4 r 2
the general-relativistic value in isotropic coordinates (computed). With the parent strain’s σ kept, the stationary
exterior is the Kerr metric in DFD’s coordinates at first
order in the spin, for every compactness; beyond that order a spinning hole’s exterior is Kerr’s by the argument
of App. WS (Proposition WS.6), and a star’s carries its
own quadrupole, as in general relativity.
Proof. The scalar ψ is a varied field of the action of
Sec. II: its Euler–Lagrange equation (App. AN, the
variational fork {δS/δψ, δS/δxµ , δS/δhTT }) is, in the
µ → 1 band, the flat-space Poisson equation ∇2 ψ =
−(8πG/c2 )ρ sourced by the rest-mass density. The
shift Ni obeys the linearized Einstein–Hilbert momentum constraint (Theorem WS.8): its transverse gravitomagnetic Poisson equation ∇2 Ni = −16πG ji⊥ , with
j ⊥ = (δ − ∇∇−2 ∂)(ρv), gives the far-zone value −4
of general relativity (Sec. IV D, Eqs. (77), (81)). The

337
mass monopole gives ψ = 2GM/(c2 r). The angularmomentum dipole solves ∇2 N ϕ -component as the standard magnetic dipole, fixing the lowered cross term
gtϕ = eψ r2 sin2 θ N ϕ = −(2GJ/c2 r) sin2 θ; the far-zone
coefficient is the PPN value −4 (dV + dW = − 72 − 12 = −4
at γ = 1, Eq. (81)), so the dipole agrees with the generalrelativistic gravitomagnetic field. At first order in J the
source of ψ is ρ alone, so the exterior is ψ = 2u. At O(J 2 )
the Einstein–Hilbert term adds the energy of the gravitomagnetic field to the source of ψ, which gives the exterior
term of the statement, and the interior matter quadrupole
enters as in the Hartle–Thorne slow-rotation framework;
at O(JM ) the same coupling enters the shift’s own equation and gives the factor 1 − GM/c2 r (computed).

2.

Frame dragging

Theorem AX.2 (Frame-dragging field, (T)). The framedragging angular velocity ωLT = −gtϕ /gϕϕ of the rotating
exterior, with the shift of Theorem AX.1 corrected at
O(JM ), is

2GJ 
GM
ωLT (r) = 2 3 1 − 3u + O(u2 ) ,
u = 2 , (AX4)
c r
c r
the general-relativistic value in isotropic coordinates (computed). At large r it is the general-relativistic Lense–
Thirring rate 2GJ/(c2 r3 ) (Eq. (100)), so all weak-field
gravitomagnetic tests (LAGEOS, Gravity Probe B) are
reproduced. The first-order shift alone gives the closed
form (2GJ/c2 r3 ) e−2u , which agrees at leading order but
not at O(u), so it is not a DFD signature; near the photon
sphere and the ISCO frame dragging is Kerr’s (App. WS).

3.

Ergoregion structure

The exterior of a spinning hole is Kerr’s (App. WS;
argued beyond first order in the spin): it has Kerr’s equatorial ergosurface, at the Boyer–Lindquist radiusp2GM/c2
for every spin, and Kerr’s horizon at GM/c2 (1+ 1 − χ2 ),
χ = Jc/GM 2 , so superradiance and the Penrose process
are as in general relativity. The first-order metric of Theorem AX.1 is its weak-field form and does not reach either
surface.

4.

Spin-dependent ISCO

Theorem AX.3 (Spin-dependent ISCO, (T)). The equatorial innermost stable circular orbit and the accretion
efficiency are Kerr’s at first order in the spin, and for
a spinning hole beyond it by the argument of App. WS
(Proposition WS.6); at zero spin the orbitp
sits at the areal
radius 6GM/c2 and the efficiency is 1 − 8/9 = 5.719%.

5.

Spin-corrected QPO law

Theorem AX.4 (Spin-corrected epicyclic/QPO law,
(T)). With the shift corrected at O(JM ) (Theorem AX.1),
the equatorial radial-epicyclic-to-orbital frequency ratio
carries the small-spin series
 κ 2
= 1 − 6u + 6u2 + O(u3 )
Ω


+ a 8 u3/2 − 12 u5/2 + O(u7/2 ) + O(a2 ),
(AX5)
(u = GM/c2 r in isotropic coordinates, a = J/M c,
units GM/c2 = 1), and at fixed orbital frequency, x =
(GM Ω/c3 )2/3 , its spin-linear part is 8a x3/2 − 4a x5/2 , the
Kerr value (computed). The leading frame-dragging term
8a u3/2 is general relativity’s spin–orbit coupling at 1.5PN,
and no DFD-specific spin–orbit term appears at the next
order: the +8a u5/2 of the first-order shift becomes Kerr’s
−12a u5/2 with the shift corrected at O(JM ). A comparison with Kerr’s Boyer–Lindquist series, whose spin-linear
part is 8a u3/2 alone, has to be made at fixed x, since
the u5/2 coefficient depends on the radial coordinate. The
static part is Schwarzschild’s, 1 − 6x at fixed x.
Theorem AX.5 (Nodal and periastron Lense–Thirring
precession, (T)). The relativistic-precession QPO frequencies, formed from the three equatorial fundamental frequencies (orbital Ωϕ , radial κ, vertical Ωθ ), carry,
with the shift corrected at O(JM ), the spin-linear terms
2a x3/2 , with no x5/2 term, in νnodal /νϕ = 1 − Ωθ /Ωϕ
and −4a x3/2 − 10a x5/2 in νperi /νϕ = 1 − κ/Ωϕ , at fixed
orbital frequency x = (GM Ωϕ /c3 )2/3 (units GM/c2 = 1),
the Kerr values (computed); at fixed isotropic u the u5/2
coefficients are −3 and −6, again Kerr’s. The leading
nodal coefficient 2a x3/2 is the Kerr/GR Lense–Thirring
rate, and no DFD-specific nodal or periastron term appears at the next order: the −a u5/2 and −16a u5/2 of the
first-order shift take Kerr’s values with the shift corrected
at O(JM ).

6.

Photon ring and shadow displacement

Theorem AX.6 (Spin-displaced photon ring). The photon ring and the shadow of a spinning hole are Kerr’s
(App. WS; argued beyond first order in the spin): to first
order in the spin the equatorial prograde/retrograde critical
impact parameters are
√ GM
J
b± (a) = 3 3 2 ∓ 2a + O(a2 ),
a=
, (AX6)
c
Mc
so the apparent shadow centre is displaced by 2a.

7.

Observational confrontation and falsifiers

The spin terms above are general relativity’s (argued
beyond first order in the spin), so spin measurements of

338
black holes (the ngEHT shadow asymmetry, QPOs, LISA
ringdowns) test DFD exactly as they test Kerr.

8.

Status

Status AX.7. Theorem AX.1 holds at first order in the
spin and leading order in GM/c2 r, where the DFD scalar
and gravitomagnetic sectors obey separate linear Poisson
equations (App. AN); beyond that the Einstein–Hilbert
term couples them (Proposition WS.5), with the O(J 2 )
field-energy term in ψ and the O(JM ) factor 1 − GM/c2 r
in the shift (computed). What stands: the leading spin
terms (the far-zone −4, the Lense–Thirring rate, LAGEOS and Gravity Probe B, the 1.5PN spin–orbit terms
8a u3/2 , 2a u3/2 and −4a u3/2 ), and at the next order
frame dragging (1 − 3u) and the spin-linear QPO and
precession terms, all general relativity’s. With the parent
strain’s σ kept the whole stationary exterior of a spinning
hole is Kerr’s (App. WS; argued beyond first order in the
spin): the ergoregion and the horizon, the spin-dependent
ISCO (Theorem AX.3) and the shadow displacement 2a
(Theorem AX.6) are Kerr’s, and the exterior carries no
DFD-specific spin signature.
a. Checks. The u5/2 spin terms of the first-order shift
(+8a u5/2 in the QPO ratio, −a u5/2 and −16a u5/2 in the
precessions) take Kerr’s values with the shift corrected at
O(JM ) (computed).

Appendix AY: Cross-Sector Relations: Flavor
Closures, Hierarchy Rigidity, and Large-Number
Structure

This appendix collects four cross-sector results, each
machine-verified (sympy/mpmath, dps ≥ 30) against the
printed, pre-locked DFD constants; none introduces a
new free parameter. Status labels follow the manuscript
convention, and the scope of each result—exact identity versus cross-sector consistency versus tautological
restatement—is stated explicitly so that no entry is read
as more than it is.

1.

AY.1

The Gatto–Sartori–Tonin closure from
locked integers

Theorem AY.1 (Gatto–Sartori–Tonin closure, (D)).
The DFD-locked Cabibbo integer λ = |Vus | = 31α
(App. AO)√and the locked down-sector mass map mf =
Af αnf (v/ 2) with (Ad , nd ) = (6, 25 ) and (As , ns ) =
( 67 , 32 ) (App. K; both prefactors data-selected, App. FM)
give
md
Ad nd −ns
ms
1
= 19.58, (AY1)
=
α
= 7α,
=
ms
As
md
7α
and thereby reproduce
the celebrated Gatto–Sartori–Tonin
p
relation |Vus | = md /ms to
r
|Vus |
31α
312 α
p
=√
=
= 1.00091,
(AY2)
7
7α
md /ms
i.e. to 0.09%.
Proof. λ = 31α is given by the selected CP 2 line-bundle
cohomology window {2, 3, 4}: 31 = h0 (O(2)) + h0 (O(3)) +
h0 (O(4)) = 6 + 10 + 15 (App. AO). The down-sector
prefactors Ad = 6, As = 6/7 and exponents nd = 5/2,
ns = 3/2 are the printed √
finite-Yukawa-ledger values
(App. K). The common v/ 2 and the factor α3/2 cancel in the ratio, leaving md /ms = (6∇ · 67 ) α1 = 7α
(machine-verified,
into the Gatto
p dps = 30). Substituting
1/2
form, |Vus |/ md /ms = 31α/(7α)
= (312 α/7)1/2 ;
with α−1 = 137.036 one has 312 α = 7.0128, so the ratio
is 1.00091. The relation would be exact iff 312 α = 7,
i.e. α−1 = 961/7 = 137.286; DFD sits 0.18% from that
integer identity.
Status AY.2. Cross-sector consistency, not an exact identity: the Gatto relation is reproduced to 0.09%, equivalent to the integer near-identity 312 α ≃ 7.013. The
exact-making coefficient would be 7.013, not the locked
7 = b0 , so DFD lands off -exact. The up-sector
dichotomy
p
is reproduced too: up-Gatto |Vcb | ≃ mc /mt fails by
∼ 2× in DFD (0.47) exactly as it fails in nature (0.477).
ms /md = 19.58 matches FLAG ∼ 19.9(±1) at < 0.5σ.
Look-elsewhere: one canonical textbook relation. Tier:
D.

339
2.

AY.2

The TM1 solar-mixing angle,
parameter-free

Theorem AY.3 (TM1 solar angle and Jarlskog invariant,
(D)). Retaining the first tribimaximal column (trimaximal1, TM1) together with µ–τ reflection symmetry, and using
the locked reactor relation sin2 θ13 = 3α (flavor master
operator, App. AT), the solar angle and Jarlskog invariant
are fixed with no continuous parameter:
cos2 θ12 cos2 θ13 = 23
sin2 θ12 =

(exact, all orders in θ13 ), (AY3)

1 − 9α
= 0.31841
3(1 − 3α)

(θ12 = 34.35◦ ),
(AY4)

LO

sin2 θ12 −−→ 13 − 2α,
|JPMNS | = 21 c213 s13 c12 s12 = 0.03371

(δCP = −π/2).
(AY5)

Proof. TM1 fixes the first
√ PMNS column to its tribimaximal value (2, 1, 1)T / 6, preserved by the residual
Klein generator. First-column normalization gives the
exact identity cos2 θ12 cos2 θ13 = 2/3 (sympy: residual 0).
2/3
Hence sin2 θ12 = 1 − 1−sin
; substituting sin2 θ13 = 3α
2θ
13
2
gives sin θ12 = (1 − 9α)/(3(1 − 3α)), with leading term
1
3 − 2α. Maximal θ23 = π/4 and δCP = −π/2 follow from
µ–τ reflection: the TM1 cos δ numerator vanishes identically in θ13 at θ23 = π/4 (sympy-verified). The Jarlskog
then evaluates to |J| = 0.03371.
Status AY.4. Parameter-free given the locked tribimaximal base, sin2 θ13 = 3α, maximal θ23 , and δ = −π/2 (all
already in the corpus). The new content is the TM1 solar
closed form and the 2/3 invariant. sin2 θ12 = 0.318 versus
NuFIT 6.0 0.307(+12
−11 ) is +0.95σ (inside 3σ). Falsifier: a
confirmed non-maximal θ23 , or sin2 θ12 outside [0.30, 0.34],
breaks the TM1 lock. Tier: D.
3.

AY.3

Mass-scale hierarchy rank

Theorem AY.5 (Mass-scale hierarchy rank, (T)). The
seven DFD mass-scale α-exponents
57
1
(xv , xMR , xΛDFD , xm3 , xH0 , xa0 , xΛtop ) = (8, 3, 19
2 , 14, 2 , 29, 2 )

satisfy exactly five independent rational linear relations
(rank 5 over Q), leaving two free directions: the direction [ 52 , 1, 3, 4, 0, 0, 0] in the hadronic/electroweak exponents, and the normalization of the cosmological-clock
block {H0 , a0 } relative to the hadronic/electroweak block.
Adjoining the cube-law relation xH0 = 3xΛDFD (App. O)
welds the second direction, reducing the hierarchy to a
single generator α.
Proof. The five solid relations are the four-sector lock
2xΛDFD = 2xv + xMR (19 = 16 + 3), the MR -cube iden1
tity 3xMR = xΛDFD − xΛtop (9 = 19
2 − 2 ), the seesaw
relation xm3 = 1 + 2xv − xMR (14 = 1 + 16 − 3), the

1
MOND-step xa0 = xH0 + xΛtop (29 = 57
2 + 2 ), and
1
the single-brick value xΛtop = 2 . The 7-column coefficient matrix has rank 5 over Q (sympy), with nullspace
spanned by [ 52 , 1, 3, 4, 0, 0, 0] and the clock-block direction
[0, 0, 0, 0, 1, 1, 0]. The cube law xH0 − 3xΛDFD = 0 pairs
nontrivially with the clock direction and raises the rank
to 6.

Remark AY.6 (xv = 8 is derived, not a residual axiom).
The base exponent xv = 8 of the harmonic ladder is
derived, not an axiom: xv = 8 = dim(CP 2 ×S 3 )+1 = 7+1.
The internal dimension X = 7 is derived by minimality
(Corollary AB.3: dim CP 2 = 4 plus dim S 3 = 3), and
the +1 is the radial/Higgs normal mode. The harmonic
ladder therefore closes at the exponent level; the only
sub-theorem-grade
residues that remain are the fitted
√
2π prefactor and√the “+1 = radial mode” identification.
We do not claim 2π is derived.
Remark AY.7 (The elimination identities are tautological, not independent tests). Because every observable
above is a single power of the one generator α, eliminating α between any√pair yields a monomial identity—e.g.
(MR /MP )8 = (v/ 2πMP )3 , (H0 tP )2 = C (MR /MP )19 ,
(a0 /2aP )3 = C 3/2 (MR /MP )29 (C = 1 for the boundmatter a0 , Postulate ZS.11). These “syzygies” are equivalent restatements of the individual α-exponents: each
reduces to αk = αk and carries no information beyond
them. They are satisfied identically and therefore cannot
serve as independent falsifiers. The testable content is
(i) the rank-5 integer structure above and (ii) the individual α-power predictions confronted with data elsewhere
in this work. The reduction to a single generator is contingent on the cube law, which is established only to
effective-field-theory grade (App. O); it is recorded here
as such, not as a closed theorem.
4.

AY.4

Large-number structure

Proposition AY.8 (Dirac large number as an exact
α-exponent, (T)). The Eddington–Dirac second large
number—the ratio of the Hubble time to the electron
atomic time—is an exact power of the gauge coupling,
√
1/H0
me c2
N2 ≡
= 23 π C −1/2 α−18 = 3.66 × 1038 ,
=
2
ℏ/me c
ℏH0
(AY6)
21
with integer exponent −18 = − 57
2 + 2 .
From the locked electron map me /MP =
Proof.
2√
21/2
π
α
(App. K) and the Hubble hierarchy H0 tP =
3
√
C α57/2 (App. O; C measured
√ on the early clock),
N2 = (me /MP )/(H0 tP ) = 23 π C −1/2 α21/2−57/2 =
2√
−1/2 −18
α . The exponent is exact; numerically
3 πC
3.66 × 1038 . The electron’s atomic time is a bound-matter
clock, so with the Hubble time read on the same clocks,
1/H0loc (Postulate
ZS.11), the coefficient is C = 1 and
√
N2 = 23 π α−18 = 3.43 × 1038 with no measured factor.

340
Remark AY.9 (Reframing, not a new prediction). This
recasts Dirac’s famous coincidence N2 ∼ 1039 as an exact
integer exponent of the α-tower rather than an accident.
It follows algebraically from the already-locked me and
H0 exponents, so it is a structural restatement, not an
independent prediction; the +3% offset from the measured
value is the locked electron-mass leading-order error, not
a free parameter.

Appendix SL: The Supernova–Lensing Sign
Discriminator

This prediction has been tested, and it failed
The sign test set out below has been run. Shah et al.
(DES Collaboration), Detection of weak lensing
magnification of supernovae and constraints on dark
matter haloes, MNRAS 532, 932 (2024)
[arXiv:2406.05047] [191], correlate the DES-SN5YR
Hubble residuals against DES Y3 Gold foregrounds and
“detect the presence of lensing at 6.0σ significance . . . the
first detection with a significance level above 5σ,” and
show that “the scatter of SNe Ia around the Hubble
diagram is reduced by modifying the standardisation of the
distance moduli to include an easily calculable de-lensing
(i.e., environmental ) term.” Magnification by foreground
matter, together with a scatter-reducing de-lensing term,
fixes the ordinary focusing sign: a supernova behind a
foreground overdensity is brighter, i.e. the residual is
negative for positive convergence, d(δµ)/dκfg < 0.
By Falsifier SL.7 as printed below, this falsifies
DFD’s optical-distance dressing DL → DL e∆ψLOS
and with it the positive-slope prediction
sDFD ∈ [+7, +28]. That dressing is retired. The
measurement predates, and outweighs, the +0.7σ
existing-data assessment recorded throughout this
appendix, which is convergence-map-noise limited.
What is not retired: DFD’s core. The Remark
following Theorem SL.1 already recorded that the
positive sign is theorem-grade only given the
optical-distance axiom, and that standard achromatic
lensing of the same n > 1 clump gives the opposite sign.
DFD’s physical metric g̃00 = −c2 e−ψ , g̃ij = e+ψ δij is
PPN γ = 1 at first order and therefore predicts the
ordinary d(δµ)/dκ = −5/ ln 10 = −2.17. The measured
sign is consistent with DFD’s core optical metric. The
failed step is the separate assumption that a longer
optical path dims a steady source: a longer path is a
propagation (Shapiro) delay, whereas luminosity distance
is fixed by flux and wavefront area, and the same n > 1
clump was in effect counted twice, once as focusing and
once as an independent dimming. Eq. (SL2) additionally
identifies a line-of-sight potential with a convergence,
which is a transverse second derivative of an integrated
potential, so no universal positive scalar connects them.
What replaces it. DFD’s surviving discriminator in
this channel is the amplitude, scale and redshift
dependence of its convergence mapping, not the sign; the
slope itself is d(δµ)/dκDFD = −5/ ln 10. The text below
states the pre-registered prediction and the criterion that
decided it.

This appendix isolates the single cleanest falsifiable separation between DFD and the ΛCDM concordance model
that is testable with standard-candle data: the sign of
the correlation between standardized Type Ia supernova
(SN Ia) Hubble residuals δµ and the foreground weaklensing convergence κfg along each line of sight. Under
DFD’s retired optical-distance dressing the vacuum optical index n = eψ made a SN seen through a foreground
overdensity appear farther (positive residual); in ΛCDM
gravitational magnification makes the same SN appear
brighter / closer (negative residual). The two predictions are opposite-signed, and the DFD sign is a theorem
given that dressing. The slope magnitude is sDFD ≃ +11
mag per unit κ (band [+7, +28] from the µ-law boost),
versus sΛCDM = −5/ ln 10 = −2.17 mag per unit κ, a
definite sign flip with a ∼13 mag/κ physical separation.
We give the√proofs and a pre-registered power forecast
(σs = σint /[ N σκ,eff ]). Status: the test has been run
and the DFD sign failed — DES detected the opposite,
ordinary focusing sign at 6.0σ (MNRAS 532, 932), which
by the falsifier below retires the optical-distance dressing
while leaving DFD’s core metric, whose first-order prediction is the standard −5/ ln 10, untouched. See the box
above.

1.

The line-of-sight optical screen and its sign

The DFD optical postulate is that the refractive index
of the vacuum is n = eψ , with ψ the optical-density field
that satisfies the screened Poisson equation


8πG
∇· µ(g/a0 ) ∇ψ = − 2 ρ,
c
(SL1)
√
x
,
a0 = 2 α cH(z),
µ(x) =
1+x
the same interpolation function and acceleration scale
fixed in Appendix AP and used for the optical σ8 analysis
of Appendix AE. Matter raises ψ (since the source ρ > 0
and the operator is positive), so along a line of sight that
crosses a foreground overdensity a photon accumulates a
strictly positive excess optical density
Z
κstruct (χ)

 = ggeo Q κfg > 0,
∆ψLOS = dχ W (χ)
µ g/a0
1
Q≡
,
µ(x̄)
(SL2)
where κfg is the foreground convergence the line of sight
samples, W is the optical window, ggeo is the orderunity geometric kernel factor (the optical-to-lensing window overlap, SL 2), and Q = 1/µ(x̄) is the deep-field
µ-law boost evaluated at the screen acceleration ratio
x̄ = gψ /a0 .
Theorem SL.1 (DFD SN-lensing sign, (T)). Under
the optical-distance dressing DL → DL e∆ψLOS (retired;
see the box above), the slope of the standardized SN Ia
distance-modulus residual with respect to the foreground

341
convergence is strictly positive,
sDFD ≡

5
d(δµ)
=
ggeo Q > 0 ,
dκfg
ln 10

(SL3)

with central value sDFD ≃ +11 mag per unit κ and band
[+7, +28]. The sign is independent of the magnitude
inputs ggeo , Q, both strictly positive.
Proof. The distance modulus is µ = 5 log10 (DL ) + 25.
The optical metric stretches the photon path uniformly
along the line of sight, DL → DL e∆ψLOS , so the residual
against an unscreened (matter-only) model is


5
δµ = 5 log10 e∆ψLOS =
∆ψLOS ,
ln 10
5
= 2.17147 (exact, the magnitude system).
ln 10
(SL4)
Inserting Eq. (SL2), δµ = (5/ ln 10) ggeo Q κfg ; differentiating in κfg gives Eq. (SL3). Since the magnitude prefactor 5/ ln 10 > 0, the field-equation operator is positive
(µ > 0 for x̄ > 0, hence Q > 0), and the kernel overlap
ggeo ∈ [1, 2] > 0, the product is positive: ∆ψLOS > 0 for
an overdense line of sight, so δµ > 0. The sign is therefore
forced by the postulate, not fitted; only the magnitude
depends on the inputs.
Remark SL.2 (Sign is theorem-grade given the optical-distance axiom). The positivity of sDFD follows by definition
of the magnitude scale once DL → DL e∆ψLOS is granted.
This optical-distance dressing is the DFD axiom under
test: standard achromatic lensing of the same n > 1
clump gives the opposite sign (solid-angle focusing brightens; surface brightness is conserved), so “object looks
farther/dimmer” is not derivable from n = eψ alone—it
is a separate optical-distance postulate. The wording
“theorem” is therefore exact given that postulate, and it
is precisely this postulate the sign test probes. This is
the correct posture for a discriminator: the experiment
adjudicates the axiom.

2.

The µ-law boost and the geometric kernel

The magnitude of sDFD is set by two order-unity-to-few
factors.
a. The µ-law boost Q = 1/µ(x̄). The screened operator in Eq. (SL1) enhances the response to a fixed source by
Q = 1/µ(x̄) in the low-acceleration regime. The screen acceleration ratio x̄ = 0.24 is the value that closes the BAO
differential in the anisotropic-screen analysis (it solves
xneed ≃ 1/(Qneed − 1)), sitting between the deep-MOND
limit (x̄ →
√ 0, Q → ∞) and the Hubble–EFE / σ8 scale
(x̄ = 1/(2 α) = 5.85, Q = 1.17):
0.24
µ(0.24) =
= 0.19355 =⇒ Q = 5.167 (central),
1.24
x̄ ∈ [0.18, 0.45] ⇒ Q ∈ [3.22, 6.56].
(SL5)

√
Status: the gate a0 = 2 α cH is derived; x̄ ≃ 0.24 is
selected by the void-dominated line-of-sight within that
derived deep-MOND→EFE range—the single residual
axiom of the magnitude (not the sign).
b. The geometric kernel ggeo . The convergence the
optical screen actually samples, κstruct , and the measured convergence κfg both integrate the same foreground density field and so are strictly proportional,
with an order-unity coefficient set by the overlap of
the optical window Wopt (χ) and the lensing efficiency
Wlens (χ) = χ(χs − χ)/χs :
• kernel-matched limit (Wopt ≈ Wlens ): ggeo = 1;
• full PPN-γ = 1 doubling (ψ = −2Φ/c2 , the 1PN
light-bending coupling DFD inherits): ggeo = 2.
c. Final
slope. Combining,
sDFD
(5/ ln 10) Q ggeo = 2.1715 × 5.167 × ggeo :

=

case
Q (x̄) ggeo sDFD (mag/κ)
kernel-matched, central 5.17 (0.24) 1.0
+11.2
nominal
5.17 (0.24) 1.5
+16.8
full γ = 1 doubling
5.17 (0.24) 2.0
+22.4
weak boost (x̄ = 0.45)
3.22
1.0
+7.0
strong boost (x̄ = 0.18)
6.56
2.0
+28.5
The central value sDFD ≃ +11 mag/κ (kernel-matched,
x̄ = 0.24) matches the conservative slope bD = 10 hardcoded in the power forecast (preflight forecast.py);
the plausible band is sDFD ∈ [+7, +28], dominated by the
µ-law boost Q and the kernel factor ggeo ∈ [1, 2].
3.

The standard (ΛCDM/GR) baseline

Proposition SL.3 (ΛCDM lensing-magnification slope,
(T)). In GR/ΛCDM the slope of the standardized SN Ia
residual with respect to the foreground convergence is robustly non-positive,
d(δµ)
5
=−
= −2.17 mag per unit κ ,
dκfg
ln 10
(SL6)
and no physical (κ, γ) makes it positive.
sΛCDM =

Proof. Weak-lensing magnification of a standard candle is
µmag = 1/[(1 − κ)2 − |γ|2 ] ≈ 1 + 2κ for κ, γ ≪ 1. A SN behind a foreground overdensity (κ > 0) is magnified, hence
brighter; standardized as a fixed-luminosity candle it is
read as “too close,” giving a negative distance-modulus
residual
δµΛCDM = −2.5 log10 µmag = −2.5 log10 (1 + 2κ),
(SL7)
d(δµ)
2.5 · 2
5
=−
=−
= −2.17.
dκ 0
ln 10
ln 10
This is exact and convention-fixed; it matches the value
bL = −2.17 used in the forecast pipeline. Keeping the
shear term and going to second order only makes the
local slope more negative (−2.18 at κ = 0.005, −2.29 at

342
κ = 0.05). Flux conservation ⟨µmag ⟩ = 1 fixes the mean
residual, not the slope; Malmquist/magnification-bias selection can dilute |sΛCDM | toward zero but never flip its
sign, since every selected SN behind an overdensity still
has δµ = −2.5 log10 µmag < 0. A coherent foreground
monopole is removed by the residual-mean marginalization in the pipeline, leaving pure line-of-sight lensing.

survey (κ map)
N
σκ,eff σs
Z
N3σ
DES-era proxy (current)
1,378 0.0002 16.4 0.74σ 22,400
DES-era proxy (best case) 4,000 0.0002 9.6 1.27σ 22,400
DES-era, if true map now 1,378 0.0025 1.39 8.8σ 162
LSST Y1 (good wl κ)
30,000 0.0013 0.60 20.4σ 648
LSST Y1 (deep κ)
30,000 0.0019 0.40 30.6σ 288
Roman (deep, low-noise κ) 20,000 0.0025 0.37 33.3σ 162

Corollary SL.4 (The slope sign is a clean DFD/ΛCDM
discriminator). The DFD slope (Theorem SL.1) and the
ΛCDM slope (Proposition SL.3) are opposite-signed, with
physical separation
sDFD − sΛCDM = 10 − (−2.17) = 12.17 mag per unit κ

Read-out. The bottleneck is the map, not the SNe.
Map quality enters linearly
(Z ∝ σκ,eff ) while the SN
√
count enters only as N : the ∼12× true-vs-proxy map
gap would need ∼150× more SNe to compensate. With
the current noisy proxy one would need ∼22,000 SNe
to reach 3σ; with a true convergence map, only ∼160.
Both LSST Y1 and Roman clear 3σ (20–33σ on paper)
the moment a real deep convergence map covers the SN
footprint.

(conservative bD = 10).
(SL8)
A measured slope d(δµ)/dκfg > 0 falsifies the ΛCDM lensing prediction and confirms the optical-distance dressing;
a slope ≤ 0 falsifies DFD’s optical-distance dressing (this
is what was measured — see the box at the head of this
appendix). The sign is the discriminator; the magnitude
band only sets the available statistical power.
Remark SL.5 (The one confound: correlated foreground
dust). The ΛCDM lensing sign has no escape—but a
different effect shares the DFD sign: correlated foreground dust/extinction. Foreground galaxies that produce κ > 0 also redden and dim the SN, giving a positive
δµ of order O(1) mag/κ, comparable to the −2.17 lensing term. This does not flip the GR lensing prediction,
so it is a detection-cleanliness requirement (reddeningcorrected/NIR samples, host-color cuts), not a sign error.
A positive measured slope is a clean DFD result only after
dust is controlled—this is the real systematic threat.

4.

Pre-registered power forecast

Theorem SL.6 (Sample size for a 3σ sign test, (P)).
For N SN Ia with intrinsic scatter σint and an effective
convergence-map noise σκ,eff (the rms spread of the regressor κfg degraded by map noise), the standard error of
the fitted slope and the DFD/ΛCDM separation are
σint
σs = √
,
N σκ,eff
√
(sDFD − sΛCDM ) N σκ,eff
sDFD − sΛCDM
=
,
Z=
σs
σint
(SL9)
so the number of SNe for a 3σ test is

2
N3σ = 3 σint /((sDFD − sΛCDM ) σκ,eff ) .
With σint = 0.13, the physical gap 12.17, and the anchor
true
that 648 SNe with a true map give 6σ (which pins σκ,eff
=
0.0025):

5.

Status and pre-registration

Status: the prediction failed; the pre-registered assessment follows
DES measured the opposite sign at 6.0σ (MNRAS
532, 932, 2024; box at the head of this appendix),
retiring the optical-distance dressing. The
pre-registered assessment that follows is what was
predicted before that measurement was taken
into account. This is not a detection. The
existing-data result is +0.7σ (joint), under-powered and
limited by convergence-map noise, not by theory.
Executed joint analysis (rotation-null calibrated, overlaps
priced): Four usable datasets (the DESI subset has ∼4
SNe and is unusable) lean DFD-positive at +0.14 to
+0.60σ each (Pantheon+×DES-Y3 +0.60,
DES-SN5YR×DES-Y3 +0.14, Pantheon+×BOSS +0.57,
Pantheon+×Planck-MV +0.22, coordinate-fixed), joining
to +0.68σ under the coherent-rotation null covariance —
and +0.2σ if the SDSS redshift-truncated counter-lean
(logged as robust at ∼ − 1σ in the same analysis) is
included. A reading of five datasets at ∼1–1.5σ each does
not hold: rotation nulls inflate naive errors ∼2.5×. No
detection, no falsification; the sign lean is a direction
indicator only. The slope-sign prediction (Theorem SL.1,
Corollary SL.4) is theorem-grade given the retired
optical-distance dressing; its magnitude band is the only
place the x̄ ≃ 0.24 axiom enters. The operative
existing-data evidential case for DFD is not this single
test but the combined preponderance tabulated in the
Confrontation Matrix (App. AG).

Falsifier SL.7 (Pre-registered SN-lensing sign test). Prediction (pre-registered). Regressing standardized SN Ia
Hubble residuals δµ on the foreground weak-lensing convergence κfg over a deep, dust-controlled (reddeningcorrected / NIR) SN sample with a true convergence map
yields a positive slope, sDFD ∈ [+7, +28] mag/κ (central
2
+11), at ≥ 3σ once N σκ,eff
reaches the forecast threshold
(Theorem SL.6; LSST Y1 ∼ 650 well-mapped SNe, or
Roman ∼ 160). Falsification. A measured slope ≤ 0 at

343
≥ 3σ, after dust control, falsifies the DFD optical-distance
dressing (Theorem SL.1); it is the GR/ΛCDM prediction
sΛCDM = −2.17 (Proposition SL.3). The sign—not the
magnitude—is the verdict. This criterion was met:
DES measured the negative sign at 6.0σ (MNRAS
532, 932, 2024); the dressing is retired. See the
box at the head of this appendix.
Status SL.8. Theorem SL.1 (the positive DFD slope)
and Proposition SL.3 (the non-positive ΛCDM slope) are
closed consequences of, respectively, the retired opticaldistance dressing and the GR magnification relation; the
slope sign is theorem-grade. The slope magnitude carries
the µ-law boost band [+7, +28], with the single residual
axiom x̄ ≃ 0.24 (the void-dominated screen value, dataselected within
√ the derived deep-MOND→EFE range; the
gate a0 = 2 α cH is derived). This appendix established
a rigorous, pre-registered prediction (Falsifier SL.7) which
has since been adjudicated against DFD by DES at 6.0σ
(box at the head of this appendix); the operative existingdata case for DFD remains the combined-evidence preponderance of App. AG (trials-factor accounting: Sec. CE 2).
The one systematic is correlated foreground dust (samesigned as DFD), a detection-cleanliness requirement, not
a sign error.

Appendix TP: The CMB Third Peak: a No-Go for
Zero-Dark-Matter, a Dark Sector with No Fitted
Density
1.

What this appendix settles, and what it does not

This appendix states the integrated resolution of the
question can DFD reproduce the height of the CMB third
acoustic peak without any cold-clustering matter at all?
The answer, proved at theorem grade and reconfirmed
here by independent CAMB reruns, is no. Within DFD’s
own printed 4D spectrum there is no field — and no optical, screening, sound-speed, echo, or gravity-enhancement
device — that lifts the third-peak height to the observed
value without a pressureless component clustering at recombination with Ωh2 ≈ 0.10–0.12.
The DFD result is the complement of that no-go, and
it is stronger than the thing the no-go forbids:
DFD does not avoid the cold-clustering charge
— it derives the field that carries it. The
χ-matter field (App. AV) is forced by the fixed
internal topology; its flux period and mass
come from α-powers with no fit (the count
158 and the stiffness are stated), and its relic
abundance is derived, conditional on the fluxperiod identification, the asserted premise P4′
(Lemma AV.27) and two stated conventions,
Ωχ h2 = 0.11928 (−0.60σ; Thm AV.13) — the
one number ΛCDM must read off the data,
DFD predicts.

The result is therefore a dark sector with no fitted density, not a no-CDM third peak. ΛCDM fits
ωc = 0.12 as one free density with no theory of its value;
DFD predicts the dark-matter particle’s mass (32.005 eV)
and flux period (9.46 × 1011 GeV, radian decay constant
1.51 × 1011 GeV) from fixed integers and α; the amount
is likewise derived (⟨θ2 ⟩ = 0.073 ⇒ Ωχ h2 = 0.11928,
Thm AV.13), with no misalignment angle as an input.
This appendix collects (i) the height no-go, (ii) the derivedabundance theorem with full provenance, (iii) the falsifiable χ-versus-CDM distinguishing predictions, and (iv) a
status box naming the one open tension.

2.

Part I — the height no-go (theorem-grade)

The rigorous statement and four-step proof are given as
the Cold-Clustering Wall theorem in App. J 9; we restate
it here in summary form and record the independent numerical reconfirmation, because it is the central negative
result of the whole dark-sector program.
Proposition TP.1 (Height no-go, summary of App. J 9).
Lifting the third-peak height ratio from the baryon-only
value (H3 /H1 )b ≈ 0.20 to the observed (H3 /H1 )obs ≈
0.44, holding the peak locations kn rs = nπ and the baryonloading ratio structure fixed, requires a driving potential
Φ(k, η) with a non-decaying (standing, Φ →const) subhorizon contribution through last scattering, sourced by
a species that is cold-clustering (w ≃ 0, c2s ≃ 0) at recombination with matter–radiation equality preceding recombination, zeq ≳ zrec . The minimal such abundance
is Ωc h2 ≳ 0.023 (standing-well floor), and matching the
observed height sharpens it to Ωc h2 ≈ 0.10–0.12. No
radiation-pressure-supported component (c2s > 0) and no
time-only modification of the homogeneous background
can supply the lift.
a. Why the two evasion classes are closed (the rigorous
core). The proof in App. J 9 closes exactly the two ways
a no-CDM device could try to succeed:
1. Pressure-supported devices self-destruct
their own well. Any component with c2s > 0
in radiation/pressure domination sources a potential that obeys Φ(x) = 3Φ0 (sin x − x cos x)/x3 ,
whose sub-horizon envelope ∼ 3Φ0 cos x/x2 → 0.
A numerical check confirms the decay: |Φ/Φ0 | =
0.024, 0.0027, 0.0013 at x = 10, 20, 40. No standing
well ⇒ no height lift. This excludes deep-MOND
gravity enhancement, the cs (ψ) sound-speed envelope, the line-of-sight ψ/Jacobian screen, and the
forced sound-horizon echo in one stroke.
2. Time-only / background devices cancel in ratios. An achromatic boost Φ → g Φ multiplies every
peak by g 2 , leaving all ratios Hn /H1 exactly invariant (verified to machine precision). Backgroundonly knobs therefore cannot move H3 /H1 ; the only

344
quantity that moves it is zeq , which is the cold
abundance, not a free dial.
Proposition TP.2 (The decisive discriminator: clustering, not background-w). A component with the correct background equation of state w → 0 but a finite
sound speed (an axion-fluid with c2s ̸= 0, non-clustering)
at Ωh2 = 0.12 produces zero third-peak lift: an independent CAMB run gives H3 /H1 = 0.205, identical to
baryon-only. Only a cold, clustering component (c2s → 0)
reaches 0.445. The third peak demands a clustering
charge; a pressureless background is not enough.
This is the single cleanest empirical statement of the wall.
b. Independent reconfirmation of the three escape caps
(this work). Three under-explored levers were re-run
from scratch in CAMB on the C = 1 background (H0 =
72.09, ωb = 0.02237, ns = 0.9667). The current measuredclock background has H0early = 67.52, ωb = 0.0222730,
and ωχ = 0.1192806; the archived runs are retained only
because the height-ratio no-go is insensitive to that clock
shift. Each lever caps far below the target, confirming
the no-go is not an artifact of an unexplored channel:
1. Scale-dependent high-k power (AQUAL tilt
/ ψ-driving proxy): H3 /H1 ≤ 0.206; positive
running makes it worse (0.169), because a smooth
power-law tilt shifts the whole Silk envelope and
cannot inject phase-coherent, peak-localized driving.
2. Radiation-budget lever (remove radiation
rather than add matter): Neff → 0 (the physical
floor) reaches only zeq = 902, H3 /H1 = 0.258 — the
radiation to remove runs out; lowering TCMB raises
zeq but drops H3 /H1 by wrecking the damping tail.
This is a new cap.
3. Warm-relic mimic (deceptive amplitude, broken pattern): an ∼11 eV neutrino species tuned
to Ωh2 = 0.12 can fake H3 /H1 ≈ 0.45 but ruins
the rest of the pattern (H3 /H2 ≈ 0.83 versus the
cold 0.98). Cold clustering is required for the full
[1, 0.45, 0.44].
Status TP.3 (The zero-dark-matter third peak is closed).
“CMB third-peak height with zero cold-clustering matter”
is a theorem-grade no-go within DFD’s printed 4D spectrum (App. J 9, Prop. TP.1). Three independent reruns
reproduced the wall and found no wrongly-dismissed channel. No fresh no-CDM mechanism reached H3 /H1 ∼ 0.45;
the best partial channels cap at H3 /H1 ≲ 0.26. DFD’s
standing is not that it dodges the cold charge but that
it derives the field supplying it (Part II). DFD does not
claim a no-CDM peak.
3.

Part II — the derived cold abundance

The complement of the no-go is the live DFD result.
The full derivation is App. AV; we collect the provenance

here and grade it exactly.
Theorem TP.4 (Derived cold dark-matter abundance,
modulo one angle). The DFD dark-matter density is fixed
by the misalignment chain
P = fχ = M̄P α3 = 9.46 × 1011 GeV
(the flux period; frad = P/2π),
√
√
mχ = 158 Λ2 /frad = 2π 158 M̄P α13 = 32.005 eV,
√
Ωχ h2 = 1.62 κrad 2π θi2 = 1.635 θi2 ,
(TP1)
so that the observed cold density ΩDM h2 = 0.12 is
reproduced
at a single O(1) misalignment angle θi =
p
0.12/1.635 = 0.271 (period and exact relic map per
Lemma AV.6 and Rem. AV.12, App. AV). The period
exponent dχ = 3 = dimC sl(2, C) and the mass exponent
13
√ = 2 · 8 − 3 are forced from the α-tower; the prefactor
158, 158 = 3(kmax −dim M7 )−1 = 3·53−1, is one stated
count of modes on CP 2 , not derived; the uniform zeromode rule would give 171 (App. AV). The cold (w = 0,
c2s → 0) shape is θi -independent; θi sets amplitude only.
(θi is not an input: the finite SU (2)60 CS vacuum gives
⟨θ2 ⟩ = 0.073 ⇒ Ωχ h2 = 0.11928, −0.60σ, Thm AV.13;
the chain above is the θi -parameterized form.)
a. Provenance, term by term (zero fit except θi ).
• Flux period P = fχ = M̄P α3 — the d = 3 specialization of the per-mode Gaussian determinant
(Lemma O.5), with dχ = 3 the complex dimension
of the weak gauge algebra sl(2, C) (the weak-sector
sibling of the dH = 8 colour block that sets the
Higgs scale). Exact Lie theory for the exponent;
the O(1) coefficient, the stiffness, is a stated input
(App. AV). That this coefficient is the period of
the unit-winding flux coordinate (not the radian
decay constant, which is frad = P/2π) is the period
identification, Lemma AV.6.
√
• Mass mχ = 2π 158 M̄P α13 = 32.005 eV —
the weak-block see-saw scale Λ2 /frad with the
electroweak-scale numerator Λ = M̄P α8 √giving the
net power 16 − 3 = 13; the multiplicity 158 is an
integer spectral count (the weak-block multiplicity
158 = 3(kmax − dim M7 ) − 1), not the loop factor
16π 2 it lies 0.05%
√ from. Forced exponent (13), stated
prefactor (the 158 integer count is one stated input: no single counting rule gives it, App. AV), with
one named physics assumption (the three-form is
shift-symmetry-protected and lifted by the see-saw
rather than sitting at the Planck-scale moduli mass;
Rem. AV.10 in App. AV). The electroweak–dark
scale link is v = 4πΛ, whose 4π carries the fitted
Higgs prefactor (Rem. AV.9).
√
• Abundance Ωχ h2 = 1.62 κrad 2π θi2 — standard
vacuum misalignment for the derived (P, mχ ), with
oscillation onset 3H(Tosc ) = mχ at Tosc ≃ 8.7 ×
104 GeV. The coefficient is computed, not fit —

345
including√
the exact radiation-era onset factor κrad =
Γ( 14 )2 /(6 3π) = 0.4026 (Rem. AV.12); θi is the
single free initial condition.
b. Contrast with ΛCDM. ΛCDM takes ωc = 0.12
from the data as one free parameter with no theory of its
value. DFD predicts mχ = 32.005 eV and P = 9.46 × 1011
GeV with no fitted parameter (the count 158 and the
stiffness are stated); in the θi -parameterized form above it
then carries one O(1) initial condition (the same one every
misalignment relic, including the QCD axion, carries) to
set the amplitude — and with the finite SU (2)60 CS
vacuum result (⟨θ2 ⟩ = 0.073, Thm AV.13) even that angle
is fixed, giving Ωχ h2 = 0.11928 with no free input. The
count is DFD: no fitted continuous input (the count
158, the stiffness and two conventions are stated;
App. FK) versus ΛCDM: one free input (a density).
χ has no photon coupling (E = 0; App. AV, Step 7), so it
leaves no decay photons and no conversion line, and any
dark-matter photon line at 16.0 eV (77.5 nm, two-photon
decay) or 32.005 eV (38.74 nm, conversion) falsifies it.
Corollary TP.5 (The third-peak height is then inherited,
by dynamical identity). Because χ is cold, non-thermal,
and pressureless
on all CMB scales (its Jeans wavenump
ber kJ ∝ mχ H exceeds the third-peak scale by ∼13–14
orders — and mχ /3H ∼ 3 × 1029 at recombination), it
enters the photon–baryon perturbation kernel identically
to a cold-dark-matter component of the same Ωh2 . The
CAMB result at the derived ωχ = 0.11928 (run on the
measured clock, which fixes the peak positions) therefore
is the χ result: the peak heights land at H3 /H2 = 0.977
(0.0σ), H3 /H1 = 0.444 (+0.5σ), H2 /H1 = 0.454 (+0.5σ)
against the Planck 2018 bandpowers, and on the same
clock the plik-lite likelihood places the derived abundance
at −0.98σ, as App. ZS reports (Thm. AV.31, App. AV).
Remark TP.6 (Caveat: the height is inherited, not independently re-derived). Thm. AV.31 is a CAMB run with
the derived ωχ inserted as a cold component, followed by
the dynamical-identity argument that χ enters the kernel
identically. The identity is physically sound (Cor. TP.5),
but the peak height is inherited from the standard coldmatter computation, not independently computed from χ
Boltzmann dynamics; the measured-clock fit of App. ZS
likewise inserts ωχ = 0.11928 as a cold component. The
DFD result is the derived abundance and the peakheight ratios it forces, not an independent height
calculation. The acoustic peak positions (θ∗ ) are a separate question, set by the clock coefficient measured from
BAO and θ∗ , with no screen (App. ZS) — they are not
claimed here. Relatedly, the Ωχ /Ωb = 16/3 coincidence is
not the mechanism (App. J 9 context; it is kept only as a
labeled coincidence, 16/3 · ωb = 0.1188, and appears in no
summary section): the live mechanism is the misalignment
chain of Thm. TP.4.

4.

Part III — distinguishability: χ versus a
free-parameter CDM

In its raw clustering, χ is observationally identical to
ordinary cold dark matter — it does not free-stream, selfinteract, or drift in mass on any accessible scale. Those
channels cannot separate the two. The verified list of
where DFD-χ does or does not differ from vanilla CDM:
• Free-streaming cutoff kcut — no. The misalignment condensate is a coherent zero-momentum field,
not a 32 eV thermal relic; kJ ∼ 1013 /Mpc, so it
clusters exactly as cold as CDM on every observable
scale. Indistinguishable.
• Self-interaction σ/m — no. The quartic λ4 =
(mχ /frad )2 ≃ 5 × 10−38 gives σ/m ∼ 10−58 cm2 /g,
≳ 55 orders below SIDM bounds; de Broglie wavelength ∼60 µm, no soliton/core. Indistinguishable.
• Time-varying mass — no. mχ is geometric (αlocked), not G-locked; no distinctive drift beyond
the generic axion mass turn-on. Indistinguishable.
• Isocurvature βiso — passes Planck, with no
inflation. DFD has no inflaton and no de Sitter stage at H⋆ ≃ 8πfχ ≈ 25 fχ (both on the α3
rung): such a stage would kick the flux coordinate
by δθ = H⋆ /2πfχ = 4 full periods per e-fold and
wash out the relic (Thm. SM.3), so no tuned preinflation θi = 0.27 exists. The relic is the finite-CS
vacuum expectation, which gives βiso ≈ 0 (a clean
Planck pass, for the opposite reason a pre-inflation
axion passes); a low-scale optical engine adds only a
quadratic isocurvature tied to the tensor ratio, negligible for r ≲ 10−10 (App. AV). The only structural
relic that survives is the NDW = 1 domain-wall case
fixed by the potential’s single minimum per period
(App. AV): with a single non-degenerate vacuum
the walls are unstable and collapse before dominating, a sharp parameter-free structural prediction of
the S 3 origin of χ. (A global cosmic-string tension
Gµ = (fχ /M̄P )2 = α6 is not a DFD prediction: χ
is the real coefficient of a single harmonic threeform with no U (1) winding and no string core, see
App. AV.)
• Late-time growth f σ8 : carried by χ with
Qχ = 1. DFD’s growth carrier is the cold χ relic
with Qχ = 1 under the adopted Rest-Mass Channel,
which confines the baryons’ polarization to bound
low-acceleration systems: unbound linear perturbations grow with Geff = G, and the polarization of
bound systems raises σ8 by a factor of only 1.001–
1.005 (App. GR; computed). The Geff /G = 1.02–
1.17 enhancement and its directional form, from
the constructions of App. AE and App. AC, do
not act on unbound linear perturbations, so latetime growth does not separate DFD from CDM at
current precision.

346
Falsifier TP.7 (χ-versus-CDM tests, with magnitudes).
DFD-χ is falsified by any of the following: (i) any confirmed coupling of dark matter to photons or to other
Standard Model fields, at any strength: a haloscope,
light-shining-through-walls or direct-detection signal, or a
dark-matter photon line at 16.0 eV (77.5 nm) or 32.005 eV
(38.74 nm) (χ has no photon coupling, E = 0; App. AV);
(ii) linear growth that departs from the Qχ = 1 growth
of the Rest-Mass Channel’s bound-system confinement
by more than the 0.1–0.5% rise in σ8 that the polarization of bound systems gives (App. GR); (iii) a measured
CDM isocurvature βiso > 0.038, which would falsify the
no-inflation finite-CS χ relic. The surviving structural
discriminator is the parameter-free NDW = 1 domainwall case fixed by the potential’s single minimum per
period (App. AV); a global-string tension Gµ = α6 is not
predicted (no U (1) winding, no string core; App. AV).

5.

Status and the one open tension

Status TP.8 (Integrated standing of the third-peak / dark–
sector question). A dark sector with no fitted density,
not a no-CDM peak.
1. Zero-dark-matter third peak: closed (no-go,
theorem-grade). No DFD field or device lifts
H3 /H1 to 0.44 without a cold-clustering charge
Ωh2 ≈ 0.10–0.12 at zeq ≳ zrec (App. J 9; reconfirmed by three independent reruns). Best partial
no-CDM channels cap at H3 /H1 ≲ 0.26.
3
2. Derived period and mass.
√ P = fχ13= M̄P α and
the exponents of
√ mχ = 2π 158 M̄P α = 32.005 eV
are forced, its 158 a stated input.

3. Height inherited, not re-derived. The peak
height uses the standard cold-matter CAMB result
via the dynamical-identity argument (Cor. TP.5,
Rem. TP.6); the result is the abundance, not the
height computation.
4. Abundance: derived, conditional on the fluxperiod identification, the stated count 158
and P4′ . DFD’s own scales remove θi (no inflaton,
no chosen angle); the relic amplitude is then the
finite SU (2)60PCS/WZW vacuum Casimir expectation, ⟨θ2 ⟩ = j |S0j |2 C2 (j)/[k(k + 2)] = 0.073 ⇒
√
Ωχ h2 = 1.62 κrad 2π ⟨θ2 ⟩ = 0.11928 (−0.60σ from
Planck; App. AV, Thm AV.13). A “∼44× overshoot” would come from reading the period-1 flux
coordinate in radians (π 2 /3 is the uniform variance
of a period-2π angle; the unit conversion 4π 2 is
96.5% of the gap in log measure) and from a continuum measure choice — the finite-vacuum value
0.073 in turns2 is what closes it (Lemma AV.6,
App. AV). The amplitude is forced — operator
(Casimir) proven, measure conditional on P4′ , normalization k(k + 2) from two stated conventions,

the edge of χ’s derived Z2 (θmax = 12 ) and the Sugawara shift (K = k + 2). The lone non-DFD input
is the standard cosmological
relic-redshift map, ex√
act at onset (1.62 κrad 2π, shared with all DM;
Rem. AV.12). Grade: derived, conditional on the
flux-period identification (Lemma AV.6), the stated
count 158 and P4′ (Lemma AV.27), downstream of
the (settled) peak-height no-go.
Remark TP.9 (Where this sits in the confrontation
matrix). In the master claims confrontation matrix
(App. AG), the third-peak entry is therefore correctly
read as a tie-by-inheritance plus a parsimony edge: DFD
matches the Planck height because χ is dynamically CDM,
and edges ΛCDM only on free-parameter count (one derived density with stated inputs versus one fitted density).
It is not a unique CMB prediction.

Appendix QM: Status of the Quantum-Mechanics
Derivation Program: What DFD Supplies and What
Remains Imported

This appendix records, at theorem grade, the current
status of the long-standing question “does DFD derive
quantum mechanics, or import it?” The ledger note at
Sec. XVII (the quantum framework—complex Hilbert
space, the unit i, the action scale ℏ, unitary evolution—
originally an adopted input) and the Born/inner-product
ledger of the Extended Derivations (App. AH, math foundations) stand. After an exhaustive, independently crosschecked attack on the most promising lead (the CP 2
Kähler complex structure J as the quantum i), the net
result is:

347
DFD does not yet derive quantum mechanics
in full—but the gap is reduced to a short,
sharply isolated import list. The CP 2 Kähler
J supplies the complex structure of the
internal microsector (a geometric partial
result), while the Dirac β is supplied by the
Yukawa mass term that couples the two
chiralities (Prop. QM.2). In §QM 4 the
field-level Schrödinger i, the Hilbert inner
product, and unitary free evolution are
derived, given the linearized stability that
Thm. QM.10 assumes, from a unique forced
complex structure J⋆ —unique because DFD
has one physical Hamiltonian
(Ashtekar–Schilling) and one absolute-time
foliation (no Bogoliubov ambiguity about the
vacuum). The value of ℏ is the unit of action,
fixed by the choice of units as in every
theory; the interacting operator-ordering
(variable-N ) map is the renormalized
dynamics common to every quantum field
theory; Born frequencies follow by typicality
(Rem. QM.8), and single-outcome selection is
the measurement postulate shared by every
quantum theory, not a claim of DFD.
This sharpens the “not yet fully derived” verdict into
a list of items DFD shares with every quantum theory,
not “all of QM.” The advances below follow from ψ + χ
positivity, the No-mixing theorem, the Ashtekar–Schilling
compatibility theorem, and DFD absolute time—no fit.

1.

Closures (what DFD supplies)

Theorem QM.1 (The Dirac β is available on the Cl(4, 0)
enlargement; sharpens Sec. XVII). Let Cl(4, 0) be the real
Clifford algebra with generators e1 , . . . , e4 , {ea , eb } = 2δab .
Then the assignment αi = ei (i = 1, 2, 3), β = e4 is a
valid Dirac set:
β 2 = αi2 = +1.
(QM1)
In particular Cl(4, 0) provides the fourth anticommuting
generator that the S 3 carrier Cl(3, 0) cannot, since the S 3
Clifford element ω is central in Cl(3, 0) and hence cannot
serve as β.
{β, αi } = 0,

{αi , αj } = 2δij ,

Proof. Direct from the defining relations; verified by explicit faithful matrix representation (M2 (H), an 8 × 8
real / 4 × 4 complex irrep) in the companion script
qm derivation check.py, which confirms all anticommutators vanish and all generator squares equal +⊮.
This upgrades the imported-input paragraph of Sec. XVII
from “a real carrier is equally admissible / β unavailable
on Cl(3, 0)” to a checked statement that β is available
precisely upon enlarging to Cl(4, 0). The enlargement
Cl(3, 0) → Cl(4, 0) is not forced by the S 3 carrier alone;
it is supplied by the mass term (Prop. QM.2).

Proposition QM.2 (The Dirac β is the Yukawa mass
block). For left- and right-handed spinors on the Cl(3, 0)
carrier with kinetic operators
a Yukawa
 ±c σ · p and

c σ · p mc2
mass m coupling them, H =
satisfies
mc2 −c σ · p
H 2 = (p2 c2 + m2 c4 )⊮,
 thedispersion of the master wave
0 ⊮
equation, and β =
obeys {β, αi } = 0, β 2 = ⊮.
⊮ 0
No 2 × 2 matrix anticommutes with all three σi , so a
massive spin- 12 field needs both chiralities, and Cl(4, 0) is
generated by the mass term.
Proof. Direct multiplication; the space of 2×2 matrices anticommuting with σ1 , σ2 , σ3 has dimension 0 (computed).
Minimal coupling of H gives g = 2 and antiparticles.
Theorem QM.3 (CP 2 Kähler J supplies the internal-microsector complex structure). Let (CP 2 , ωFS , J) be
the Fubini–Study Kähler manifold, J 2 = −⊮, ∇J =
0. Berezin–Toeplitz quantization of CP 2 at level 9
produces the 55-dimensional holomorphic-section space
H 0 (CP 2 , O(9)); with the five sections of O⊕5 it gives
(60)
the 60-dimensional microsector C60 = Hmicro (the
microsector-finiteness theorem, Ext. Deriv., App. AH).
On this internal factor the Kähler J is the operative complex structure (i = J), and it is precisely this holomorphic
structure that makes the corpus’s particle-physics results
(e.g. the closed form α = 1/137, the fermion exponent
ladder, the tree-level strong-CP result) go through. To this
narrow extent, part of “i” is geometric.
Proof. J 2 = −⊮ and ∇J = 0 on Fubini–
Study are standard and are checked numerically in
qm derivation check.py at a generic point. The microsector is the section space H 0 (CP 2 , O(9) ⊕ O⊕5 ) of
¯
the statement; “holomorphic” there is defined as ker ∂,
60
¯
and ∂ is built from J. Hence the complex structure of C
is literally J. The central role of this holomorphic structure for the internal observables is documented across
Appendix F (F) and the α and mass derivations.
Remark QM.4 (The two J’s must never be conflated). The
corpus carries two distinct objects both written J. (i) The
Kähler complex structure of CP 2 , J 2 = −⊮, linear, used
here. (ii) The Connes real/charge-conjugation structure
2
JDFD of the DFD spectral data, JDFD
= −⊮, antilinear,
KO-dimension 4 (the spectral-data theorem, Ext. Deriv.,
App. AH). These have opposite linearity. An antilinear
JDFD can never be the linear quantum i. Neither is the
Schrödinger i.

2.

What remains imported, and why (the precise
obstruction)

Proposition QM.5 (CP 2 Kähler J is not the field-level
quantum i). The Kähler J of Theorem QM.3 cannot
serve as the quantum imaginary unit appearing in the
canonical commutator [ψ̂(x), π̂(y)] = iℏ δ(x − y), in the

348
path-integral phase eiS/ℏ , or in unitary time evolution.
Three independent obstructions hold:
1. Wrong tensor factor.
J acts on the 4-realdimensional tangent space of the compact internal CP 2
fibre. The Schrödinger i must act on the external,
infinite-dimensional field state space over the R1,3 base
(the R of R1,3 × CP 2 × S 3 ). J has no action there.
2. Circular where it does act. Berezin–Toeplitz quantization uses J to select the holomorphic polarization;
“holomorphic = complex” is defined relative to J. So
extracting “i = J” on C60 returns the i inserted by the
polarization choice—no free lunch.
3. Not central over the full algebra. The only lift
to the external state, ⊮ ⊗ J, satisfies (⊮ ⊗ J)2 = −⊮
and commutes with purely external observables, but
[ ⊮ ⊗ J, B ⊗ C ] ̸= 0 for any observable C that probes
CP 2 non-holomorphically. The quantum i must be
central over the entire observable algebra (this is exactly
why QM is complex, not real or quaternionic). ⊮ ⊗
J is central only after restricting to the holomorphic
subalgebra—which is obstruction 2 again. Moreover
c1 (CP 2 ) = 3H ̸= 0, so on the tangent bundle J is
point-dependent, not a single global scalar operator.
Proof. (1) is the factorization R1,3 × CP 2 × S 3 with ψ
a real scalar on the R1,3 base (Sec. I); the field phase
space is built over the base, disjoint from T (CP 2 ). (2) is
the definition of ∂¯ via J (proof of Theorem QM.3). (3)
The centrality failure and the non-triviality c1 = 3H are
verified numerically in qm derivation check.py: a fresh
toy operator C acting non-holomorphically on the CP 2
factor yields ∥[⊮ ⊗ J, B ⊗ C]∥ > 0.
Proposition QM.6 (Cl(4, 0) supplies no i: the complexifier is the external commutant H). The center of Cl(4, 0)
is R (one-dimensional). The pseudoscalar ω4 = e1 e2 e3 e4
satisfies ω42 = +1 (not −1) and is not central. Hence
Cl(4, 0) contains no central square root of −1. The complex structure that makes the spinor module complex is
the external right action of the commutant H (real dimension 4, by Schur), i.e. an imported i outside the Clifford
algebra.
Proof. Computed by brute-force null-space of the
commutator map on the 16-dimensional algebra in
qm derivation check.py: dim Z(Cl(4, 0)) = 1; ω42 =
+⊮; [e1 , ω4 ] = 2e2 e3 e4 ̸= 0 so ω4 is non-central; and the
real commutant of the irrep is 4-dimensional (∼
= H).
Proposition QM.7 (Schrödinger evolution: Hamiltonian derived, first-order complex form imported). From
˜ g̃ −
the real, 
second-order DFD master wave equation □
2 2
2
m c /ℏ Ψ = 0 (Theorem AN.1, Eq. (AN3)), the nonrelativistic limit yields the Hermitian Hamiltonian HNR =
|p|2 /2m − 12 mc2 ψ. This H is derived. However, the
first-order complex form iℏ ∂t Ψ = HΨ is not derived:
2
the standard reduction Ψ = e−imc t/ℏ ϕ with ∂t2 ϕ dropped
inserts i and ℏ before they are derived, the cross-term
2i(mc2 /ℏ)∂t ϕ being exactly the Schrödinger left-hand side.

Proof. The HNR derivation is the weak-field expansion
in the proof of Theorem AN.1. The insertion of i, ℏ
in the order-reduction ansatz is verified symbolically in

2
qm derivation check.py by expanding ∂t2 e−imc t/ℏ ϕ
and isolating the i-proportional cross-term.
Remark QM.8 (Born rule and Gleason are not non-circular
here). The Born-rule equivariance theorem (Ext. Deriv.,
App. AH, Born ledger) is a theorem conditional on the
guidance-law postulate (the DFD guidance law, labelled
“completion postulate, not forced”) and on a self-adjoint
Hamiltonian acting on a complex Ψ = ReiS/ℏ —i.e. it
presupposes a complex Hilbert structure, which DFD
supplies from the forced J⋆ (h = g + iω). Gleason’s
theorem is likewise circular in this context: it assumes the
projection lattice of a complex Hilbert space of dimension
≥ 3, exactly the structure in question. The Fubini–Study
measure is natural on the internal CP 2 factor, whereas
Born |Ψ|2 is needed on the external configuration space;
it therefore repackages rather than removes the guidance
postulate. The device that does work is typicality (Ext.
Deriv., App. AH, the typicality / clean Born closure),
which secures Born frequencies given the complex Ψ and
self-adjoint H (the latter forced via J⋆ )—a consistency
result built on the derived inner product.

3.

The sharpened open problem (single attackable
object)

Proposition QM.9 (Reduction to one global compatible complex structure). By the Ashtekar–Schilling/Kibble
geometric formulation, ordinary quantum mechanics is
equivalent to real symplectic dynamics (M, ω) equipped
with one global compatible complex structure J⋆ : then
g(·, ·) = ω(·, J⋆ ·) is the Hilbert metric, and any quadratic
J⋆ -preserving Hamiltonian generates unitary Schrödinger
flow. DFD already supplies the symplectic dynamics
and an absolute-time positive-frequency split Jpf with
2
Jpf
= −⊮ whenever the linearized spectrum is positive.
Consequently the entire “derive QM” problem collapses
to a single target: prove that DFD forces a unique global
compatible J⋆ on the field phase space.
Status of proof. The Ashtekar–Schilling/Kibble equivalence
√ is standard. That Jpf is H-dependent (built as
A/ −A2 from the generator) and equals ±⊮-rooted
only on a stable spectrum is verified numerically in
qm derivation check.py. What is not yet established
is a DFD field-stability theorem forcing positivity of the
linearized ψ + χ + hT T spectrum on the physical sector;
absent that, J⋆ exists but is not forced/unique, and ℏ’s
numerical value is still inserted.

349
4.

The J⋆ -uniqueness target via the physical
Hamiltonian (the one-H theorem)

Comparing the positive-frequency complex structure Jpf
across two different Hamiltonians (different couplings
gA ̸= gB ) gives ∥Jpf (A) − Jpf (B)∥ ≈ 0.22 ̸= 0, but that
is the wrong question. Different couplings are different
physical theories; their J⋆ ’s should differ. DFD has one
physical Hamiltonian with fixed couplings. The correct
question is whether that one H admits a unique compatible J⋆ . It does.
Theorem QM.10 (One physical H forces a unique global
J⋆ ). Let (M, ω) be the DFD field phase space with the
canonical symplectic form, and let A = ω −1 Hess H be
the linear generator of the quadratic part of the single
physical DFD Hamiltonian about the vacuum. Suppose
the linearized spectrum is stable, i.e. −A2 is positive on
every mode with k ̸= 0 (argued for DFD: the ψ + χ + hT T
Hessian is block-diagonal by the No-mixing theorem of
Sec. V/Eq. (106) with χ background 0 at the misalignment vacuum, each block positive with ψ via convex W
(Appendix U), χ gapped by m2χ > 0, hT T the healthy wave
operator; under Axiom E ψ has no temporal kinetic term
and is fixed by its source). Then:
1. There is exactly one complex structure J⋆ on (M, ω)
satisfying J⋆2 = −⊮, [J⋆ , A] = 0, ω(J⋆ ·, J⋆ ·) = ω, and
g := ω(·, J⋆ ·) positive-definite; it is J⋆ = A (−A2 )−1/2
(the polar/sign part of A).
2. Equivalently, in the energy eigenbasis A is blockdiagonal over harmonic k-planes of frequency ω(k) > 0;
in each plane exactly one of the two admissible sign
choices yields a positive-definite metric. The global J⋆
is the direct integral of these unique per-mode choices.
Proof. Statement (1) is the Ashtekar–Schilling/Weinstein
compatibility theorem applied to a fixed stable generator: a positive-definite −A2 has a unique positive square
root, so J⋆ = A(−A2 )−1/2 is well-defined, squares to
−⊮, commutes with A, preserves ω, and gives g ≻ 0;
uniqueness follows because any other compatible J ′ commuting with A is block-diagonal over the eigenplanes of
A (spectral theorem), and the positive-definiteness of g
removes the per-plane sign ambiguity. Both statements
are verified numerically in physical H Jstar.py: for the
DFD-flavoured Hessian exactly 1 of the 2n sign choices
is positive-definite, and across 200 random stable Hamiltonians (including generic configuration-space mixing)
J⋆ = A(−A2 )−1/2 is the unique compatible positive structure in 200/200 trials. The ≈ 0.22 figure is reproduced
as the distance between J⋆ of two different theories, and
the same-H distance is identically 0.
Theorem QM.11 (Absolute time removes the Bogoliubov ambiguity about the vacuum). In generic relativistic QFT the global J⋆ is not unique even for a fixed H:
inequivalent foliations give Bogoliubov-rotated positivefrequency splits (∥JA − JB ∥ ̸= 0; Unruh/Hawking). About
the vacuum, where H does not depend on time, DFD removes this last infinite-dimensional ambiguity because its

time is an absolute parameter t on a preferred foliation
(Sec. II F, item “Time”). The positive-frequency split is
taken with respect to this single global t; there is no family
of inequivalent foliations and hence no Bogoliubov freedom.
The per-mode uniqueness of Theorem QM.10 therefore
lifts to a unique global J⋆ on the field phase space. On
a time-dependent background, such as the expanding universe, the one foliation does not remove the Bogoliubov
freedom: J⋆ is time-independent only for the conformally
rescaled field, so it selects the conformal vacuum, and the
instantaneous vacua of other field variables differ from it
by a Bogoliubov rotation.
Proof. About the vacuum the Bogoliubov ambiguity is
parametrized by the choice of timelike Killing/Cauchy
foliation; with a single absolute t the parameter set is a
point. Verified numerically in physical H Jstar.py: a
foliation-mixing angle θ ̸= 0 gives ∥JA − JB ∥ > 0, and
DFD fixes θ = 0 by construction (one foliation). On an
expanding background the rescaled tensor field v = a h has
a time-independent generator in the radiation era, so J⋆ is
its conformal vacuum; the instantaneous vacuum of h at
conformal time η0 differs from it by |βk |2 = 1/(4k 2 η02 ).
Corollary QM.12 (Three ledger rows move imported→derived). Combining Theorems QM.10
and QM.11 with the positivity of the linearized ψ +χ+hT T
spectrum that Theorem QM.10 assumes (argued, not
proved), the free/kinematical structures are forced, not
imported: (i) the field-level complex unit i = J⋆ ; (ii) the
Hilbert inner product ⟨·, ·⟩ = g + i ω with g = ω(·, J⋆ ·);
(iii) unitary free Schrödinger evolution (any J⋆ -preserving
quadratic H generates a one-parameter unitary group).
Interactions do not require their own global complex
structure: they are evolved on the same kinematical
Fock space built from J⋆ . This interaction-picture
step is exact on any finite-mode or preferred-frame
sector where Stone–von Neumann equivalence holds (in
particular the finite C60 microsector, and finite-time
dynamics — DFD’s absolute-time foliation removes the
boost-reconstruction horn of Haag’s theorem, so the
finite-time interaction picture exists as in non-relativistic
many-body theory). On the noncompact R3 matter
continuum, however, the intertwiner V : Ffree → Fint
need not exist (Haag): the interacting vacuum/ordering
map remains imported (Rem. QM.13), a wall shared with
all realistic 4D QFT. QM does not require the interacting
flow to be globally linear. One physical interaction does
break global linearity: the gravitational self-source, where
the classical ψ is sourced by the c-number ⟨ρ̂⟩ = m|Ψ|2 ,
gives the nonlinear Schrödinger–Newton evolution of the
open-problems section (semiclassical self-gravity). DFD
does not take this semiclassical reading: the operator
completion of the same action (App. ZU) forces the linear
side: every completion term is translation-invariant
or proportional to N̂ , so the self-gravity sideband on a
trapped crystal’s center-of-mass mode is exactly zero (all
orders in αψ , zero parameters), and the Schrödinger–
Newton frequency shift is the falsifier of the completion,

350
not a DFD prediction. The free qualifier in (iii) remains
essential in the semiclassical reading above; under
the operator completion, exact unitary linearity is not
confined to the test-particle limit.
Remark QM.13 (Exactly what remains imported—the
residue). This reduces the corpus’s stated “single biggest
gap” (a field-stability theorem forcing a unique global
J⋆ ) to the stability of the linearized spectrum, which is
argued, not proved. It is not a full derivation of QM. Two
items remain, and neither is a free parameter:
1. The value of ℏ is the unit of action: J⋆ fixes the
structure ⟨·, ·⟩ = g + iω, and the absolute scale relating the symplectic ω to the metric g (the canonicalcommutator normalization [q̂, p̂] = iℏ) is a choice of
units, as the value of c is. Every DFD prediction is a
dimensionless ratio, so no theory derives this number.
At the finite-spectral level the Berezin–Toeplitz level
k = 9 fixes a relative ℏeff ∼ 1/k on the internal CP 2
factor.
2. The interacting normalization/ordering map is
the renormalized dynamics common to every quantum field theory: it uses couplings already counted,
scheme conventions carry no physical parameter, and
its non-perturbative construction in four dimensions is
an open mathematical problem shared with the Standard Model.
Net: the QM ledger improves from “field-level i, Hilbert
product, and unitary evolution all imported” to “all three
derived from a unique forced J⋆ , with the Dirac algebra
from the mass term; ℏ is the unit of action, and the
interacting dynamics and the measurement postulate are
those of every quantum field theory.”

5.

Matter–ψ infrared class and dressed
(Faddeev–Kulish) asymptotic states

Theorem QM.14 (Soft-ψ universality class; IR-finite
dressed S-matrix). (i) The matter–ψ coupling of Eq. (14)
makes ψ a massless mediator with Newtonian 1/r tail
(Sec. II C); the static field profile of a hard source is
ψ̃(k) ∝ 1/k 2 , so the field-profile (dressing-kernel) amplitude f obeys |f (k)|2 ∼ 1/k 4 —the infrared class of
soft photons in QED and soft gravitons in perturbative
gravity. (ii) Hence the free→interacting Fock intertwiner
fails for DFD exactly as for QED (van Hove/Haag, universal for massless mediators): the per-mode coherent
displacement scales as k −3/2 , the dressing kernel fails
Hilbert–Schmidt at both ends (∥K∥HS = ∞; IR: massless
ψ, UV: point vertex), and the dressed sector is unitarily
inequivalent to free Fock space. (iii) Because the soft kernel is in the QED class, the standard resolution applies
verbatim: Faddeev–Kulish/Kibble dressed (coherent-state)
asymptotic states, on which interacting dynamics is well
defined and the dressed S-matrix is IR-finite order by
order (Kulish–Faddeev 1970; Kibble 1968; gravity: Ware–
Saotome–Akhoury 2013).

Status of proof. (i)–(ii): computed directly for the linearized point vertex; intertwiner failure uses only
∥K∥HS = ∞ (Haag/van Hove). (iii) is the cited standard
construction, transplanted, not rederived; in particular
the t → ±∞ Møller/FK limit is part of the import.
Remark QM.15 (Ledger: derived / imported / open). Derived: the universality-class identification and ∥K∥HS =
∞ (Thm. QM.14); the intertwiner failure is a shared
feature of massless mediators, not a DFD pathology, so
the FK construction applies to matter–ψ. Imported:
the FK machinery itself (standard QFT technology); and
the dressing reference choice: J⋆ and the absolute-time
foliation do not canonically fix the FK cloud—the ambiguity is per-hard-particle (BMS-supertranslation-type,
at null infinity) and a bulk foliation does not collapse it.
On IR structure DFD sits at exact QED parity. Open:
the nonperturbative interacting vacuum. Absolute time
removes Haag’s boost-reconstruction horn at finite times
(Cor. QM.12), but the interacting vacuum/ordering map
of Rem. QM.13(2) stays imported—the universal Wightman/Haag wall shared with every realistic 4D QFT.

6.

Status box

Status QM.16 (QM-derivation ledger).

351
QM ingredient

status

Internalmicrosector i (=
Kähler J)
Dirac β (4th
anticommuting
gen.)
Nonrel.
Hamiltonian HNR
(Hermitian)

Geometric, inserted
Thm. QM.3
by the polarization
(Prop. QM.5)
Derived: the Yukawa Prop. QM.2
mass block coupling
the two chiralities
Derived
Thm. AN.1

Field-level /
free-Schrödinger i
(= J⋆ )
Hilbert inner
product g + iω
Unitary free
Schrödinger
evolution
IR-finite S-matrix
(dressed sector)

Derived (forced
Thm. QM.10, QM.11
unique J⋆ ), given the
argued stability
Derived (from J⋆ )
Cor. QM.12

Value of ℏ (action
scale)
Interacting ordering/normalization
map
Born rule |ci |2
Single-outcome
selection
Net: full QM
derived?

where

Derived (from J⋆ )

Cor. QM.12

Derivedapplicability (FK
imported)

Thm. QM.14

unit of action (a
choice of units)
shared with every
quantum field theory

Rem. QM.13(1)
Rem. QM.13(2)

conditional
Rem. QM.8
(typicality)
measurement
Rem. QM.8
postulate, shared with
every quantum theory
The free-field core this appendix
(given the argued
stability) and the
Dirac algebra are
derived; the rest is
shared with every
quantum field
theory

The field-level i, the Hilbert inner product, and unitary
free Schrödinger evolution are forced by a unique global
compatible complex structure J⋆ (Thm. QM.10), made
unique on the field phase space by DFD’s absolute-time
foliation (Thm. QM.11) and the positivity of the linearized
ψ + χ + hT T spectrum, which is argued, not proved.
Two items remain, as in every quantum field theory:
ℏ, the unit of action ([q̂, p̂] = iℏ), and the interacting
operator-ordering/normalization map; the Born value is
conditional (typicality), and single-outcome selection is
the measurement postulate.

7.

Falsifier

Falsifier QM.17 (Of the closures and of the obstruction).
The closure claims fail if any of the following hold (each
is checkable in qm derivation check.py):
1. Cl(4, 0) β closure fails if some {ea , eb } ̸= 2δab or some
e2a ̸= +⊮ in the constructed representation. (Checked:
holds.)
2. The obstruction is wrong (a hidden derivation
exists) if either (a) Cl(4, 0) has a central element

squaring to −1 (it does not: dim Z = 1, ω42 = +1), or
(b) ⊮ ⊗ J commutes with every external-plus-internal
observable (it does not: a non-holomorphic C gives
nonzero commutator). Either would upgrade the status
to a real field-level i.
3. The sharpened reduction is vacuous if no DFD
field-stability / coercivity theorem can in principle
2
force Jpf
= −⊮ globally; conversely, exhibiting such
a theorem converts the import to a derivation up to
the scale ℏ and removes the “imported” label from the
field-level i.
4. The J⋆ -uniqueness closure fails (checkable in
physical H Jstar.py) if either (a) more than one of
the 2n sign choices yields a positive-definite Hilbert
metric g = ω(·, J⋆ ·) for a stable physical H (it does
not: exactly 1/2n , and 200/200 random stable Hamiltonians give a unique J⋆ = A(−A2 )−1/2 ), or (b) DFD
admits more than one time foliation so a Bogoliubovrotated JB with ∥J⋆ − JB ∥ ̸= 0 is equally physical (it
does not: absolute time t is a single preferred foliation,
Sec. II F). A genuine counterexample on either point
would reopen the field-level i as imported.
5. The residue is overstated if a forced derivation
of ℏ’s numerical value or of the interacting ordering
map is exhibited from ψ + M7 + α alone—this would
remove an import and would not be an obstruction
but a further result; conversely, if either supposed
“derived” kinematical row (i, inner product, unitary
free evolution) is shown to secretly reintroduce ℏ or an
ordering choice, that row reverts to imported.
A demonstration that the corpus secretly uses the linear
Kähler J and the antilinear Connes JDFD interchangeably
(Rem. QM.4) would also falsify the bookkeeping and must
be corrected, not weaponized.
Remark QM.18 (One-line claim). The QM-derivation program stands at: geometric i in the internal particle sector
is homegrown (the Dirac β is the Yukawa mass block), and
the free-field core—the field-level i = J⋆ , the Hilbert inner
product g + iω, unitary free Schrödinger evolution—is
derived to the extent the corpus establishes (forced by the
unique global compatible J⋆ of Thms. QM.10–QM.11);
what remains is ℏ, the unit of action (Buckingham–
Pi irreducible), and the interacting many-body / full
Hamiltonian-dependent map; the Born value stays conditional (typicality) and single-outcome selection is not
derived.

352
Appendix PK: The DFD-Native Production Galaxy
P (k) Pipeline

Status PK.1 (Numerical run on the one-clock C = 1 background). The window/convolution/covariance pipeline
result below was run only on the explicitly frozen H0 =
72.09, Ωm ≃ 0.274 background. Its mathematical pipeline
and its conclusion on that background (acceptable, not
preferred on the full shape) stand; its printed σ8 = 0.820,
S8 = 0.784, AP distances, and χ2 values are not outputs
of the measured clock (H0early = 67.52). The source scripts
and BOSS covariance inputs are absent from this Overleaf archive, so a full identical windowed rerun cannot
be reproduced here. The current upstream normalization is R8 = 11.848 Mpc, σ8 ≃ 0.8063, Ωm ≃ 0.312,
S8 ≃ 0.822. On the current background the linear shape
differs from Planck ΛCDM by 0.6% in broadband tilt
over k = 0.03–0.2 h/Mpc and by 0.4% in the BAO scale,
against 10% and 7% on the H0 = 72.09 background of the
fit below (computed). The full-shape comparison on the
current background therefore reduces to Planck ΛCDM’s;
a windowed rerun, not yet made, would confirm it.

1.

PK.1

Purpose: closing the pipeline half of item
F

Appendix GR (GR) closed the cosmological
growth/velocity regime question: linear growth is
carried by χ-matter with Q ≡ Geff /G = 1, GR-like,
giving the single primary prediction at the calibrated
As , matched by 32πα5 . On the current measured-clock
background the forward values are σ8 (0) = 0.8063,
Ωm ≃ 0.312, and S8 ≃ 0.822; the 0.820/0.784 and
0.790/0.755 branches belong to the C = 1 background
(see GR.5/GR.7). It explicitly left open the pipeline
obligation of open-problems item F: every prior DFD
power-spectrum confrontation stopped at a linear
Kaiser multipole-ratio inversion (β = f /b from P2 /P0 )
on mock-derived ratios.
None forward-modelled a
window-convolved, covariance-weighted, full-spectrum χ2
against real data.
This
appendix
documents
the
script
(dfd class/dfd pk pipeline.py) that does.
It
takes the frozen DFD cosmology to a predicted galaxy
P (k) multipole vector, convolves it through the published
survey window, and scores it against the real Beutler et al.
BOSS DR12 measurements under a real Patchy-mock
covariance, with a matched ΛCDM run through the
identical code path.

Summary of this appendix
The production pipeline is built, runnable, and
self-validating (ΛCDM recovers χ2 /ν ≃ 1.0 on the
mock-covariance bins), with self-consistent Patchy P0 +P2
mock covariances built for all four BOSS DR12 bins
(NGC/SGC ×z1,3 ; each passes the single-mock-vs-mean
validation at χ2 /dof ≃ 0.55–0.93). On the one-clock
C = 1 background (H0 = 72.09), under a maximal-frozen
protocol (no σ8 renormalization, As =32πα5 frozen,
identical nuisance freedom for both models, fixed
model-predicted Alcock–Paczynski dilation; two
independent reruns agree to all digits): in the clean linear
regime k ≤ 0.15 h Mpc−1 DFD and Planck-ΛCDM are
statistically indistinguishable (DFD χ2 /dof = 1.059,
p = 0.34, vs 0.967); at k ≤ 0.20 DFD trails by
∆χ2 = +21.5 over 124 degrees of freedom (1.311 vs 1.137)
— acceptable but disfavored, at the edge of the ≤ 1.3 band
(PK.11). The deficit is a broadband shape effect: the
turnover sits at slightly too large a scale, the direct
signature of that background’s lower Ωm = 0.274 at
higher H0 = 72.1. Identical 6-term broadband
marginalization for both models reduces the gap to +9.7
(DFD 1.012). An AP-marginalised figure of
χ2 /dof ≃ 1.27 holds only under the retired σ8 →0.790
normalization; the frozen-protocol numbers above are the
result. A false-positive guard confirms AP is a physical
d.o.f. and not a χ2 eraser: a deliberately broken
ωc = 0.05 cosmology stays at χ2 /dof ∼ 28 even with free
AP. Verdict on the C = 1 background:
indistinguishable at k ≤ 0.15; acceptable but
disfavored at k ≤ 0.20 — not a full-shape “loss”,
not a tie. The score is reported as computed at every
protocol level; the sign of ∆χ2 never flips. The
comparison has not been rerun on the measured-clock
background.

a. Neutrino-consistency
cross-check
on
the
C
= 1 background. The maximal-frozen protocol above sets mν
= 0 for both models
(dfd class/dfd pk pipeline.py, CAMB call) —
a symmetric simplification, not a hidden asymmetry. Re-running with each model’s own consistent
neutrino sector (DFD: the normal-ordering chain
of App. X, (m1 , m2 , m3 ) = (2.34, 8.96, 50.16) meV,
Σmν = 61.46 meV, ων = 0.00066; ΛCDM: the Planckbaseline 60 meV, ων = 0.00065; ων carried in the AP
background) gives ∆χ2 = +23.0 at k ≤ 0.20 (DFD
161.4/124 = 1.302 vs ΛCDM 138.4/124 = 1.116; AP-off
base +39.0), with one-sided brackets +20.4 (DFD-only
massive) to +24.1 (ΛCDM-only massive). The frozen
mν = 0 value of +21.5 is therefore ≈ 1.5 χ2 units
generous to DFD, not biased against it; it is kept as
the registered protocol number, with +23.0 as the
physically-consistent cross-check. Consistency anchors:
σ8 (0) shifts 0.8235 → 0.8114 (DFD) and 0.8226 → 0.8110
(ΛCDM, matching the published 0.8111); Ωm : DFD
0.2705 → 0.2718, ΛCDM 0.3138 → 0.3152 (matching
Planck 0.3153); the recomputed AP factors match the
logged run to |∆| ≤ 6 × 10−5 .

353
2.

PK.2

The linear matter transfer function and power spectrum
are produced by CAMB (version 1.6.6) at the frozen C = 1
DFD parameters: H0 = 72.09, ωb = 0.02237, ωχ = 0.1182
(not the derived abundance 0.11928, App. AV) (the χmatter field of Appendix AV supplies the cold component ωcdm ), ns = 0.9667 — the measured-clock background (H0early = 67.52 ± 0.09, ωχ = 0.11928; App. ZS)
moves these inputs, and the stage numbers below are the
C = 1-background runs — with the primordial amplitude
As rescaled so that the realised present-day amplitude
matches the GR appendix normalisation,
As −→ As



σ8target
σ8CAMB (0)

2

,

σ8target = 0.820 (at As = 32πα5 ).

(PK1)
This yields Plin (k, z=0) on 10−3 ≤ k ≤ 2 h/Mpc. The
primary normalization of these runs is the carried σ8 (0) =
0.820 (S8 = 0.784), the value the amplitude 32πα5 gives
on this background (on the measured clock the same
amplitude gives the σ8 = 0.806 of record, App. GR); the
alternative corpus value σ8 = 0.790 (S8 = 0.755) is an
explicitly contingent low-arm (asserted, not re-derived;
GR.5). The BOSS full-shape verdict below is insensitive
to this choice: the deficit is a broadband shape effect
(set by this background’s Ωm = 0.274 and H0 = 72.1),
not amplitude — re-normalising σ8 over 0.79–0.82 moves
the combined χ2 /dof by < 0.03 (PK.8), so the headto-head numbers stand at either normalization. The
matched ΛCDM twin uses Planck-2018 parameters (H0 =
67.36, ωc = 0.12, ns = 0.9649, σ8 = 0.8111). Figure 23
shows the resulting linear spectrum and the Q=1 growth
history; Fig. 24 shows the BAO distance ladder of the
same frozen background (the geometry whose predicted
DV /rd shift reappears as the AP dilation of PK.9/PK.11;
BAO likelihood scored in App. CL).

3.

PK.3

4.

Stage 1 — linear matter power at the
frozen DFD cosmology

Stage 2 — DFD growth carrier (χ-matter,
Q = 1)

Per Proposition GR.4, the linear-growth carrier is χmatter at Q = 1 (GR-like, scale-independent on linear
scales), so the linear growth obeys the standard ODE
d2 D  3 d ln H  dD 3 Ωm (a)
+
+
−
Q D = 0,
Q = 1,
da2
a
da
da
2 a2
(PK2)
on the DFD background H(z). Because Q = 1, CAMB’s
own D(z) at these parameters is this solution; the
pipeline reads CAMB’s redshift outputs directly and extracts f (z) = d ln D/d ln a and σ8 (z). A CAMB arrayordering convention (the σ8 (z)/f σ8 (z) arrays are returned
in decreasing-z order, reversed relative to the ascending
power-spectrum redshifts) is accounted for; the realised
values f (0.38) = 0.681, f (0.61) = 0.763 are consistent
with Ωm (z)0.55 and with the GR appendix.

PK.4

Stage 3 — nonlinear correction

The mildly nonlinear regime is supplied by CAMB’s
HMcode (mead2020) halo-model prescription, Pnl (k, z).
On the fitted range k ≤ 0.20 h/Mpc the boost is modest — rising from ∼ 2% at k = 0.1 to a factor ∼ 4.5
only by k = 1 — but it is included in full rather than
truncated to the linear spectrum. This is a standard (not
DFD-native) nonlinear closure; a DFD-native nonlinear
correction remains open (PK.9).

5.

PK.5

Stage 4 — galaxy redshift-space
multipoles

The galaxy spectrum in redshift space uses linear bias
b1 , Kaiser redshift-space distortions, and a Lorentzian
Fingers-of-God damping:

2
b1 + f µ2
Pg (k, µ) = 
2 Pnl (k) + Pshot , (PK3)
1 + 12 (kµσv )2
with f = f (z) fixed by the model growth (PK.3) and the
nuisances {b1 , σv , Pshot } profiled per bin. The Legendre
multipoles are
Z 1
Pℓ (k) = (2ℓ + 1)
Pg (k, µ) Lℓ (µ) dµ,
ℓ = 0, 2,
0

(PK4)
evaluated by 16-node Gauss–Legendre quadrature, giving
the monopole P0 and quadrupole P2 .
6.

PK.6

Stage 5 — survey-window convolution

The finite survey footprint mixes the true multipoles.
The fine theory vector [ P0 (kfine ), P2 (kfine ) ] (1000 points
each, dk = 0.001) is convolved through the published
Beutler 200 × 2000 window matrix W ,


dmodel
= W200×2000 t2000 ,
t = P0fine , P2fine ,
200
(PK5)
returning the measured 200-vector [P0 (100), P2 (100)] on
the dk = 0.01 grid. The matrix’s undocumented column
layout (input ordered as two multipoles over 1000 fine-k,
not five multipoles over 400) was reverse-engineered and
validated against the Patchy mock mean: a windowed
CAMB spectrum reproduces both the mock P0 and P2 .
Setting W = 0 makes χ2 explode (∼ 2 × 104 ), confirming
the window is wired into the fit.

7.

PK.7

χ2 , covariance, and the validation data

For each cap–redshift bin the score is the profiled,
Hartlap-corrected covariance χ2 over 0.03 ≤ k ≤
0.20 h/Mpc,

⊤ Nmock − p − 2 −1


χ2 = d − m
C
d − m , (PK6)
Nmock − 1

354

FIG. 23. Linear matter power and growth at the frozen DFD parameters of the one-clock C = 1 background with H0 = 72.09
(CAMB forward run at DFD’s inputs, Stage 1–2 of this pipeline; with Qχ = 1 and the polarization confined to bound systems the
linear problem is the standard one, so the Boltzmann code needs no modification; script dfd class/figures/make matter pk.py).
Left: linear matter power spectrum P (k) at z = 0 for DFD versus the Planck-2018 ΛCDM twin, with S8 annotated. Right:
the growth history f σ8 (z) carried by χ-matter at Q = 1 (Eq. (PK2)) overlaid on the BOSS/eBOSS redshift-space-distortion
measurements, with the ΛCDM curve for contrast; the dotted ‘EFE-growth’ curve is an enhanced-growth construction that does
not act on linear perturbations (App. GR), and the panel labels give the C = 1 background’s values.

FIG. 24. BAO/H(z) distance ladder on the one-clock C = 1
background with H0 = 72.09 (CAMB forward run at DFD’s
inputs; script dfd class/figures/make bao hubble.py): the
predicted DV /rd , DM /rd and DH /rd curves, using the derived
rdrag , overlaid on the 12 consensus BAO data points (error
bars), with the Planck-ΛCDM curves for contrast and the
BAO χ2 annotated (likelihood-level scoring in App. CL).

minimised over {b1 , σv , Pshot } (p = number of fitted data
points; the Hartlap factor de-biases a mock-estimated
inverse covariance). The ΛCDM twin is fit through the
identical code (same window, same covariance per bin,
same k-mask, same three nuisances); only the input cosmology differs.

a. Validation data. Real, published Beutler et al.
deconvolved BOSS DR12 galaxy P (k) multipoles
(ps1D BOSS DR12 {NGC,SGC} {z1,z3}, zeff = 0.38, 0.61),
with the published 200 × 2000 survey-window matrices.
b. Covariance status. The published 200 × 200
Patchy covariance has a correct monopole block (its
P0 errors match the mock scatter to ∼ 1%) but a misnormalised quadrupole block (its P2 errors are ∼ 5–45×
smaller than the true mock scatter; a single mock against
the mock mean returns χ2 /ν ≃ 32–47 with it). To
use the quadrupole we rebuild the covariance directly
from the Patchy mock suite. For the legacy-input run
only the NGC z1 mock covariance had been rebuilt (231
mocks; data-vs-mock-mean χ2 /ν = 1.87, ΛCDM bestfit χ2 /ν ≃ 1.0, validating the whole pipeline). The remaining three bins fall back to a P0 -only fit with the
verified-correct published P0 covariance — a standard,
conservative choice that drops the (unusable) quadrupole
rather than trusting a mis-normalised error.

8.

PK.8

Legacy-input run: DFD vs ΛCDM

The run in this subsection uses legacy inputs on the
one-clock C = 1 background. The result of record on
that background is the maximal-frozen protocol result
of PK.1/PK.11 at the derived inputs: ∆χ2 = +21.5
over 124 degrees of freedom (1.311 vs 1.137) at k ≤
0.20. Table CXIX reports the per-bin and combined
χ2 /dof from the legacy run; Fig. 25 overlays the windowconvolved best-fit spectra on the data.

355
TABLE CXIX. [Legacy-input run on the one-clock C = 1
background; the maximal-frozen result of record on that
background is ∆χ2 = +21.5 (1.311 vs 1.137), PK.1/PK.11.]
Full-shape BOSS DR12 P (k) scores from dfd pk pipeline.py
(window-convolved, Patchy covariance, Hartlap-corrected, nuisances {b1 , σv , Pshot } profiled, 0.03 ≤ k ≤ 0.20 h/Mpc). NGC
z1 uses the self-consistent P0 +P2 Patchy mock covariance; the
other three use the published P0 -only block.
DFD χ2 /ν ΛCDM χ2 /ν

bin

z

mult.

Nbin

NGC z1
NGC z3
SGC z1
SGC z3

0.38 P0 +P2
0.61
P0
0.38
P0
0.61
P0

34
17
17
17

1.65
2.14
0.93
1.54

1.01
1.56
0.79
1.07

85 (ν = 73)

1.59

1.09

combined

The combined scores of the legacy-input run are
χ2DFD /dof = 115.8/73 = 1.59,
χ2ΛCDM /dof = 79.2/73 = 1.09,

(PK7)

∆χ2 = +36.5 (legacy input).
The best-fit nuisances are physical for both models (b1 ≃
1.95–2.1, σv ≃ 1–4 Mpc/h). That ΛCDM scores ≃ 1.0
on this real data validates the pipeline; that DFD scores
≃ 1.6 is the adverse result. A diagnostic confirms the
deficit is broadband shape, not amplitude: re-normalising
DFD’s σ8 up to the ΛCDM value 0.8111 does not improve
the fit (1.586 → 1.608), so no bias/amplitude knob can
absorb it — it is driven by the C = 1 background’s lower
Ωm = 0.274 and higher H0 = 72.1.
9.

PK.9

Status: what is production-grade and
what remains

Production P (k) pipeline — Status
Production-grade:
• End-to-end forward model: CAMB linear+HMcode
→ χ-matter Q=1 growth → Kaiser+FoG
multipoles → published 200 × 2000 window
convolution → Hartlap-corrected covariance χ2 ,
run identically for DFD and ΛCDM.
• Validated against real published Beutler BOSS
DR12 multipoles; window layout reverse-engineered
and verified against the Patchy mock mean;
self-consistent P0 +P2 Patchy mock covariances
built for all four bins; on NGC z1 ΛCDM recovers
χ2 /ν ≃ 1.0.
• Reproducible head-to-head score (Eq. (PK7),
Table CXIX), stress-tested under a maximal-frozen
protocol (no σ8 renormalization, As =32πα5 frozen,
identical nuisance freedom, fixed model-predicted
Alcock–Paczynski dilation; two independent reruns
agree to all digits): at k ≤ 0.20 h Mpc−1 DFD
trails Planck-ΛCDM by ∆χ2 =+21.5 over 124
degrees of freedom (χ2 /dof = 1.311, p = 0.012, vs
1.137, p = 0.14) — acceptable but disfavored, at

the edge of the ≤ 1.3 band; at k ≤ 0.15 the two are
statistically indistinguishable (1.059 vs 0.967,
∆χ2 =+7.7, pDFD = 0.34). A “competitive ≃ 1.27”
figure holds only under the corpus σ8 →0.790
normalization, which the frozen protocol forbids;
with frozen As the deficit is real, concentrated in
the NGC bins as a low-k (0.03–0.08) broadband
overshoot (turnover at too large a scale, the direct
signature of the C = 1 background’s lower Ωm at
higher H0 ), with a secondary NGC z1 BAO-band
phase mismatch; high-k slopes are comparable.
Identical 6-term broadband marginalization for
both models reduces the gap to +9.7 (DFD
χ2 /dof = 1.012). The claim is valid only for
k ≤ 0.20: at kmax =0.25 both models fail
(χ2 /dof ∼ 5), i.e. the minimal Kaiser+FoG model,
not either cosmology, breaks first. Not a full-shape
loss; not a tie either. The deficit belongs to the
H0 = 72.09 background: on the current
background the linear shape differs from Planck
ΛCDM’s by 0.6% in tilt (computed), and a
full-shape fit that excludes (Ωm , h) = (0.312, 0.675)
at more than 5σ would falsify it.
Still open (not yet production-grade):
• Quadrupole on all bins — closed. All four bins
carry self-consistent Patchy P0 +P2 mock
covariances (300 mocks each for SGC z1 , NGC z3 ,
SGC z3 ; 231 for NGC z1 ), each validated at
single-mock χ2 /dof ≃ 0.55–0.93 (builder
build mock cov.py). The fixed-template P0 +P2
combined score on the C = 1 background (DFD
1.30 vs ΛCDM 1.11) is narrower than the P0 -only
fallback (1.59 vs 1.09); the maximal-frozen protocol
score above is the result.
• RSD model. The nuisance model is minimal (linear
bias + Kaiser + Lorentzian FoG); a full
TNS/EFT-of-LSS one-loop counterterm treatment,
as in published BOSS full-shape analyses, is not
yet implemented.
• Geometry. The Alcock–Paczynski dilation is folded
in as the fixed model-predicted shift (under the
maximal-frozen protocol above); the remaining
geometry gap is the explicit wide-angle M -matrix
only (the v1 averaged window partially absorbs it).
• Nonlinear closure is standard HMcode, not
DFD-native; and eBOSS QSO (z = 1.52) and the
raw DESI DR1 catalogs (which would need a P (k)
estimator from scratch) are not yet included.
On the C = 1 background the qualitative verdict — DFD
acceptable, ΛCDM preferred on pre-recon full shape, gap
closing as kmax tightens toward the linear regime — is
robust across covariance choices, AP treatment, and
broadband marginalization; the sign of ∆χ2 never flips.

10.

PK.10

Reproducibility

Reproducibility
The pipeline is a single script
(dfd class/dfd pk pipeline.py, CAMB 1.6.6 +

356

FIG. 25. DFD-native production galaxy P (k) pipeline against real BOSS DR12 data. Window-convolved best-fit monopole P0
(and, left panel, quadrupole P2 ) for DFD (red dashed) and ΛCDM (blue solid) overlaid on the Beutler measurements (black/grey
points, Patchy-mock error bars), with residual sub-panels ∆P0 /σ and the ±1σ band shaded. Left: NGC z = 0.38 with the
self-consistent P0 +P2 Patchy mock covariance (the bin that validates the pipeline, ΛCDM χ2 /ν ≃ 1.0). Right: NGC z = 0.61,
P0 -only. DFD tracks the data acceptably but overshoots ΛCDM at low k, the broadband-shape signature of the one-clock C = 1
background’s lower Ωm and higher H0 .

numpy/scipy; the public Beutler et al. BOSS DR12 files
it reads are not in this archive). Run
python3 dfd class/dfd pk pipeline.py
It loads the four BOSS DR12 bins, runs CAMB for DFD
and ΛCDM, fits each bin, and prints the per-bin and
combined χ2 /dof of Table CXIX (∼few seconds). The
companion make fig pk pipeline.py regenerates Fig. 25
(fig pk pipeline.png) by reusing the same fit. The
Patchy mock covariance for NGC z1 is cached in
mock cov cache.npz. Caveat: no χ2 literal appears in
the source; every score comes from a live
scipy.minimize over a live CAMB P (k). Perturbation
checks confirm a real fit (scaling the model, the data, or
the covariance moves χ2 as expected;
W =0 ⇒ χ2 ∼ 2 × 104 ; a deliberately broken cosmology
ωc = 0.05 explodes to χ2 /ν ≃ 21).

11.

PK.11

Falsifier

Production P (k) pipeline — Falsifier
DFD’s prediction is falsified if, on the full-shape galaxy
P (k):
1. Decisive exclusion. A complete full-shape
analysis (all four bins with self-consistent P0 +P2
covariances, plus eBOSS QSO) returns a DFD
χ2 /dof that is unacceptable in absolute terms (≫ 2
with physical nuisances) while ΛCDM remains ≃ 1
— i.e. DFD is not merely worse but ruled out.
2. Shape, not nuisance. The DFD spectrum, with
a TNS/EFT model and AP marginalisation,
cannot be brought to χ2 /dof ≲ 1.3.

357
Appendix YM: Status of the DFD Strong Sector:
Confinement Scale and the Yang–Mills Mass Gap

This appendix gives the theorem-grade status of Density
Field Dynamics with respect to (a) the QCD confinement
scale and (b) the Clay pure Yang–Mills mass-gap problem [189]. The discipline throughout is to state each
result at exactly its true grade: a derived quantity is a
parameter-free consequence of the locked DFD ledger; a
validated quantity is confirmed by executed lattice computation but requires one external scale input; an inherited quantity is computed through the generator-closed
Euclidean/lattice path integral with DFD-fixed inputs,
exactly as in standard lattice QCD, and is not analytically
proved here. The single most important sentence of this
appendix is the last row of Table CXX: the full Clay
pure Yang–Mills mass-gap proof is not claimed; it
remains open.

1.

The confinement scale from the α-tower

The closed-form value of the fine-structure constant
is fixed first, under two stated choices, from the finite
microsector (kmax = 60, the input Ngen = 3, Tr(Y 2 ) =
10), giving α−1 = 137.036 to 21 ppm from the integers;
the bracket coefficient is open (§X). Nothing in the strong
sector is permitted to retune α; it is the fixed input of
every chain below.
Theorem YM.1 (Confinement scale from the selected
α-tower exponent). Given the locked α−1 = 137.036, the
input Ngen = 3, and the Planck mass MP = 1.220890 ×
1019 GeV, the DFD confinement scale and its MS fiveflavour partner are fixed with zero continuous free parameters:
ΛDFD = MP α19/2 = 61.20 MeV,
(YM1)
√
(5)
ΛMS = 4π ΛDFD = 216.95 MeV,
and the running coupling at the Z pole follows by dimensional transmutation,
αs (MZ ) = 0.1187,

(YM2)

landing 0.8σ from the PDG world average 0.1180 ± 0.0009.
The same MP α19/2 chain run to three active flavours gives
ΛQCD,3 = 332 ± 20 MeV, independently cross-checked by
the τ -decay coupling αs (m2τ ) = 0.3186 versus the PDG
value 0.3187 (< 0.1%).
Proof. ΛDFD = MP α19/2 is the locked ledger value of
App. AT, reproduced numerically in App. YM’s companion script: with α = 1/137.036 and MP = 1.220890 ×
1019 GeV,√MP α9.5 = 61.20 MeV to four significant figures. The 4π is the selected proper-time→ MS matching
(5)
factor, giving ΛMS = 216.95 MeV. Dimensional transmu(5)

tation of the four-loop QCD beta function from ΛMS up
to MZ yields αs (MZ ) ≃ 0.118. The three-flavour running
and the τ -sector cross-check are recorded in Ext. Deriv.,

App. AH (τ sector). No step introduces a continuous fit
parameter.
a. Provenance of the exponent 19/2. The exponent
has two convergent internal anchors. First, 19 = 3 + 6 +
10 = h0 (O(1)) + h0 (O(2)) + h0 (O(3)) is the CP 2 linebundle cohomology sum — the same
√ integer 19 that appears in the baryon tower (MN = 19 Λ5 ≃ 946 MeV vs.
57
939, +0.7%). Second, 19
2 = 6 = (kmax − Ngen )/(2Ngen )
57
ties it to the same α count that fixes the exponent in
GℏH02 /c5 = Cα57 (with C measured on the early clock,
C = 1 on bound-matter clocks, Postulate ZS.11).
Status YM.2 (Grade of the confinement scale). α−1 =
137.036 is exact under two stated choices (§X C): the
topological integers alone give it to 21 ppm; the bracket
(5)
coefficient is open. αs (MZ ), ΛDFD , ΛMS , ΛQCD,3 , and
θ̄ = 0 at tree level (strong-CP, with θYM = 0 fixed by
the spectral action given that nothing is added to it for
the gauge fields; no axion) are fixed with no continuous
parameter. One open lemma remains: the exponent 19
2 is
rigidity-selected (the unique catalogue form in the empirical window) and its first-principles derivation is contingent on the cube-law Proposition AT.4, not independently
proved (App. AT, Status AT.24). The confinement
scalp
√
19/2
ing ΛDFD ∝ MP α
and σ ∝ ΛQCD kmax /Ngen are
derivation-grade; the string tension and the glueball mass
are QCD outputs at DFD’s Λ, and their ratios to Λ are
lattice numbers, as in the Standard Model (§YM 2).

2.

The open item: string-tension normalisation

The companion volume (Ext. Deriv., App. AH, notes
to the QCD-confinement and glueball theorems) records,
and we restate here, that the printed string-tension normalisation chain does not close as written:
s
√
332 √
ΛQCD,3 kmax √
20 · 4π MeV
· 4π =
2π
Ngen
2π
= 837.7 MeV ̸= 440 MeV.
(YM3)
The required bridge factor ∼ 1.86 appears in no preceding formula. There is additionally a flavour-scheme
bookkeeping issue: the value 210 MeV used in the glueball
√ chain is numerically the five-flavour MS number
( 4π ×61.20 = 217 MeV), whereas the three-flavour value
is ΛQCD,3 = 332 MeV. The
√ conclusion, already stated by
the corpus, is: the scaling σ ∝ ΛQCD and m0++ ∝ ΛQCD
stand at derivation
grade; the absolute normalisation con√
stants of σ = 440 ± 25 MeV and m0++ = 1.69 GeV are
not closed-form DFD outputs: once DFD fixes Λ, both
follow from QCD, their ratios to Λ being pure numbers
of the Yang–Mills path integral computed on the lattice,
as in the Standard Model. The printed closed-form chain
is not a DFD result and is not used.

358
3.

The validated part: executed SU(3) lattice

The DFD QCD targets were confronted with executed quenched pure-gauge SU (3) Wilson-action runs
(Apps. AQ, AS). The boundary between DFD output
and external input is sharp, and it is the reason for the
“validated, not derived” label.
a. Output (DFD-independent, dimensionless). The
scale-free
string tension at the largest box (204 , β = 6.2) is
√
r0 σ = 1.178, at the lower edge of the standard quenched
window 1.18–1.19.√ The best quenched estimate across
β = 5.8, 6.0, 6.2 is σ = 470 ± 10 MeV at the convention
below. Plaquettes match the benchmark to better than
0.1% at four (β, V ) points; the 0++ glueball is a m =
0.81(8) ⇒ 1.71 ± 0.17 GeV; the Nf = 2 Wilson-HMC
engine passes all algorithmic checks.
b. Input (external, not from DFD). Converting any
dimensionless ratio to MeV requires the Sommer scale
r0 = 0.5 fm. This single external ruler
√ is what the lattice
√
MeV numbers ride on. With it, r0 σ = 1.178√⇒ σ ≃
464.9 MeV, meeting the DFD ledger band σ DFD =
440 ± 25 MeV at its upper edge (∼ 1σ, ∼ 5%).
Corollary YM.3 (Lattice
√ consistency of the α-tower
scale). The ledger target σ DFD = 440 ± 25 MeV, whose
normalisation is not a DFD output (§YM 2), is consistent
at the ∼
√ 1σ/∼ 5% level with the executed quenched SU (3)
result σ ≃ 465–470 MeV (at r0 = 0.5 fm). The lattice
run contains no√DFD input: the lattice fixes the dimensionless ratio r0 σ = 1.178 from first principles, but the
MeV central value depends on the external r0 .
4. Two native gaps: an unconditional
compact-manifold gap and a conditional loaded-frame
gap

The strongest DFD-native step toward a positive lower
bound is the conditional infrared-gap theorem of App. AR,
restated here in status form. It is a real, valid spectral
theorem — and it is correctly scoped well short of the
Clay statement.
Proposition YM.4 (Conditional DFD-loaded-frame infrared gap; correctly scoped). Place spatial Yang–Mills
on the DFD-loaded frame hψ = e2αψ δ. The Hodge–
Weitzenböck identity injects the loaded-frame Ricci tensor into the transverse fluctuation operator LA0 ,ψ . If a
strictly positive physical-sector curvature floor holds (Assumption AR.2: Richψ (a, a) + RA0 (a, a) ≥ ΛDFD |a|2 ),
then by Rayleigh–Ritz
λ1 (LA0 ,ψ ) ≥ ΛDFD > 0 and
√
mDFD-YM = λ1 > 0 (App. AR, Thm. AR.3). On the
compact internal manifold CP 2 ×S 3 the positive Ricci curvature (S 3 : Ric = (2/r2 )g; CP 2 : Einstein with Ric > 0,
b1 = 0, so Bochner kills harmonic one-forms) forces a
strictly positive lowest transverse one-form mode independently of the assumption. This compact-manifold
Bochner gap is unconditional — a separate theorem
(positive Ricci + b1 = 0 on CP 2 × S 3 ), distinct from and

not requiring Assumption AR.2; only the flat-R4 /hadronic
extension is conditional.
Status YM.5 (Why this is not the Clay gap). Proposition YM.4 is valid mathematics and an internal-coherence
result. It falls short of the Clay Yang–Mills mass gap
in three explicit ways: (1) Scale. The realised floor is
geometric/curvature-set:
∼ MP on the internal manifold,
√
∼ α MP ≃ 1018 GeV for the finite Toeplitz spectrum
√
(̸=0)
(App. AP, KK gap λmin ∼ αMP ), and a ∼ 10−30 eV
curvature term on the galactic deep-field annulus — 18
to 39 orders of magnitude from the hadronic confinement scale. It is a UV/curvature floor, not the QCD
gap. (2) Arena. It lives on a curved/loaded space and
bounds a fluctuation operator, not pure Yang–Mills on
flat noncompact R4 with a constructed measure (the Clay
arena). (3) Conditionality. The geometric (Ricci) half is
proved; the nonabelian gauge-algebraic floor sign RA0 is
assumed (Assumption AR.2), not derived. Hence: a real
curvature-driven spectral gap, not the prize.

5.

The ledger

Table CXX collects the derived, validated, and inherited
statuses of the strong sector in one place; the final row
states explicitly that the Clay pure Yang–Mills mass gap
is not claimed.

359
TABLE CXX. Derived / validated / inherited ledger for the DFD strong sector. The final row is the key scope statement: the
Clay pure Yang–Mills mass gap is not claimed.
Quantity

Status

Value / basis

−1

Two stated choices
21 ppm (integers); −0.005 ppm (open c)
Selected (1 open lemma)
61.20 MeV
Selected
216.95 MeV
Selected (from the selected scale)
0.1187 (0.8σ vs PDG)
Selected, τ -checked
332 ± 20 MeV
Derived (tree level; θYM = 0 fixed by strong-CP, structural
the spectral action given that nothing
is added to it for the gauge fields)
√
σ = 440 ± 25 MeV
Target (norm. open)
scaling derived; const not closed
m0++ = 1.69 GeV
Target (norm. open)
scaling derived; const not closed
IR√floor λ1 > 0 (curvature floor) Proved (conditional)
∼ 10−30 eV; not the QCD gap
Validated (r0 = 0.5 fm)
∼ 1σ vs DFD band
r0 σ = 1.178 → 465 MeV
0++ glueball 1.71 GeV
Validated (r0 = 0.5 fm)
overlaps 1.69 target
Nonperturbative confinement
Inherited
generator-closed lattice sector
Clay pure-YM mass gap
Not claimed
open
α = 137.036
ΛDFD = MP α19/2
√
(5)
ΛMS = 4πΛDFD
αs (MZ )
ΛQCD,3
θ̄ = 0 (no axion)

6.

Scope of the claim

QCD / Yang–Mills mass gap: what DFD does and
does not claim
Fixed with no continuous parameter (the
exponent 19/2 selected). The confinement scale is
fixed by the DFD α-tower:
(5)
ΛDFD = MP α19/2 = 61.20 MeV, ΛMS = 216.95 MeV,
αs (MZ ) = 0.1187 (0.8σ vs PDG), ΛQCD,3 = 332 MeV
(τ -checked), θ̄ = 0 at tree level (θYM = 0 fixed by the
spectral action given that nothing is added to it for the
gauge fields; no axion). The Standard Model leaves
ΛQCD a free input; DFD fixes it. Open lemma: the
exponent 19/2 is rigidity-selected (cube-law contingent),
and the overall O(1) string-tension normalisation does
not close as printed (§YM 2).
Native theorem (small). A conditional, tiny geometric
infrared floor on the DFD-loaded frame, meff ∼ 10−30 eV
(Prop. YM.4; App. AR). Structural coherence, not a
hadronic gap.
Validated (external
r0 = 0.5 fm).
Executed quenched
√
√
SU (3) lattice: r0 σ = 1.178 ⇒ σ ≃ 465 MeV (∼ 1σ vs
DFD band), plaquette < 0.1%, 0++ glueball 1.71 GeV,
Nf = 2 Wilson-HMC validated.
Inherited / not claimed. A proof of the Clay pure
Yang–Mills mass gap on flat noncompact R4 ; an analytic
first-principles derivation of all hadron masses; a
continuum/infinite-volume extrapolation.
Nonperturbative confinement is the generator-closed
lattice sector, computed as in standard lattice QCD —
the same “tie/inheritance” status held by String theory
and LQG on the DFD rival scoreboard. The Clay
statement remains open.

7.

Falsifier

Falsifier YM.6 (Strong-sector scale). The confinementscale chain is falsified if any of the following holds, with
α and Ngen frozen at their locked values: (i)√a continuum, infinite-volume lattice determination of σ at the
physical point (with
√ an independently measured r0 ) lands
outside MP α19/2 4π · [2–3-loop running] by more than
the
√ combined uncertainty, i.e. the α-tower scale misses
σ = 440 ± 25 MeV by ≫ 2σ once the normalisation
constant is closed; (ii) αs (MZ ) moves outside 0.1187± its
DFD uncertainty by ≫ 3σ against the world average; (iii)
the strong-CP angle is measured nonzero, |θ̄| > 10−15 ,
excluding the tree-level θ̄ = 0 with its Standard-Modellevel loop floor; (iv) the cube-law lemma underlying 19/2
is shown inconsistent with the microsector count, removing the only anchor for the confinement exponent. None
of these would touch the conditional spectral theorem
(Prop. YM.4), which is a mathematical statement; and
none would constitute a claim on the Clay problem, which
DFD does not assert.
Summary. DFD does not solve the Yang–Mills mass-gap
problem. From α it fixes the confinement scale, tested by
αs (MZ ) = 0.1187 (0.8σ); the string tension then follows
from QCD’s lattice ratios, as in the Standard Model. The
new mathematical result is a curvature-driven positive
energy gap on DFD’s internal geometry, at the Planck
scale rather than the QCD scale.

360
Appendix FM: Charged-Fermion Mass Sector:
Forced Content and the Within-Multiplet
Degeneracy Obstruction
1.

Scope and Summary

This appendix is the calibrated, conservatively stated
status of the charged-fermion mass spectrum. Two machines exist in the corpus: the pure-topology A5 walksum (Appendix √
Y) and the closed-form prefactor table
nf
mf = Af α v/ 2 (Appendix K, Table XCII). They give
very different evidential grades, and the gap between them
is the result.
Summary
The charged-fermion mass spectrum other than the top
and the charm is not derived. What is forced (no mass
input, parameter-free) is: (i)√
the exponent of the
√
electroweak scale v = MP α8√ 2π (its prefactor 2π is
fitted); (ii) the top,
√ mt = v/ 2 (A=1, n=0); (iii) the
charm, mc = α v/ 2 (A=1, n=1); (iv) the spinc
line-bundle exponent ladder nf on CP 2 (removes the
gross hierarchy pattern); and (v) a family of
equal-exponent integer ratios in which α and v cancel and
only locked integers (Nf , b0 , Ngen , dim CP 2 ) enter —
among them mt /mb = Nf b0 = 42 (the top pole mass
over mb (mb ); with both masses at one scale, Mt , the
ratio is 59.7). These remove 2 of the Standard
Model’s 9 charged Yukawas relative
√ to v. The
remaining 7 (the prefactors {2/3, 2, 8/3, 6, 6/7, 1/42}
and the lepton/quark kernel→sector dictionary) are
fitted: their generation↔sector assignment is selected
for consistency with the observed masses, not forced by
the core action. The within-multiplet degeneracy of
Proposition Y.1 is not broken by forced structure.

2.

What is forced

a. Overall scale (exponent derived, prefactor fitted).
The exponent of the Yukawa normalization is fixed with
no mass input,
√
v = MP α8 2π = 246.09 GeV (−0.05%),
(FM1)
v
√ = 174.01 GeV,
2
from the four-sector scale lock (Thm. Z.3).
b. Top and charm (2 species, A=1). With no prefactor,
v
mt = √ = 174.0 GeV (+0.72%),
(FM2)
2
v
mc = α √ = 1.270 GeV (−0.01%).
(FM3)
2
These are the only two absolute masses below the scale
itself that require no data-selected coefficient. They are
forced predictions
√ (the SM predicts neither). The leadingorder mt = √
v/ 2 = 174.0 GeV (+0.72%) is dressed
to (1 − α)v/ 2 = 172.74 GeV (0.52σ) by the (1 − α)

EW/QED factor (App. AT); the refined value is the one
used in the prediction tables.
c. Exponent ladder (gross hierarchy removed). The
bare exponents follow from spinc line-bundle degrees on
CP 2 ,
kf + k H
nbare
=
,
nf = nbare
+ ∆nf ,
(FM4)
f
f
2
e
with kH = +1 (H-coupling) or −1 (H-coupling)
and
degrees (kτ , kµ , ke ) = (1, 2, 4), (kt , kc , ku ) = (1, 3, 6),
(kb , ks , kd ) = (1, 2, 4). A single shift ∆nb = −1 (colorvertex saturation on S 3 ) completes Table XC. The Higgs
localization width εH = Ngen /kmax = 3/60 is derived
(Thm. H.4). This skeleton carries the order-of-magnitude
hierarchy with no mass input.
d. Forced equal-exponent ratios. Whenever two
species share the same exponent nf , the factors α and v
cancel in their mass ratio, leaving a pure number built
from locked integers. These are parameter-free predictions that break the observed near-coincidences at the
few-percent level:
TABLE CXXI. Forced equal-exponent mass ratios.
α
and v cancel; only locked integers Nf =6, b0 =7, Ngen =3,
dimR CP 2 =4, dimC CP 2 =2 enter. No coefficient fit. (The
value is forced; see Sec. FM 3 for why the assignment that
deploys it is not.)
Ratio
Forced value
Origin
Observed Error
mt /mb Nf b√
6
×
7
41.33 +1.62%
0 = 42
√
mτ /mc
2
dimC CP 2
1.399 +1.08%
2
2
me /md 1/Ngen
= 1/9
1/Ngen
0.1094 +1.54%
2
mu /md Ru /Rd = 4/9
dimR CP 2 /Ngen
0.4625 −3.91%
mµ /ms b0 /Nf = 7/6 QCD b0 = 7, Nf = 6 1.1312 +3.13%

Mean |err| = 2.3%, robust across the PDG light-quark
band. The mt /mb = 42 comparison divides the top
pole mass by mb (mb ), where the data give 41.33; with
both masses at one scale, Mt , the ratio is 59.7, from
four-loop QCD running of mb (mb ) = 4.18 GeV with
the same pole mass converted to MS at four loops and
αs (MZ ) = 0.1180 (two-loop conversion and running give
59.4, and the directly measured MS top mass, 162.5 GeV,
gives 59.0 at two loops and 59.5 at four; computed), so it
awaits the same one-scale test as the double ratio D.
e. The determinant-free Nc3 double ratio (the circular
scale cancels). The weakness of the absolute-mass construction is that the overall within-multiplet enhancement
ηq is the cube root of the observed lepton/quark mass
determinant (Sec. FM 3), i.e. circular. That circularity
can be algebraically removed. Because the Ldet generation
weights (+1, −1, 0) are traceless, the double ratio
(md /me ) (mb /mτ )
= Nc3 = 27,
(FM5)
(ms /mµ )2
has the imported scale enter with net exponent 1−2+1 =
0, so ηq cancels identically, and the surviving inputs are
Nc = 3 and the traceless O(3) spinc weight set. Because
ηq runs, it is common to the three generations only at one
D ≡

361
renormalization scale: there the data give D = 33.1 ± 0.8
between 2 GeV and the top mass and 29.3 ± 0.7 at Λtop ,
while PDG’s mixed convention gives 27.7, a value that
moves from 21 to 31 with the light-quark reference scale
(App. Y). The weights (+1, −1, 0) give the sibling ratio
(mµ /ms )/(me /md ) = 9 at any scale, against 10.35 ± 0.15
2
observed; the value (b0 /Nf )Ngen
= 21/2 = 10.5 comes
from the prefactor table, not from the weights. The
circular scale is removable, and the texture is a pattern,
not a precision identity; the absolute light masses remain
open (Sec. FM 3).
Internal-consistency note. The identity D = Nc3 = 27
is exact for the Ldet traceless weight assignment. Reconstructing D instead from the explicit closed-form prefactor table of App. K returns D = 28.26 (+4.7% from 27).
The residual confirms that those prefactors are leadingorder selections rather than an exact realization of the
determinant-free identity — consistent with the 1.42%
mean mass error and the four-discrete-bit selection caveat
documented in App. K.
3.

(b) A forced modulus-splitting kernel — not supplied. Distinct phases are not distinct masses.
The required object is a forced, color-/hyperchargedependent weight coupling the family irrep to the
SU (3)c ×U (1)Y factor (a Georgi–Jarlskog-type Clebsch). The bare A5 family rep carries no color index, so it cannot, by itself, produce the residual
mµ /ms ≃ 1.13 split. That color-weighted Clebsch
is the exact, minimal, forced object that DFD does
not yet possess.
Why the 7/6 does not count as a forced break.
The corpus reaches mµ /ms = b0 /Nf = 7/6 (Table CXXI)
only because the strange quark is assigned As = 6/7 and
the muon Aµ = 1. That value is forced; the assignment
(which species carries the Qd -channel) rests on the lepton/quark kernel→sector dictionary, which Appendix Y
itself states is “not derived from the core DFD action.”
Under pure topology (the A5 walk-sum of Appendix Y,
the no-fit ceiling) the muon and strange land in the same
exponent bin and are exactly degenerate, with the muon
∼ 23% from the observed value. Therefore:

The within-multiplet degeneracy: theorem-grade
obstruction

Degeneracy verdict
The within-multiplet/cross-sector µ–s degeneracy is not
broken by forced structure. The 7/6 value is real, but it is
deployed through a fitted generation↔sector assignment.
The forced ceiling for the muon is 23.4% (pure-topology
A5 walk); the +2.7% table value is achieved only with
the asserted dictionary and is not a parameter-free result.

We promote Proposition Y.1 to a theorem and identify the exact missing object. The mechanism is Schur’s
lemma.
Theorem FM.1 (Minimal-connector mass degeneracy).
Let the charged-fermion Yukawa be the gauge-equivariant
b on the minmatrix element of the bare Higgs connector H
imal internal module M6 (C) with the (3, 2, 1) block strucb intertwines irreducible
ture (Appendix Y). Because H
gauge factors, HomG (VL , VR ) is one-dimensional (Schur),
so the Yukawa restricted to any fixed gauge multiplet is a
scalar. Hence every state in one gauge multiplet shares
one mass: no species splitting is possible from the minimal connector. Splitting requires dim HomG ≥ 2, i.e. the
gauge irrep must appear with multiplicity ≥ 2 carrying a
non-scalar Yukawa block.
Proof. The within-multiplet element factorizes as
Ea5 E56 = Ea6 = h2 for every color a ∈ {1, 2, 3} (Appendix Y, Eqs. (Y17)–(Y18)), independent of the species
label f beyond multiplet type. By Schur’s lemma an
intertwiner between irreducibles is a scalar multiple of a
fixed isomorphism; the only freedom is that scalar, which
is species-blind. □
Theorem FM.2 (Minimal forced object required to split
the spectrum). To lift the degeneracy of Theorem FM.1
within forced structure, two ingredients are needed:
(a) A family index with distinct phases — supplied.
The A5 three-dimensional irrep restricted to a Z3
(3-cycle) splits with eigenvalues {1, ω, ω 2 } (tr =
χ3 ((123)) = 0). This is the rank-3 generation projector and it lifts the gauge-only degeneracy at the
level of phases.

4.

Scorecard

Table CXXII grades every species. “Forced?” is yes
only if the absolute mass requires no data-selected prefactor (A=1) and no asserted sector assignment.
TABLE CXXII. Charged-fermion
scorecard. Predicted from
√
mf = Af αnf v/ 2 (Appendix K); errors vs. PDG central
values. “Forced?” marks parameter-free, assignment-free
predictions.
f
t
c
b
τ
s
µ
d
u
e

nf
0
1
0
1
1.5
1.5
2.5
2.5
2.5

Af
1
1
1/42
√
2
6/7
1
6
8/3
2/3

Predicted
174.0 GeV
1.270 GeV
4.143 GeV
1.796 GeV
92.98 MeV
108.5 MeV
4.749 MeV
2.111 MeV
0.528 MeV

Observed
172.76 GeV
1.270 GeV
4.18 GeV
1.777 GeV
93.4 MeV
105.66 MeV
4.67 MeV
2.16 MeV
0.511 MeV

Forced?
yes (+0.7%)
yes (−0.0%)
no (−0.9%)
no (+1.1%)
no (−0.5%)
no (+2.7%)
no (+1.7%)
no (−2.3%)
no (+3.3%)

The
√ full-table mean |err| is 1.45% (App. K’s table, with
v/ 2 = 174.1 GeV and its observed values, gives 1.42%),
but this is achieved with seven data-selected prefactors. A look-elsewhere check shows that an alphabet

362
√
of {small rational × 2a × rational} prefactors with halfinteger α-powers yields, on average, ∼4–5 candidate (A, n)
pairs within 2% of any target mass. A per-fermion 2%
hit is therefore statistically cheap, and the closed-form
table is best read as “structured but not forced” for the
light species.

5.

Reading the related appendices

• Appendix K: the G, Qd , Ru,d , Dℓ factors are each
derivable in isolation, but they are applied with perspecies on/off switching and a generation↔sector assignment that is selected against the masses. Count:
2 of 9 forced (t and c relative to v).
• Appendix Y: the statement “µ/s degenerate at
4.7×10−4 , muon 23.4% off” is the correct forced ceiling; it is the no-fit result, not a defect that the table
overrides. The Ldet = O(3) twist of Appendix Y,
Sec. Y 9 (Prop. Y.2) supplies the pattern-level structure behind the within-multiplet spread (3ngen from
Nc = 3), but its normalization ηq is imported and
the sign dictionary is open, so it does not alter
this appendix’s “not parameter-free / forced-ceiling
muon 23.4%” conclusion.
• Appendix AT: its closed-form ratios miss data by
1–8% and are the same prefactors relabeled via Λ5 .
Status

Status FM.3 (Charged-fermion mass sector). Forced
(beyond the SM, no mass input):
the exponent
√
8
of the overall
scale v = √MP α 2π (prefactor fitted);
√
mt = v/ 2; mc = αv/ 2; the CP 2 spinc exponent
ladder; and
√ the equal-exponent ratios mt /mb = 42,
mτ /mc = 2, me /md = 1/9, mu /md = 4/9, mµ /ms =
7/6 (mean |err| ≈ 2.3%; the mt /mb comparison divides the top pole mass by mb (mb ), and at one scale,
Mt , the ratio is 59.7). Yukawas removed: 2 of
9 at forced grade (t and c relative to v), plus the
gross hierarchy pattern. Open / not forced: the
absolute light/intermediate masses√e, u, d, µ, s and b, τ
require the prefactor table {2/3, 2, 8/3, 6, 6/7, 1/42}
and the lepton/quark kernel→sector dictionary, whose
generation↔sector assignment is fitted. Degeneracy:
the within-multiplet µ/s degeneracy of Theorem FM.1
is not broken by forced structure; forced-ceiling muon
error is 23.4%. Minimal missing object: a forced color/hypercharge-weighted family Clebsch (Georgi–Jarlskog
type), Theorem FM.2. Until it is derived, the spectrum
other than the top and the charm is open, and “masses
derived” is not claimed.

Appendix LS: Line-of-Sight Optical-Screen
Amplitude: Full 2-Halo and Realization Computation

Status LS.1 (Scope). The halo/realization calculation
below is retained as a conditional model of local or
anisotropic refractive structure. It is not a global, probeselective shift of CMB/BAO/SN distances or H0 : exact conformal invariance leaves θ∗ unchanged, a homogeneous mode is pure gauge, and the current inhomogeneous mean is far too small. All numerical runs using
H0 = 72.09, Ωm = 0.274, and σ8 = 0.820 belong to the
C = 1 background. No statement below may be used to
resurrect the optical-screen Hubble reconciliation of the
Extended Derivations or a global peak-location screen;
the local/anisotropic correlation observable stands.

1.

LS.1

Statement and Scope

Under the retired optical-distance dressing (App. SL)
the DFD optical metric (n = eψ ) would make the apparent
distance to a background source depend on the matter the
line of sight (LOS) threads, through a refractive screen
Z
D
E
1
(χ),
∆ψBAO (z) =
dχ WBAO (χ) κstruct (χ) µ(g/a
0)
µ(x) =

x
,
1+x

√
a0 (z) = 2 α c H(z),

(LS1)
where κstruct is the projected matter fluctuation correlated
with the galaxy tracer and the factor ⟨1/µ⟩ is the DFD
weak-field (deep-MOND) boost in the splashback/filament
skin, with x = glocal /a0 . The retired global-screen program on the one-clock C = 1 background (Appendix AU
and the four-case decomposition) required this screen to
carry an amplitude ∆ψBAO ≃ 0.030 to move θ∗ by +1σ
while holding the BAO fit (χ2 /dof = 0.78); the joint
late-time data prefer a slightly larger ≃ 0.039.
A single-halo NFW+splashback toy returns an excess
≃ 0.034 with un-tuned ingredients, bracketing the target
without fixing it to three digits. This appendix computes
the amplitude with two independent realistic-structure
computations and reports the result with a full, conservative error budget.
Scope and non-claims
No parameter is tuned to 0.030 or 0.039. Every
ingredient is a standard, pre-registered literature object
(Tinker08 mass function, Tinker10 bias, Duffy+08
concentration, standard
√ HOD, CAMB P (k), DFD
cosmology, a0 (z) = 2 α cH(z)). The target appears only
in printed ratio lines. We report whatever the calculation
gives, central value and band, including the regimes
where it falls short.

363
2.

LS.2

The two factors and what is firm

The amplitude factorizes as ∆ψBAO = κstruct × Q, with
Q ≡ ⟨1/µ(g/a0 )⟩. The calculation settles the first factor
and exposes the second.
a. κstruct
(firm). In
Strand
1
(los screen 2halo.py) the projected matter convergence is a Limber RMS of the halo-model power
spectrum Pmm = P1h + P2h , with P1h the NFW-u(k)
one-halo term integrated over the Tinker08 mass function
and P2h the Tinker10-bias two-halo term times the linear
P (k). This derives σκ = 0.0027 (one-halo share 81%)
and the effective galaxy bias beff
g = 2.00 (HOD-weighted
Tinker10, not set), giving
κstruct = beff
g rcorr σκ = 2.00 × 0.70 × 0.0027 = 0.0038
(LS2)
(one-halo 0.0031 plus two-halo 0.0007). The lognormal realization (Strand 2, los screen mock.py) independently measures σκ = 0.0027 ± 0.0004 on the resolutionindependent 4 Mpc/h-smoothed convergence map at a
mock seed σ8 = 0.79 (the screen output σκ ∼ 0.003
is normalization-insensitive, so this seed value does not
propagate into the prediction; the current measured-clock
forward normalization is σ8 = 0.8063), and a BOSSanchored estimate (Strand 3) gives the per-LOS RMS
0.0029. All three methods agree:
κstruct ≃ 0.003–0.004 is robust and measured, not assumed.

b. Q (governed by an undetermined floor). The local self-acceleration in the filament/splashback skin is
deep-MOND (gself /a0 ∼ 0.01–0.2), so Q is dominated by
whatever floors the argument of µ – the MOND externalfield-effect (EFE) value xext = gext /a0 . The action does
not fix xext at LOS-skin scales: Appendix AE shows
the FRW background gives |∇ψ̄| = 0 (cosmological deepMOND, xext → 0), while Appendix
AC adopts the Hubble√
EFE hypothesis xext = 1/(2 α) ≃ 5.85 (named there as
a testable hypothesis the action does not produce). The
LOS screen lives between these two corpus-named regimes.

3.

LS.3

Derived amplitude: central value and band

a. Fiducial (deep-MOND, xext → 0). At the cosmological deep-MOND floor that Strand 1/2 use by default
(x = max(g/a0 , 10−4 )), the full 1h+2h halo model gives
a ρ(trace)-weighted boost Q = 13.5 and
fid
∆ψBAO
= 0.052,

band over pre-registered variations = [0.035, 0.071].
(LS3)
The lognormal realization independently returns 0.040 ±
0.004. The band is built by varying one standard axis
at a time over its literature range, summarized in Table CXXIII.

TABLE CXXIII. Strand 1 halo-model amplitude at the cosmological deep-MOND floor (xext → 0), varying one pre-registered
axis at a time. Nothing tuned; the target appears only in the
final ratio column.
axis

value

beff
g

κstr

Q

∆ψ

mass function
Tinker08
2.00 0.0038 13.5 0.052
mass function Sheth–Tormen 2.03 0.0039 13.5 0.052
concentration
×0.7
2.00 0.0038 14.5 0.055
concentration
×1.4
2.00 0.0039 12.6 0.049
splashback Rsp
1.0 R200m
2.00 0.0038 9.2 0.035
splashback Rsp
2.0 R200m
2.00 0.0038 18.5 0.071
HOD log Mmin
12.0
1.73 0.0033 17.2 0.057
HOD log Mmin
13.0
2.29 0.0044 11.3 0.050
infall slope
r−1.0
2.00 0.0038 13.8 0.053
infall slope
r−2.0
2.00 0.0038 13.2 0.051

b. The
floor
sensitivity. Strand
3
(halo los 2halo efe.py) holds κstruct fixed and
maps the amplitude as a function of xext with two
independent boost estimators (a halo-profile integral and
a lognormal-PDF path integral), which agree across the
grid. The result (Table CXXIV) is a ∼10× swing:
TABLE CXXIV. Screen amplitude vs the undetermined EFE
floor xext (κgal = 0.0052 from the BOSS-anchored estimate).
The amplitude reaches 0.030 only for xext ≲ 0.11 and 0.039
only at xext ≃ 0.03; at the corpus Hubble floor it is 5× short.
xext

regime

Q

∆ψ

→ 0 cosmological deep-MOND (App. AE) 11.7–13.5 0.040–0.052 (Strands 1–2)
0.031
in-range edge
7.5
0.039
0.107
in-range edge
5.8
0.030
0.24
screen-module floor
4.1
0.021
5.85
Hubble-EFE (App. AC)
1.17
0.006

At the corpus screen-module floor xext = 0.24, the band
over all un-tuned structure ingredients is [0.017, 0.033] –
the target 0.030 sits at the upper edge, and 0.039 is not
reached. The dominant uncertainty is therefore not the
halo ingredients (they move the answer by only ±0.006)
but the order-of-magnitude swing from the undetermined
xext .

364
4.

LS.4

What the 2-halo calculation settles

Summary
The 2-halo calculation confirms the machinery and
the structure factor (κstruct ≃ 0.003–0.004, three
independent methods), and it improves on the toy by
replacing a hand-set κgal = 0.006 and a single profile with
a derived Pmm = P1h + P2h and a population-averaged
boost. But on the main question – “is the amplitude
comfortably ≥ 0.030?” – it weakens the clean
single-halo reading. The single-halo toy’s in-range result
was an artifact of an implicit xext = 0. The full
calculation shows that the amplitude is a steep function
of the undetermined EFE floor: it brackets 0.030 only for
xext ≲ 0.11 (a narrow weak-field corner), does not
robustly reach 0.039, and falls 5× short at the
corpus-named Hubble floor.

The result, with the floor stated explicitly:
∆ψBAO = κstruct × Q(xext ) ∈ [ 0.006, 0.052 ] as xext : 5.85 → 0

(LS4)
(with κ between 0.0038 and the BOSS-anchored 0.0052),
in range (≃ 0.020–0.030) only for the weak void-skin
floor xext ≃ 0.1–0.24. The amplitude is not pinned by
structure; it is pinned by the EFE floor, which the corpus
does not derive at LOS-skin scale. Figure 26 shows both
the halo-model build-up and the floor sensitivity.
Reproducibility
All numbers from dfd class/, deterministic (camb ≥1.6,
dfd params.py: H0 = 72.09, α = 0.00730, Ωm = 0.274,
Ωχ h2 = 0.120):
• los screen 2halo.py – Strand 1, halo model
(∼6 s): fiducial 0.052, band [0.035, 0.071],
beff
g = 2.00, σκ = 0.0027.
• los screen mock.py – Strand 2, lognormal
realization: 0.040 ± 0.004, σκ = 0.0027 ± 0.0004,
⟨1/µ⟩HOD = 11.7.
• halo los 2halo efe.py – Strand 3, EFE-floor
scan: amplitude 0.052 → 0.006 as xext : 0 → 5.85;
band at xext = 0.24 is [0.017, 0.033].
• make fig los screen.py – regenerates
fig los screen.png.

Status LS.2 (LS: LOS optical-screen amplitude). Full
2-halo: yes – Pmm = P1h + P2h from Tinker08 MF +
Tinker10 bias + Duffy+08 c(M ) + analytic NFW u(k),
with a cross-checking lognormal realization; not a relabeled single-halo toy. Tuned: no. Structure factor
κstruct : firm, ≃ 0.003–0.004. Amplitude: not pinned by
structure – governed by the undetermined EFE floor xext ;
central spans ≃ 0.006–0.052. Brackets 0.030: yes, but
only marginally and only for xext ≲ 0.11. Reaches 0.039:
no, except at the near-divergent deep-MOND edge. Net
effect of the 2-halo calculation: confirms machinery,
improves rigor, weakens the clean “comfortably exceeds
target” claim. The amplitude is settled only up to the

EFE floor.
Falsifier LS.3 (LS). The screen mechanism is falsified
if either (i) the structure convergence is shown to be
σκ ≪ 0.003 at z ≃ 0.5 (it is currently 0.0027–0.0029 across
three methods), removing the first factor; or (ii) the EFE
floor at LOS-skin scale is pinned ≥ 2 (e.g. the corpus
Hubble-EFE value 5.85 is shown to apply to cosmological
filaments), collapsing Q ≤ 1.5 and ∆ψBAO ≤ 0.008, ∼ 4×
below the target. The sign test of Appendix SL has been
decided against the optical-distance dressing on which
this screen rests.

365

FIG. 26. Left (a): cumulative LOS screen ∆ψBAO (< χ) from the full halo model at z ≃ 0.5, decomposed into 1-halo (dark) and
2-halo (light), reaching the fiducial 0.052 in the deep-MOND (xext → 0) limit; target 0.030 (dashed) and data-preferred 0.039
(dotted) marked. Right (b): the sensitivity – amplitude vs the undetermined MOND external-field floor xext . The amplitude
collapses by an order of magnitude between the cosmological deep-MOND regime (xext → 0) and the corpus Hubble-EFE floor
(xext = 5.85); the shaded band is the narrow corner where 0.030–0.039 are met.

Appendix CL: Publication-Grade CMB Likelihood
and Bayesian Model Comparison
1.

CL.1

Scope and the publication-grade primary
CMB

Remark CL.1 (Scope: the measured-clock result). DFD’s
CMB result is the no-screen fit of App. ZS, not the fittedscreen confrontation recorded in this appendix: with the
spectrum-shape inputs carried by derived or adopted-andpriced structure (ωb at its conventional-point value on the
adopted Axiom E-transport′ , its coefficient a calibration,
ωχ = 0.11928, the flux-period value, Σmν , Neff , and the
tilt ns = 1 − 2/60 (a stated number; the mechanism is
open); per-input grades in App. ZS) and two calibrations
free (the measured clock coefficient, H0early = 67.52 ± 0.09,
and one overall amplitude), DFD fits Planck plik-lite with
no θ∗ screen at all: 100 θ∗ = 1.04110, fair ∆χ2 = +3.20 for
2 fewer effective parameters with the ωb coefficient counted
(AIC −0.8, BIC −9.6; −2.8 and −16.1 if it is not), sustained out-of-sample by SPT-3G (preferred, −0.77) and
ACT DR6 (+5.94, below the 3σ falsification threshold).
The fitted A∗ = 0.0908 screen below, and every number
computed at the C = 1 background (H0 = 72.09), are
not current results. The current dependent clustering
outputs are R8 = 11.848 Mpc, σ8 ≃ 0.8063, Ωm ≃ 0.312,
S8 ≃ 0.822, and Aϕϕ = 0.9887 (−0.80σ). Thus every later
unqualified 0.820/0.784 or AL = 1.0542 statement in this
appendix belongs to the C = 1 likelihood record, not the
measured clock. There σ8 = 0.820 is the carried normalization, the value that the amplitude As = 2.08 × 10−9
(the value of 32πα5 , a numerical match to the calibrated
amplitude; App. FK) gives on that record’s background

(H0 = 72.09, Ωm = 0.274). On the measured clock the
same amplitude and the same χ carrier give the σ8 = 0.806
of record (App. GR); the two values differ through the
background alone.
This appendix records the real-data confrontation of
DFD with the standard cosmological probes at likelihood
grade and the Bayesian model comparison against ΛCDM.
The primary-CMB step is publication-grade: the Planck
2018 high-ℓ plik lite TT/TE/EE likelihood is evaluated
with its full 613 × 613 Fortran binned covariance (not a
diagonal first pass), with the Planck calibration Aplanck
profiled, and the pipeline is cobaya-validated : feeding
the screened DFD bandpowers through cobaya’s own
PlanckPlikLite.get chi squared reproduces the hand-rolled
χ2 to the digit (ΛCDM 587.6, DFD 610.9 at Aplanck = 1),
and the pipeline returns the published ΛCDM plik lite
value as a unit test.
The DFD cosmology is run on its C = 1 frozen background parameters (not the measured clock, Rem. CL.1:
H0 = 72.09, ωb = 0.02237, ωχ h2 = 0.1199, ns = 0.9667,
As = 2.08 × 10−9 , τ = 0.054) with the optical θ∗ screen of
the retired global-screen reading (App. ZS) applied as the
θ∗ multipole rescale at one fitted amplitude A∗ = 0.0908;
no refit of the background. The screen amplitude is fitted,
∗
not
√ derived: the fitted
√ A = 0.0908 sits −26.6% off the
2 α-gate value A = 2 α Rd /(1 + Rd ) = 0.0667, and the
frozen gate value does not reproduce the fit — evaluated at
the consistent (CAMB) convention it leaves θ∗ misplaced
at +12.2σ and costs plik lite ∆χ2 = +202/ + 172 (per-bin
variants). A “frozen-amplitude validation” claiming the
gate value works as well as the fit does not hold: it rests
on a cross-pipeline calibration transplant (a toy pipeline
miscalibrated by −0.32% in absolute θ∗ compared against
a CAMB-convention measurement). The result is θ∗ as a

366
2.

CL.2

Parameter count

ΛCDM carries six free parameters: ωb , ωc , θ∗ (or H0 ),
As , ns , τ . DFD fixes some of these from its tower and
calibrates the others (current grades: App. FK):
• ns = 1 − 2/60 = 0.9667 (a stated number; the
mechanism is open),
• ωb h2 = 0.0222730 (baryon deposit on the adopted
Axiom E-transport′ ; +1 priced discrete bit; the α4
power is derived and the coefficient is a calibration,
Ext. Deriv. App. AH),
FIG. 27. DFD forward-computed CMB TT power spectrum
Dℓ = ℓ(ℓ + 1)Cℓ /2π on the one-clock C = 1 background
(H0 = 72.09, ωb = 0.02237, ωχ = 0.12, ns = 0.9667, As =
2.08 × 10−9 ), with no θ∗ screen (the annotated DFD 100 θ∗ =
1.05478 is the unscreened value) versus the Planck-2018 ΛCDM
reference. This is the DFD-CLASS/CAMB forward run of
the likelihood pipeline documented in this appendix (CL.1;
script dfd class/figures/make cmb tt.py): the first three
acoustic peaks, the third-to-first peak height ratio, and 100 θ∗
are annotated for each model.

one-parameter fitted screen. A forward re-derivation (conventions pinned by first reproducing the anchors of this
paragraph, including the frozen-gate +12.2σ and the fitted
A∗ → +0.0σ) exhausted the derivable-amplitude menu√at
the consistent CAMB
√ convention: the drag-loaded 2 α
gate (+12.2σ), the fb loading (+11.7σ drag / −1.5σ
saturated), the AQUAL f 0.72 /f 0.79 kernels (+14.6σ to
+26.1σ), the ungated screens
√ (−41.7σ/−75.4σ), and inserted gate multipliers (× 2 → −2.0σ, ×4/π → +2.9σ,
×2 → −22.1σ, ×π/4 → +19.6σ, × 12 → +29.3σ) all fail
to land on A∗ ; the
√ near misses (the saturated-loading
gate at −0.7σ, × 2 at −2.0σ) are menu selections —
a convention branch and an inserted multiplier — and
promoting one to a derivation would incur exactly the
trials charge √
of App. CE. The forced boost identity
1/µ − 1 = 2 α checks exactly (residual 0) but is zindependent, so it cannot generate the screen’s ΩΛ (z)
loading shape, which enters as a separately inserted
kernel. A single-amplitude scan confirms the trade-off:
the gate value A = 0.0667 reproduces the BAO sector
(χ2 = 9.5/9.3, matching the corpus 9.4) while misplacing
θ∗ at +12.2σ, whereas the fitted A∗ = 0.0904 lands θ∗ at
+0.0σ but degrades BAO to 12.6/13.3 (SNe essentially
unchanged, 1390.2 → 1388.6): one amplitude cannot
serve both sectors from the gate. On the frozen background with the single fitted amplitude DFD attains
χ2 /dof = 0.97 (593.5/613, amplitude-profiled) versus
ΛCDM 0.96 (585.5/613): θ∗ lands at +1.0σ and ΛCDM
is marginally preferred by ∆χ2 = +23.3/ + 8.0 (per-bin
variants) over 613 degrees of freedom (statistically minor).
DFD is an excellent absolute fit to the full Planck spectrum. Figure 27 shows the unscreened forward run on
that background.

• ωχ h2 = 0.11928 (the finite SU (2)60 Chern–Simons
vacuum relic with the flux-period identification, the
stated count 158, the asserted premise P4′ and two
stated conventions; −0.60σ),
• H0early = 67.52 ± 0.09 (the α57 invariant at the
measured clock coefficient C; the exponent is a
theorem, the coefficient a calibration),
• the screen amplitude Ascreen (fitted, A∗ = 0.0908),
used only by the global-screen confrontation
recorded in this appendix, not by the no-screen
fit of App. ZS.
The DFD side’s continuous calibration inventory is four:
the clock coefficient C (absolute scale), the As overall
coefficient (the 32π normalization), the ωb coefficient, and
the reionization optical depth τ , an astrophysical nuisance
shared with ΛCDM (App. FK, Rem. FK.8). The table
below uses kDFD = 3 (C, As coeff, τ ), which omits the
ωb coefficient; the count of App. FK is kDFD = 4, and
5 with ωχ ’s stated conventions charged as a knob. A
maximal skeptic who rejects every derivation drives the
count to kDFD = 8. In every count but the skeptic’s, DFD
frees fewer parameters than ΛCDM, which is the source
of its parameter-economy (BIC) credit. On the Planck
plik-lite likelihood alone, where the amplitude and τ enter
only through As e−2τ , the effective count falls to 3 against
ΛCDM’s 5 (App. ZS). The fit numbers tabulated below
are computed at the C = 1 background, not the measured
clock, as stated above.

3.

CL.3

Bayesian model comparison

Table CXXV collects the recorded per-sector best-fit χ2
(all from real data: plik lite full covariance, 12 BAO consensus points, the 1580-SN Pantheon+ cosmology sample
with full stat+sys covariance, and the full Planck 2018
nine-bin lensing bandpower likelihood) and the modelcomparison statistics
BIC = χ2 + k ln N,

AIC = χ2 + 2k,

(CL1)

for the two data combinations: (A) CMB+BAO+SNe
(no lensing) and (B) CMB+BAO+SNe+lensing. Deltas
are ∆ ≡ (DFD) − (ΛCDM), so a negative ∆BIC favours
DFD.

367
2
TABLE CXXV. Bayesian model comparison, DFD vs ΛCDM, on real Planck+BAO+SNe+lensing data.
√ Per-sector χ are
∗
recorded best-fit values; the θ∗ screen carries the single fitted amplitude A = 0.0908 (−26.6% off the 2 α-gate value 0.0667;
the frozen gate value fails at +12.2σ, ∆χ2 = +202/ + 172, and a “validation” of it does not hold — background not refit).
∆ ≡DFD−ΛCDM. Combination (A) excludes the Planck CMB-lensing bandpowers; (B) includes them. The from-scratch CAMB
recompute at the carried σ8 = 0.820 gives a flat CLϕϕ ratio, AL = 1.0542 ± 0.0263, ∆χ2lens = +3.9 (χ2 = 12.66/9, +2.06σ pass),
so Combination (B) at the carried σ8 is the Combination-(A) near-tie.

N χ2DFD χ2ΛCDM

∆χ2

CMB plik lite (full 613×613 cov)
613 593.5 585.5
BAO (12 consensus pts)
12
9.4
13.2
Pantheon+ SNe (1580, full cov)
1580 1390.4 1387.2
Planck lensing (9-bin, carried σ8 =0.820; CL.4)
9 12.66
8.7

+8.0
−3.8
+3.2
+3.9

Sector (real data)

Combination (A): CMB+BAO+SNe
joint χ2
joint BIC

(N = 2205, kDFD = 3, kΛCDM = 6)
2205 1993.3 1986.0 +7.3 (∼ +20 profiled)
2016.4 2032.2
−15.8

Combination (B): CMB+BAO+SNe+lensing (N = 2214, kDFD = 3, kΛCDM = 6)
(B, carried σ8 =0.820, flat CLϕϕ , CL.4):
joint χ2
2214 2006.0 1994.6
+11.4
joint BIC
2029.1 2040.9 −11.8 (mildly DFD)

a. Combination (A), no lensing. The joint χ2 gap is
small (∆χ2 = +7.3 per-sector, ≈ +20 in the amplitudeprofiled pipeline, on N ∼ 2.2 × 103 ). DFD’s parameter
economy (3 fewer free parameters, a BIC credit of 3 ln N ≃
+23) outweighs that gap: ∆BIC(A) = −15.8 (kDFD = 3)
to −8.1 (kDFD = 4). Both are negative: on the one-clock
C = 1 record DFD is competitive with, and on
parameter economy mildly favoured over, ΛCDM
on CMB+BAO+SNe. The no-lensing ∆BIC sign is
borderline and prior/count dependent; the proper Laplace
evidence ln B(DFD/ΛCDM) ≈ −1.2 to +0.8 makes this
an evidence-level tie, not a clean DFD preference.
A bare
√
“strong DFD” reading of the BIC is a 1/ N shortcut and
is not claimed.
b. Combination (B), with CMB lensing. At the carried σ8 = 0.820 (CL.4, from-scratch CAMB): the
CLϕϕ ratio is flat (below ΛCDM, shape scatter 0.3%),
AL = 1.0542 ± 0.0263, ∆χ2lens = +3.9 (χ2 = 12.66/9,
+2.06σ pass), so the lensing sector does not swamp the
parameter-economy credit; Combination (B) reverts to the
Combination-(A) near-tie (∆BIC(B) ≈ ∆BIC(A) , mildly
DFD-favoured on parameter economy, an evidence-level
tie). σ8 = 0.790 is not what the amplitude As = 32πα5
gives and is not simultaneously available with the lensing
pass.
c. The σ8 amplitude on the C = 1 background: a
shared data anomaly. The DFD amplitude chain —
As = 32πα5 (a numerical match to the calibrated amplitude; App. FK), ns = 1 − 2/60 = 0.9667, the C = 1 background’s H0 = 72.09 and Ωm = 0.274, gravity-universal
growth — yields σ8 = 0.820 and S8 = 0.784 with no
further freedom (forward CAMB at the canonical inputs:
σ8 = 0.8193, S8 = 0.7843). On that background this
would resolve the cosmic-shear tension (χ2 = 0.73 vs
ΛCDM’s 10.15; +0.4–1.3σ vs +3–4.9σ) at the cost of a
mild Planck ϕϕ clean-window pull (+3.9 χ2 ; ϕϕ alone
prefers σ8 ≈ 0.842). This lensing-high/shear-low split is

internal to the data: on identical clean-window data no
single-amplitude model fits both — the joint fit carries
an irreducible residual of ≈ 3.9 χ2 (∼ 2σ) at that background’s Ωm and ≈ 13.6 χ2 (∼ 3.7σ) at Planck’s Ωm —
and DFD’s mid value sits within 1.2 χ2 of the joint optimum while ΛCDM pays ≈ 4.9 over its own floor (27.3
vs 13.9 in total). (Scope: the ∼ 50/9 figure from the
fiducial-referenced likelihood is an artifact of applying the
CMB-linear correction — anchored to the Planck fiducial sky — to DFD’s unscreened theory TT outside the
correction’s regime (per-bin shifts up to 2.5σb , 1.0–2.6×
beyond the ±2σ ΛCDM-posterior envelope). Evaluated
with Planck’s own CMB-marginalized consext8 likelihood
— the prescribed variant for non-fiducial backgrounds —
DFD gives χ2 = 14.0/9 (AL = 1.067 ± 0.029, +2.3σ) versus ΛCDM 9.3/9: the same shared-anomaly regime as the
clean-window channel, closing the lensing item without appeal to internal judgment. DFD’s residual AL pull (+2.3σ
vs ΛCDM’s +0.8σ) remains within the shared-anomaly
interpretation; the CMB-lensing falsifier above stands.)

4.

CL.4

Summary assessment

Summary assessment (two-sided)
On the one-clock C = 1 record DFD is competitive
with ΛCDM on CMB+BAO+SNe (joint ∆χ2 ∼ +20
over 2205 points in the recorded pipeline; a near-tie) and
is favoured on parameter economy. CMB lensing.
A CMB-lensing penalty (∆χ2 = +232, AL = 1.40,
“ΛCDM decisively favoured”) does not hold. The correct
reconciliation is two separate facts, not a single inflation
factor: (a) the +232 lived at the input σ8 = 0.790 (an
alternative corpus normalization, not the one
As = 32πα5 gives) and was further distorted by a
normalisation bug in the inherited cobaya-path pipeline;

368
and (b) the pass belongs to a different input — the
amplitude As = 32πα5 , a numerical match to the
calibrated amplitude, whose CAMB-native σ8 = 0.820
gives a flat CLϕϕ ratio. (We do not quote a single “×N ”
deflation of the +232: dividing it by any factor still
leaves a fail; the resolution is that the pass is a different
σ8 arm, not a rescaled 0.790 arm.) An independent
reconstruction of the Planck 2018 9-bin lensing likelihood
from the official bandpowers/covariance/windows (no
cobaya), validated to reproduce the published ΛCDM
χ2 = 8.7/9 and AL = 1.0, and a direct CAMB recompute,
both give: at the carried σ8 = 0.820 the lensing power is
a flat deficit below ΛCDM across L = 8–400 (shape
scatter 0.3%: pure amplitude, no shape penalty); the
nine-bin likelihood returns AL = 1.0542 ± 0.0263, a mild
+2.06σ lean that passes (χ2 = 12.66/9, ∆χ2 = +3.9).
The joint accordingly reverts from “decisively ΛCDM” to
the near-tie. The cause of the (mild) deficit is structural
— the C = 1 background’s high H0 = 72.09 forces low
Ωm = 0.274 and its χ-matter growth is GR-like (Q ≃ 1),
so CLϕϕ ∝ Ωm σ82 runs ∼ 4% low — but it is mild, not a
catastrophe. The cost of using the carried σ8 = 0.820:
S8 = 0.784, so the “S8 -low” galaxy-lensing result of
App. GR softens to a mild match (KiDS-1000 +1.0σ,
DES-Y3 +0.5σ). The sharper σ8 = 0.790/S8 = 0.755 is
not simultaneously available with this pass: at σ8 = 0.790
the deficit grows to ∼ 11% and the lensing likelihood
disfavours DFD at > 3.6σ. Because DFD’s growth is
Q = 1, one σ8 sets both arms, so the lensing pass and the
S8 = 0.755 value cannot be claimed together.

A CAMB-only comparison (one spectrum evaluation
per model, no sampler) makes the cause explicit: across
the entire Planck lensing band L = 8–400 the DFD CLϕϕ
curve sits slightly below ΛCDM, by a flat amount (shape
scatter 0.3%). The signal-weighted lensing-amplitude
ratio is
ADFD
L
=
AΛCDM
L

(
0.956
0.890

(σ8 = 0.820, As = 32πα5 ),
(σ8 = 0.790, S8 = 0.755, alt. corpus norm.),

(CL2)
a ∼ 4% deficit at the carried normalization (growing to
∼ 11% only at the alternative σ8 = 0.790). Because DFD
sits flatly below, the data prefer AL = 1.0542 ± 0.0263
to match it ⇒ a small positive χ2 penalty, +2.06σ, a
mild pass at the carried σ8 = 0.820. A σ8 Ω0.25
proxy
m
would let low Ωm over-compensate low σ8 through its
weak matter weighting; the full bandpower kernel weights
matter correctly, and the direct CLϕϕ ratio is used (AL =
1.0542, +2.06σ pass).

5.

CL.5

Reconciliation of the item-4 S8 prediction

Remark CL.2 (The carried S8 = 0.784 is a mild match;
S8 = 0.755 is not a result). DFD’s structure-growth
prediction at the carried normalization (As = 32πα5 ,
σ8 = 0.820, Ωm = 0.274; on the measured clock σ8 =

0.806 and S8 = 0.822) is
p
p
S8 = σ8 Ωm /0.3 = 0.8193 0.2749/0.3 = 0.784.
(CL3)
This is a mild match to galaxy weak lensing on the low
side: KiDS-1000 (0.759 ± 0.024) at +1.0σ and DES-Y3
3 × 2pt (0.776 ± 0.017) at +0.5σ — both within ∼ 1σ, a
mild match. Because DFD’s growth is Q ≡ Geff /G = 1
(LCDM-like), this same σ8 = 0.820 also sets the CMBlensing amplitude, where it gives a flat AL = 1.0542
(+2.06σ pass, CL.4). The galaxy-lensing match and the
CMB-lensing pass are therefore simultaneously available at one σ8 = 0.820 — there is no two-sided contradiction at the carried normalization.
S8 = 0.755 is not
p a DFD result: it is just σ8 = 0.790
restated (S8 = σ8 Ωm /0.3), and σ8 = 0.790 is an alternative corpus CMB-normalization (asserted, not re-derived;
it requires shrinking As = 32πα5 by 7.2%). At σ8 = 0.790
the CMB-lensing deficit grows to ∼ 11% and the ninebin likelihood disfavours DFD at > 3.6σ. Since one σ8
(Q = 1) sets both arms, the sharper S8 = 0.755 galaxylensing number and the CMB-lensing pass cannot be
claimed together. The single prediction at the carried
normalization is the S8 = 0.784 mild match with a +2.06σ
CMB-lensing pass. On the C = 1 background DFD is a
mild low-side match to galaxy lensing and a mild pass of
CMB lensing, jointly.

6.

CL.6

Status and falsifier

Status CL.3 (CMB-likelihood package). Publicationgrade: primary CMB (plik lite TT/TE/EE, full 613 × 613
covariance, Aplanck -profiled, cobaya-validated); Planck
2018 nine-bin lensing bandpower (cobaya native CMBlikes, ΛCDM-validated at χ2 = 8.7/9, AL = 1.004±0.024);
BAO (χ2 /dof = 0.78); Pantheon+ SNe (full stat+sys covariance). Remaining for full publication maximality: (i)
the full plik TT/TE/EE with explicit foregrounds (we use
the cobaya-validated plik lite); (ii) the BAO full covariance
(we use consensus points with quoted errors); (iii) nestedsampling Bayesian evidence (PolyChord/MultiNest not
available here) — the proper Laplace evidence is the substitute and gives an evidence-level tie; with CMB lensing
at the carried σ8 = 0.820 (+2.06σ flat-ratio pass, CL.4)
folded in, the joint stays a near-tie.
Falsifier CL.4 (CMB lensing — mild pass at the carried
normalization). On the one-clock C = 1 background the
amplitude As = 32πα5 (a numerical match; σ8 = 0.820,
Ωm = 0.274, S8 = 0.784, GR-like Q = 1 growth) gives
a flat CLϕϕ ratio below ΛCDM across L = 8–400, i.e.
AL = 1.0542 ± 0.0263, a mild +2.06σ pass of the Planck
2018 nine-bin lensing likelihood (χ2 = 12.66/9, ∆χ2 =
+3.9). The falsifier has not fired at the carried
normalization. The remaining item is not a fired falsifier
(App. GR, GR.7): the “one σ8 sets both arms” lock rests
on χ clustering Newtonianly (Q = 1) on linear scales,
which the adopted Rest-Mass Channel gives (App. GR).

369
If instead the AQUAL screen leaked into linear growth, Q
would run with z and CLϕϕ would shift. This connects to
the long-standing “needs-clustering / effective-CDM” wall
of the third-peak analysis; it is the deciding dark-sector
growth item, not a present falsification.

Appendix FK: The Fitted Cosmological Knobs:
Status and Closure

This appendix states, in conservative terms, exactly
which cosmological coefficients DFD derives and which it
still fits. It collects three threads — the scalar amplitude
As , the χ relic abundance, and the global knob count
— and grades each against ΛCDM’s six free parameters
(ωb , ωc , θ∗ , As , ns , τ ). The arithmetic of every claim below
is reproduced in dfd class/fitted knobs analysis.py
(numpy only; no sampler).
Of the six ΛCDM inputs, DFD fixes the tilt ns =
1 − 2/60 as a stated number, the tilt of the ripples of
its stated smooth beginning (App. Q); the mechanism
is open, as it is for As (Prop. FK.4). It derives the α4
power of a second, the baryon density ωb h2 , and the B−L
current that carries its bias, but not its coefficient: with
the stated carrier, the mechanism gives ηB of about (2–
3) × 10−9 , three to five times Planck’s value (Ext. Deriv.
App. AH), so the coefficient is a calibration. It forces the
exponent of the expansion rate H0early (the α57 hierarchy;
its O(1) coefficient rests on one measured input, the vacuum coefficient R = ΩΛ C = 0.6042 ± 0.0027, of which
the clock C = 0.87784 ± 0.00266 is today’s early-clock
value, proven underivable from the printed corpus by the
eight-route no-go and independent of the normalization
of G, Rem. AP.6), and calibrates a third, the amplitude
As , which the formula 32πα5 matches at −0.6 to −0.8σ,
depending on the Planck data combination, without a
derivation of its power or its coefficient. What it does not
derive are the vacuum coefficient R, the amplitude As and
the coefficient of ωb h2 , and it carries the reionization optical depth τ , an astrophysical quantity shared with ΛCDM.
R, As and τ are normalizations: none of them sets the
shape of any observable DFD predicts (Rem. FK.8); the
ωb coefficient does, through the baryon loading of the
acoustic peaks.

1.

The scalar amplitude As : one calibrated input

Proposition FK.1 (The formula 32πα5 in the de Sitter
dictionary). In the single-field de Sitter dictionary
As =

H⋆2
8π 2 ε

2
W M̄P

=

1
V
,
2
24π εW M̄P4

(FK1)

with the primordial (Wheeler–DeWitt) scale on the locked
Majorana rung H⋆ ∝ MR = MP α3 (App. AT 1; no
printed stage holds the universe at that scale, Prop. FK.3)
and the one-loop QED measure εW = α/(4π), the exponent collapses exactly to
As = 32π α5 = 2.080 × 10−9

(−0.82σ from Planck),
(FK2)
the power α5 = (α3 )2 /α1 being machine-rigid (sympy
residual 0). The pair (H⋆ , εW ) that reproduces (FK2) is
a single-parameter
choice, equivalently the normalization
√
H⋆ /MR = 8π, equivalently εW = α/(4π), equivalently

370
the inflationary potential V = 3 α6 MP4 .
√
√
8π MR =
8π MP α3
Proof. Substituting H
√⋆ =
and M̄P = MP / 8π into (FK1) gives As =
(8πMR2 )/(8π 2 εW M̄P2 )
=
(MP2 α6 )/(πεW M̄P2 )
=
6
5
(8 α )/εW ; with εW = α/(4π)
this
is
32πα
.
The
√
three statements H⋆ /MR = 8π, εW = α/(4π), and
V = 3α6 MP4 are algebraically interchangeable given
(FK1) and H⋆2 = V /(3M̄P2 ); fixing any one fixes the other
two. Hence exactly one normalization degree of freedom
remains. All four identities are verified to ≥ 20 digits in
the companion script.
The rigidity is algebraic: it holds inside the de Sitter
dictionary at H⋆ . No stage in DFD freezes perturbations
at H⋆ (Prop. FK.4), so neither the power α5 nor the
coefficient 32π is a derived property of As , and 32πα5 is
recorded as a numerical match, one of about 225 grammar
candidates inside Planck’s one-sigma band (App. AT).
a. εW = α/(4π) is not derived. Two corpus routes
purport to fix εW ; neither is a derivation.
• Inversion route. Writing εW = H⋆2 /(8π 2 As M̄P2 ) =
α/(4π) recovers εW by assuming the target As =
32πα5 . This is explicitly circular and carries no
independent content.
• “Magnetic-dual” route. Setting εW = α2 αM /π with
1
the dual coupling αM = 1/(4α) gives α2 · 4α
/π =
2
α/(4π) identically: the α cancels αM ’s 1/α, leaving
the universal one-loop QED measure α/(4π) with
no surviving trace of the value 1/(4α). The same
number appears for any αM paired with its matching prefactor; the dual sourcing is a relabelling of a
known constant, not a slow-roll computation.
Theorem FK.2 (No native inflaton). The compactification CP 2 × S 3 supports no canonically normalized scalar
with a slow-roll inflaton potential. Six independent obstructions each suffice: (i) Lichnerowicz rigidity removes
graviton/metric-moduli candidates; (ii) the truncated microsector carries no Kähler, size or shape modulus (no
light flat direction), and on a continuous internal space the
volume is not held either: with the connection Laplacian
that fixes α it runs away without end, and with the Dirac
operator volume and shape sit at a maximum with η = −2
(Prop. FK.3); (iii) the only light pseudoscalar, χ (the
b3 = 1 three-form zero mode), is a compact pNGB with a
periodic—not slow-roll—potential; (iv) shift symmetry forbids the required non-periodic V (ϕ); (v) no worldvolume
scalar carries a trans-Planckian field range; (vi) the wouldbe εW reduces identically to the one-loop measure α/4π,
a relabelled constant rather than a potential slope. (Full
proof: Extended Derivations, App. AH series IV–VI.) Nor
do the nine further routes derived from the printed structure (Proposition FK.3) supply an early accelerating stage
with an exit or a source of the primordial perturbations.
Theorem FK.2 establishes that DFD has no native slowroll scalar on CP 2 × S 3 (six independent failure modes,

and nine derived routes that close). With no inflaton
M̄ 2
potential there is no V (ϕ) whose slope ε = 2P (V ′ /V )2
can be computed; εW is therefore a fitted number, not a
derived slow-roll parameter.
Proposition FK.3 (No derived early stage and no derived source of the primordial perturbations). Inside the
window that reheating to the baryogenesis temperature
TL = 1.89 × 1012 GeV (App. AV) and the χ relic leave
for any accelerating stage, 5 × 106 ≲ H ≲ 4 × 1010 GeV
(Eq. (Q13)), nine routes derived from the printed structure
give no stage with an exit and no source of the primordial perturbations (computed). (a) The spectral action.
With the plateau cutoff every internal-mass field is conformally coupled and the Einstein-frame vacuum
is lowest
√
at zero internal mass, where H ≳ 2M̄P / f0 at N = 60
and no exit exists; on a continuous internal
space
√
√ the
volume runs away along e−λϕ , λ = 14/3 < 2, for
the connection Laplacian (endless acceleration, ns = −6,
r = 12.4), or sits with the shape at a maximum with
η = −2 for the Dirac operator. (b) The Higgs. The spectral action fixes its coupling to curvature at the conformal
value ξ = −1/6 for any cutoff; its potential in
√ the canonical Einstein-frame field ĥ, 9λH M̄P4 sinh4 (ĥ/ 6 M̄P ), has
ε ≥ 4/3 everywhere, so it never inflates, and Higgs inflation would need ξ = +1.7 × 104 and an H about 330
times the relic ceiling. (c) χ and the Chern–Simons sector. A full winding of χ is a gauge identification and the
level-60 space ends at the half-period edge, so there is no
monodromy branch; χ is stable (E = 0), so it is no curvaton; as the whole signal it would be pure isocurvature;
steering reheating, its sector measure gives S/ζ ≥ 1.05
against Planck’s 0.031; a descent through its sectors costs
at least 2.9 × 1016 GeV per step against a de Sitter temperature below 1010 GeV; and no other Chern–Simons
or flux mode of the theory carries a stage. (d) Thermal and holographic fluctuations. The early plasma’s
heat capacity is extensive with no limiting temperature,
thermal contact on the matter metric is causal, and a
global thermal initial state leaves white noise, ns = 4
at As ∼ 10−82 ; warm-plasma and holographic-cell variants reach the amplitude only with two fitted numbers.
(e) The trace anomaly. The anomaly of DFD’s own fields
drives de Sitter only at H ≃ 5M̄P , above the cutoff; below
about 3M̄P the Planck-strength stiffness of ψ dominates
its four-derivative terms (below about 16M̄P the R2 term,
the only one with time derivatives), and the matter metric
has no inhomogeneous conformal mode. (f ) Anisotropic
scaling. No field acquires the Lifshitz exponent z = 3 at
high energy: the preferred foliation adds no operator with
more than two spatial derivatives, the a4 Weyl term with
the gravitomagnetic shift keeps z = 1, and the deep-field
µ-law gives ψ an amplitude-dependent z = 3/2 only in
the infrared, which freezes no mode in the radiation era.
(g) Relaxation of the internal spectrum. If the 60 eigenvalues of DF were dynamical, every stationary spectrum
of the spectral action would have one nonzero level inside
the plateau (at most two beyond it for the cutoffs tested),

371
so the 57-level spectrum of the theory is not a vacuum of
the action; free eigenvalues would leave it within a few
Planck times or, with the a0 term kept, collapse to DF = 0
in an eternal Planck-scale de Sitter phase, and the α57
rungs are self-consistent only at k = 0 and k = 57. The
spectrum is held by the rigidity premise (Theorem Q.12),
so nothing relaxes. (h) The E-transport clock. The step
A5 = Q d ln A is locally pure gauge (F = 0): it carries no
stress-energy or entropy and fires on the local e-fold count
of the matter metric, so it follows the curvature perturbation rather than creating it, and equal steps would give
ns = 1. (i) The Majorana sector. The heavy-neutrino
mass MR = MP α3 is an identification (App. P), not the
value of a field, so the sector stores no energy that could
drive a stage and end it; in the plasma the three heavy
neutrinos drop out near T ≃ MR /2.5 and lower the pressure there by about 1% (w ≥ 0.330), so the expansion
keeps decelerating (computed). The fast-light past is the
ordinary expansion read on the arena clock, too slow to
open the horizon (Rem. Q.11). Every working stage found
needs a field the theory does not contain, so the primordial
perturbations and the uniformity behind the horizon problem have no derived mechanism: they enter through one
stated starting condition, the smooth beginning of App. Q
(§XVIII A).
b. The tensor ratio does not share this knob. The
dictionary ratio 16εW = 4α/π = 9.29 × 10−3 is, by definition, the same single knob viewed twice, so it could never
corroborate εW , and it is not the observable B-mode ratio.
The primordial tensor ratio is bounded by the χ relic,
r ≲ 10−7 , independently of εW (App. SM, Cor. SM.4).
Proposition FK.4 (Status of As ). As is one calibrated
amplitude, as in ΛCDM, and its source is not in the
printed field content. A source that shares the optical
cone with the tensors has r = 8F 2 /(λ2 M̄P2 ), where F
is its decay constant and λ its conversion to curvature
√
(Cor. SM.4). Read on that cone, ψ has F = M̄P / 2 and,
on the power-law engine in the radiation era (p = 12 ),
1
3
2 < λ < 2 , so r > 1.7. At the rate the χ relic allows
(Eq. (Q13)) the observed amplitude needs a light field
with F/λ ≲ 1.5 × 1014 GeV that converts to radiation;
the only light field with so small a decay constant, χ, is
stable dark matter and seeds isocurvature alone, and DFD
has no reheating stage through which another field could
convert. On its own cone ψ never leaves the horizon:
its characteristic speeds stay below 1.07 c (Prop. III.7)
while the background it sees decelerates (Thm. Q.9), so
its fluctuations keep the blue spectrum of the initial state,
ns = 3. The de Sitter dictionary returns 32πα5 only
at H⋆ , where one e-fold moves χ through four periods
(Thm. SM.3); at the allowed rate it gives As ≲ 7 × 10−15 .
The source of As is therefore open, and As is counted
as one calibrated input, the amplitude that the smooth
beginning of App. Q states.
A derivation of As needs a source of one of two kinds: a
light field on the optical cone with F/λ ≲ 1.5 × 1014 GeV

and a conversion to radiation, or an accelerating stage on
ψ’s own cone, which needs a vacuum energy DFD does
not contain. Either would fix As and ns together.
2.

The χ abundance normalization: from open
overshoot to derived-with-conventions

Removing the misalignment angle θi (no inflaton) appears to fix the amplitude at the classical uniform-circle
variance ⟨θ2 ⟩ = π 2 /3, giving a factor-≈ 44 overshoot,
Ωχ h2 ≃ 1.62 · π 2 /3 ≈ 5.3. That is not the relic: π 2 /3 is
the classical-continuum measure, the wrong vacuum for a
finite topological mode. Step 5b of App. AV (Thm AV.13)
replaces it with the finite SU (2)60 CS/WZW vacuum expectation of the normalized quadratic Casimir,
k/2
X

C2 (j)
= 0.073
k(k
+ 2)
j=0
√
⇒ Ωχ h2 = 1.62 κrad 2π ⟨θχ2 ⟩ = 0.11928

⟨θχ2 ⟩ =

|S0j |2

(FK3)

(−0.60σ from Planck),
√
where κrad = Γ( 14 )2 /(6 3π) = 0.4026 is the exact
radiation-era onset√ factor (the sudden approximation
owed it) and the 2π is the flux-period identification
(Lemma AV.6: P = M̄P α3 is the flux period, frad = P/2π;
App. AV). A caution: the 45× reduction from π 2 /3 = 3.29
to 0.073 is not a finite-Hilbert-space effect — it factorizes
as 39.4784 (a radians2 →turns2 units conversion, 4π 2 )
× 1.17922 (a continuum measure choice) × 1.00092 (the
finite-k effect itself, i.e. 0.09%) × 0.96774 (Sugawara k/K),
per Eq. (AV14) (App. AV). The amplitude ⟨θχ2 ⟩ is the
correct period-1 (turns2 ) flux variance, a topological deliverable, not a fitted knob.
Proposition FK.5 (The relic is derived with stated conventions (Finite SU (2)60 CS Vacuum Relic), not a fitted
3
knob). With
P α the flux period (frad = P/2π),
√ P = M̄
13
mχ = 2π 158 M̄P α = 32.005 eV, and χ the compact S 3
CS/WZW flux mode at k = 60, the relic amplitude (FK3)
is forced: the operator is the unique invariant quadratic
Casimir (the geodesic angle-squared ψ 2 is non-invariant
— nonzero Haar-character coefficients for every j ≥ 12
— and over-produces by 35–150×); the measure is the
modular S-matrix vacuum row |S0j |2 , given the asserted
premise P4′ (Lemma AV.27); and the k(k + 2) denominator is the Sugawara level k + 2 = k + h∨ times the
bare CS level k, locked by C2 (k/2)/[k(k + 2)] = 14 (the
k-independent Z2 half-period edge). No correction for
an overshoot — late entropy dilution ∆ ∼ 40, a revised
mχ /fχ rung, anharmonic/string-network corrections, a
temperature-dependent mass, a shared As normalization,
a stress-tensor projection — is needed: there is no overshoot to cure. The edge θmax = 12 is fixed by χ’s derived
Z2 (χ → −χ) orientation-oddness (the same Z2 giving
NDW = 1), and the lock C2 (k/2)/[k(k + 2)] = 14 then
makes k(k + 2) the unique k-independent
denominator;
P
K = k + 2 is the Sugawara shift ( j |S0j |2 = 1 for every

372
integer K, so K is set by the shift, not normalization).
Given these two stated conventions (Rem. FK.6), the amplitude ⟨θ2 ⟩ = 0.073 is forced. The single non-DFD input
is the relic-map prefactor 1.62 — the standard radiationera H(T )/entropy redshift that every DM relic (incl.
ΛCDM’s) carries; it is operator-agnostic, so applying it to
the flux amplitude is consistent. The relic is therefore
derived, conditional on the flux-period identification, the stated count 158 and P4′ (Lemma AV.6;
the Finite SU (2)60 CS Vacuum Relic, Ωχ h2 = 0.11928,
−0.60σ), not a fitted knob. Its radiation-era rate is DFD’s
own: the Friedmann law of the Einstein–Hilbert term
(§XIX E) supplies H(T ) with the same G. χ has no photon coupling (E = 0, App. AV), so it has no decay or
conversion line.
a. No convention band. A “∼ 9–44×, conventiondependent” overshoot follows from the oscillation-onset
convention of the 1.62 prefactor (3H = m vs H = m)
applied to the wrong amplitude ⟨θ2 ⟩ = π 2 /3. With the
period-1 flux amplitude ⟨θ2 ⟩ = 0.073 and the exact onset
factor κrad replacing the sudden approximation, the band
collapses to Ωχ h2 = 0.11928 (the convention ambiguity is
resolved: κrad is the derived onset). There is no overshoot
left to be convention-dependent about.
Remark FK.6 (Where the knob went: eliminated, not
relocated). Step 5b eliminated the θi knob outright —
with no inflaton no angle is chosen, and the amplitude is
the topological finite-vacuum expectation. The powers of
α in the mass and decay constant are derived (the count
158 and the stiffness are stated inputs); the abundance is
also derived, conditional on the flux-period identification,
P4′ and the two stated conventions above, not an open
overshoot and not a relocated fitted knob. The path
from “Derived” to “proven” is the two first-principles
items of Prop. FK.5/Rem. AV.14: a native derivation of
the 1.62 prefactor and a CS symplectic-form derivation
that θmax = 12 and K = k + 2 are forced. We do not
manufacture a fix beyond those stated conventions.
√
Remark FK.7 (The 158 multiplicity: one stated input,
and why it is not 4π). The χ-mass exponent (α13 , from
the see-saw
√ 16 − 3 = 13) is derived to theorem grade; the
prefactor 158 is one stated discrete input. It is read as
3(kmax − dim M7 ) − 1, the three-dimensional weak block
times the 53 = kmax − dim M7 block, less one zero-mode,
but no single counting rule gives it: the uniform rule, all
modes less the zero modes, gives 171, and every count
invariant under a symmetry acting irreducibly on the weak
block is a multiple of 3, which 158 is not (App. AV). The
count 171 would give mχ = 33.30 eV and Ωχ h2 = 0.1217,
a 2% shift that measurements of the cold-matter density
can separate. The prefactor is not to be replaced by
the loop factor 4π (16π 2 = 157.91) it sits 0.05% from: a
multiplicity is an integer, 16π 2 is transcendental, and the
swap is motivated only by making “v 2 = 2 mχ P ” exact,
a back-solve. The electroweak–dark link is v = √
4πΛ
(Rem. AV.9), whose 4π is the fitted Higgs prefactor 2π
in another form; the burden of any future revision is on

deriving the integer count, not on the 0.027% coincidence.
3.

Knob count: DFD vs ΛCDM

Table CXXVI grades all six ΛCDM-equivalent cosmological inputs. The count of fitted or measured
continuous DFD coefficients is four: the vacuum coefficient R (today’s early-clock value C), the amplitude As , the coefficient of ωb h2 and τ (Rem. FK.8),
against ΛCDM’s six. (ωχ h2 is not in this list: it is
graded forced-amplitude-with-stated-conventions, and H0
derived-exponent / measured-coefficient; the membership
is that carried by Table CXXVI and Rem. FK.8.) For
ωb h2 the α4 power of ηB is derived, and so is the current
that carries the bias, B−L, but the coefficient is not. The
conventional-point value 0.2057 α4 (1 + 2πα) used in the
fits rests on a Gaussian carrier that the print cannot supply, and it was set after Planck’s ηB was known; with the
stated carrier and every step pinned to the print, the mechanism gives ηB of about (2–3) × 10−9 , three to five times
Planck’s value (Ext. Deriv., App. AH, Berry-holonomy
baryogenesis theorem), and agreement would need one
winding of the determinant line per e-fold, which the
stated carrier does not give. The deposit factor (1 + 2πα)
follows from Axiom E-transport′ (+1 priced discrete bit)
with the stated carrier. ωb h2 is therefore graded derived
power, calibrated coefficient: one calibrated input, as in
ΛCDM. The tilt ns = 1 − 2/60 is a stated number; the
mechanism is open, as it is for As (Table CXXVI).
a. Parameter count. With ωχ h2 graded forcedamplitude-with-stated-conventions (Prop. FK.5), H0
graded derived-exponent/measured-coefficient (the eightroute no-go) and the ωb coefficient a calibration, the
calibrated continuous count is kDFD = 4: R, As , the ωb coefficient and τ . It is 5 if ωχ ’s stated conventions (the 1.62
relic prefactor and the Z2 /Sugawara edge) are counted
as a knob. On the Planck plik-lite dataset the effective
count is 3 (R, i.e. H0 ; one overall amplitude absorbing
As e−2τ ; and the ωb coefficient), against ΛCDM’s effective
5 (App. ZS). The discrete bit of Axiom E-transport′ is
not free of charge: it is priced in the App. CE trials audit (row 9, axiom-adoption bits; the adopted statements
are listed in Table CXXVII), and its adoption was made
after Planck’s ηB was known to select its consequence at
the conventional freeze-out point; the statement is “one
data-concordant discrete bit,” never “zero assumptions.”
The conclusion is unchanged: DFD’s parameter economy
over ΛCDM holds at every reading, 4–5 < 6 globally and
3 < 5 on the CMB likelihood itself.
Remark FK.8 (The parameter claim, stated exactly). This
remark is the reference form of DFD’s parameter claim:
it is the statement below, not the unqualified phrase “no
continuous fit parameters.”
DFD fits no continuous parameter to the shape of
any observable it predicts, with one exception, the
baryon density. Every ratio, hierarchy and spectrum
shape the theory reports — the CMB peak structure,

373
Input

Grade

Basis

H0early = 67.52 ± 0.09 Derived expon., GℏH02 /c5 = C α57 : exponent 57 a theorem;
Measured coeff. R = ΩΛ C = 0.6042 ± 0.0027 measured; today’s
(1 knob: R)
early clock C = 0.87784 ± 0.00266 (eight-route no-go;
C=1: the bound-matter clock, H0loc = 72.09, Postulate ZS.11);
five concordant CMB-anchored channels; 1/H0 = 14.48 Gyr.
ns = 0.9667
Stated
ns = 1 − 2/60: a stated number; the mechanism
(0 knobs)
is open. +0.4σ from Planck 2018; the
combination of Planck, ACT DR6, lensing
and DESI gives 0.974 ± 0.003, 2.4σ above.
As
Calibrated
One calibrated amplitude; 32πα5 matches
(1 knob)
it at −0.6 to −0.8σ with its source open
(Prop. FK.4).
P
ωχ h2
Forced ampl., ⟨θ2 ⟩ = j |S0j |2 C2 (j)/[k(k+2)]
Stated conv.
⇒ Ωχ h2 = 0.11928 (−0.60σ; flux-period
√
fix, ×κrad 2π, κrad =
√
Γ( 14 )2 /(6 3π); App. χ-matter);
Casimir operator proven; measure given the
asserted P4′ (Table CXXVII); Z2 /Sugawara
edge + 1.62 relic-prefactor stated
(Prop. FK.5).
ωb h2
Derived power, ηB ∝ α4 and the B−L bias derived; with
calibrated coeff. the stated carrier,
(1 knob; +1
about (2–3) × 10−9 pinned to print,
priced discrete 3–5× Planck’s 6.13 × 10−10 ; the fits use
bit)
the conventional-point value; deposit
(1 + 2πα) by Axiom E-transport′
(Ext. Deriv. App. AH).
τ
Calibrated
Astrophysical reionization depth,
(astrophysics) as in ΛCDM.
TABLE CXXVI. Per-input grade of the six ΛCDM-equivalent cosmological inputs (coefficients at the measured clock; ωb h2
graded under Axiom E-transport′ ). Stated: ns (a stated number; the mechanism is open). Derived power, calibrated
coefficient: ωb h2 (the printed mechanism gives ηB of about (2–3) × 10−9 , three to five times Planck’s value; +1 priced discrete
bit, Axiom E-transport′ ). Derived exponent, measured coefficient: H0early (the vacuum coefficient R). Calibrated:
As , which the formula 32πα5 matches without a derivation. Forced amplitude with stated conventions: ωχ h2 , the
finite-vacuum Casimir amplitude with the flux-period identification (Lemma AV.6) and the stated count 158, Ωχ h2 = 0.11928 at
−0.60σ (Prop. FK.5; App. χ-matter), with the operator proven, the measure conditional on the asserted P4′ (Table CXXVII),
and the Z2 /Sugawara edge and 1.62 relic-prefactor as stated conventions. Calibrated continuous coefficients (4): R, As ,
the ωb coefficient and τ , the last an astrophysical quantity shared with ΛCDM. In every count DFD frees fewer parameters than
ΛCDM’s six.

ωχ h2 , the forced fermion ratios, the mixing towers, the
closed-form value of α−1 — is delivered by derived or
discretely selected structure, with no continuous quantity
adjusted against the data it is compared to. The tilt
ns = 1 − 2/60 is a stated number; the mechanism is open.
The exception is ωb h2 : its α4 power is derived, but its
coefficient is a calibration (the printed mechanism gives
ηB of about (2–3) × 10−9 , three to five times Planck’s
value), and the CMB fits use its conventional-point value
(Ext. Deriv. App. AH).
What the theory’s cosmology carries is four continuous

calibration constants:
(i) The vacuum coefficient R = ΩΛ C = 0.6042 ±
0.0027 calibrates the absolute scale. Today’s earlyclock value is C = R + 1.924 (ωm + ωr ) = 0.87784 ±
0.00266; through GℏH02 /c5 = C α57 (exponent 57 a
theorem) it fixes H0early . Measured from BAO+θ∗ ;
underivable from the printed axioms by the eightroute no-go (Rem. AP.6).
(ii) The primordial amplitude As calibrates the
scalar power; the formula 32πα5 matches it, but
no printed mechanism produces it (Prop. FK.4).

374
(iii) The coefficient of ωb h2 calibrates the baryon density, the one shape among the four, through the
baryon loading of the acoustic peaks; its α4 power
and its B−L current are derived.
(iv) The reionization optical depth τ calibrates the
late-time optical depth to recombination: an astrophysical quantity taken over from ΛCDM, degenerate with the amplitude in the CMB likelihood.
Three of the four are DFD-internal; τ is shared with the
comparison model. Against ΛCDM’s six free parameters
the global count is four (five if ωχ ’s stated conventions
are charged as a knob). On the Planck plik-lite likelihood
itself, where (ii) and (iv) collapse into the single combination As e−2τ , the count is three calibrations against
ΛCDM’s five effective parameters (App. ZS). The
particle sector adds
√ three continuous numbers, the fitted
Higgs prefactor 2π, the imported ηq and the Higgs quartic, conjectured to be 1/8 at a matching scale that is not
derived; G sets the unit (§FK 4).
The discrete inputs are priced, not free. They are
enumerated and charged in the trials audit of App. CE
(row 9, axiom-adoption bits; the adopted statements are
listed in Table CXXVII and the discrete inputs in Table CXXVIII). The form of the discrete statement is
“priced discrete bits,” never “zero assumptions.”
Scope. The count above covers the six ΛCDM-equivalent
inputs graded in Table CXXVI. Individual results elsewhere in the corpus whose normalization is imported
rather than derived — for instance the within-multiplet
ηq of App. Y — are graded as such at their own sites and
are not reported as parameter-free predictions (App. FM);
they do not enter the prediction set to which this claim
applies. The inputs of the graded predictions are listed
in Tables CXXVII and CXXVIII (§FK 4).

4.

the last an astrophysical quantity shared with ΛCDM),
three particle-sector numbers (the fitted Higgs prefactor,
the Higgs quartic and the imported ηq ) and the one dimensionful scale that every theory measures; the quartic
is conjectured to be 1/8, but its matching scale is not derived, so it counts as one continuous number. The cluster
χ amplitude is not an input: it is predicted, 1.21 ± 0.05
of the cosmic mix inside r500 at estimate grade (App. I).
b. Comparison with ΛCDM and the Standard Model.
ΛCDM with the Standard Model fits six cosmological and
nineteen particle parameters, and seven to nine more once
neutrino masses and mixing are included; it adopts about
ten structural assumptions: general relativity, quantum
field theory, the gauge group, the matter content, three
generations, the Higgs doublet, cold dark matter, a cosmological constant, inflation with an unspecified potential,
and the cosmological principle. DFD fits far fewer continuous numbers, six beyond the unit of scale and seven with
τ , which ΛCDM fits as well, and it adopts more structure; colour, the matter content and three generations
are inputs in both. Among the numbers the comparison
fits, DFD derives the χ abundance (conditional on the
flux-period identification, P4′ , the stated count 158 and
two stated conventions), the exponent of H0 and of the
dark-energy density, a0 , the closed-form value of the finestructure constant (exact under two stated choices, with
its last 21 ppm on the open bracket coefficient), the weak
mixing angle, the top and charm masses and θ̄ = 0 at
tree level (θYM = 0 fixed by the spectral action given that
nothing is added to it for the gauge fields), and, with the
discrete selections of Table CXXVIII, αs (MZ ), the CKM
magnitudes, the three neutrino masses, the PMNS angles
θ12 and θ23 and the phase δCP , each at the grade stated
where it is treated; sin2 θ13 = 3α is a stated conjecture,
and the tilt ns = 1 − 2/60 is a stated number; the mechanism is open. Its claim is fewer fitted numbers, not fewer
assumptions.

Inputs and axioms of DFD
5.

Tables CXXVII and CXXVIII list the inputs of the
graded predictions: the postulates and adopted axioms,
the identifications adopted, the discrete selections, the
continuous calibrations and the imported frameworks,
each with where it is stated and its status. A statement
that recurs in several places is listed once. An item leaves
the open list of §XVIII A when it is derived, reduced to
an entry here, or handed to a named observation.
a. The count. The tables hold two core postulates,
twenty-five adopted axioms, premises, postulates and
clauses, premise G1, which follows from the Einstein–
Hilbert static-energy identity, one G and the Rest-Mass
Channel, fourteen adopted identifications, eighteen rows
of discrete selections, which carry the fifteen discrete inputs of the fermion, Higgs and neutrino sectors among
others, and six imported frameworks. The continuous
entries are four cosmological calibrations (the vacuum
coefficient R, the amplitude As , the ωb coefficient and τ ,

Status and falsifier

Status FK.9. Four continuous constants are calibrated: the vacuum coefficient R, the amplitude
As , the coefficient of ωb h2 and the shared astrophysical τ (Rem. FK.8); only the ωb coefficient
sets a shape, and the χ relic is derived with stated
conventions. (i) As is one calibrated amplitude, as
in ΛCDM: the formula 32πα5 matches it at −0.6 to
−0.8σ, but no printed mechanism produces it, and the
no-native-inflaton theorem (Thm. FK.2) means there is
no inflaton potential to derive it from; the tensor ratio is bounded by the χ relic instead (Cor. SM.4). (ii)
The χ relic is derived with stated conventions (the Finite SU (2)60 CS Vacuum Relic, Thm AV.13) with the
flux-period identification (Lemma AV.6) (P = M̄P α3 is
the flux period ; frad = P/2π) and the stated count 158:
Ωχ h2 = 0.11928 (−0.60σ
√ from Planck),√the exact onset
factor κrad = Γ( 14 )2 /(6 3π) and the 2π period shift

375

TABLE CXXVII. Inputs of DFD (I): postulates, adopted axioms and adopted identifications. “Stated in” gives where each is
printed, and the status is the one given there.
Input
Stated in
Postulates and adopted axioms
Light follows the optical metric, n = eψ , on flat R3 with
§I B
absolute time
Test bodies move in the potential Φ = −c2 ψ/2; inside bound §I B
systems χ does so where it fills the medium’s demand
The scalar sector is an instantaneous elliptic constraint
Axiom WS.1
Rest mass sources an unscreened channel, the Einstein–Hilbert Thm. GR.10
term’s own static scalar energy; the µ-law acts only on the
polarization of bound systems
The floor law in energy form: matter moves under the
Eqs. (AV27), (AV28)
momentum-conserving energy of the two Newtonian fields,
which one energy, the χ assignment and the floor law force in
spherical symmetry; its three-dimensional filling rule is stated

Environmental confinement: inside a host field above (ln 2)2 a0 a Axiom NL.14
subsystem is Newtonian; in a weaker host the fields compose in
quadrature

The Einstein–Hilbert term acts on the scalar field only through Prop. WS.5; Prop. WS.6
the rest-mass channel (premise G1)

Gravity’s spectral trace uses the four-dimensional Dirac
Prop. WS.5
operator, while α’s uses the internal connection Laplacian
(premise G2)
Gravity’s dynamics are the Einstein–Hilbert term of the
Prop. WS.6
spectral action, in the rest-mass sector where µ → 1
The finite spectral-action clock dictionary
Axiom AP.5
The scalar acceleration action dictionary
Axiom AP.7
The primordial state: the no-boundary state with nothing
Rem. Q.14
inserted
A smooth beginning: smooth background, small regular ripples; Post. Q.15
it carries the arrow of time
One universe, two clocks: matter and the expanding universe Postulate ZS.11
keep clocks offset by one constant factor λ, λ2 = 1/C
Size of the clock gap: when matter first binds, its ψ gap is the Postulate ZS.12
microsector response read with both rates on one worldline
Axiom E-transport′ : the flux advances one lattice step per
Ext. Deriv. App. AH
e-fold
The crossover normalizer is domain-local
Axiom ZT.2

Seven vacuum axioms, which single out CP 2 × S 3
Thm. AB.1
Postulate Y.10: the det-orthogonal geometric CP offset of quark App. AO
mixing
The channel space is the group algebra C[A5 ]
App. Z
Trimaximal mixing with µ–τ reflection
Thm. AY.3
Baryon number is violated only by the electroweak anomaly; Thm. F.26; Rem. F.27
the posited soliton reading is not used

Status
Postulate
Postulate
Axiom
Adopted postulate

Adopted; the Local
Group’s matter
barycentre tests it
against the total-ψ rule,
the disc’s vertical force
against the static field
Adopted; local wide
binaries and the edge
step test it; the
outer-halo globular
clusters beyond the step
show about a third of
the predicted boost
Follows from the
Einstein–Hilbert
static-energy identity
and one G; its partition
clause is the Rest-Mass
Channel
Stated premise of the
derived term
Stated premise
Axiom
Axiom
Stated input; Ωχ h2
tests it
Stated premise
Stated postulate; form
derived, size measured,
derivation open
Stated postulate, to be
tested; not derived
Adopted; one priced bit
Optional extension;
environment data
decide it
Axioms, satisfiable
Structural postulate
Structural postulate
Structural postulate
Stated premise; τp = ∞
rests on it

376

Table CXXVII, continued.
Input
Frame stiffness scales with block rank, κr = nr κ0 ; the scalar
channel’s frame factor is n3 /n2 (the internal gauge partition)

Stated in
Thm. F.25; Axiom AP.18

The scalar channel couples through the magnetic sector, the
magnetically dominated regime E 2 ≪ c2 B 2
The four-dimensional three-form of the internal C3 ’s b0 piece
carries zero flux

Thm. G.1; App. G 2

The spectral action’s Planckian vacuum term cancels

Rem. AU.3; App. WS

App. AU

From decoupling on, the (K − K̄)2 term acts on every slice on Axiom WS.1; Rem. WS.7
which the µ-law acts somewhere; while no baryons are bound
anywhere, and at least until decoupling, gravity is the
Einstein–Hilbert term alone
The internal Dirac operator is held at its printed spectrum, and Thm. Q.12; Ext. Deriv.
only the inner fluctuations of End(E) are physical (rigidity,
App. AH
layer 7 of the single-vacuum theorem)
Adopted identifications
The ground-state principle, µ = P(n=0): it gives ν = Z1
Rem. BS.4
eψ as the S 3 weight ratio, minimal level response,
App. N
saturation-union composition
The deep-field limit with unit slope: µ(s) = s + O(s2 ) in App. N, Assumption N.7; Thm. BS.9
2
and gobs
→ gbar a0 in the interpolation-function theorem
The temporal kinetic function obeys K ′ (∆) = µ(∆)
Eq. (Q8)

fχ = M̄P α3 is the flux period

Lemma AV.6

The 2π/α winding instanton changes sector
Thm. AV.24
The generation space is the fixed kernel of the vacuum operator App. F
The electroweak group is the structure group of the isotropy
Rem. F.2
bundles at the vacuum point of CP 2 , with the hypercharge the
direction diag(1, 1, −2) normalised as 6Y , acting on the twisting
bundles and not by isometries
On the soliton reading, baryon number is the S 3 winding
App. F 10
The Majorana scale, MR = MP α3
App. P
One α per flavour connector
App. AO
The ρ-shifted Casimir ∆ + 1 as the operator of α’s trace on
§X C
S 3 = SU (2)
α’s trace as the CP 2 ζ(0) times the S 3 heat trace, with the
§X C
heat-kernel regulator
The weak mixing angle’s trace weights: hypercharge over one
generation, SU (2) over one doublet

Thm. Z.1

Status
Motivated assumption
in App. F 9, printed as
an axiom in App. AP;
ka , a0 and the weak
mixing angle rest on it
Stated premise; ka rests
on it
Stated premise; a
nonzero flux adds its
own vacuum energy
Stated premise;
uncancelled, it would
gravitate as Λ does
Stated clause; with it
k = 0 is derived

Stated premise; the
57-level spectrum rests
on it
Stated input
Assumptions
Assumption
Adopted identification;
the dust branch rests on
it
The χ abundance rests
on it
Asserted
Conditional
Identification

Identification
Identification
Asserted
Stated choice
Stated choice; the
closed form is exact
under the two
Stated choice

377
TABLE CXXVIII. Inputs of DFD (II): discrete selections, continuous calibrations and imported frameworks.
Input
Discrete selections
Three generations, Ngen = 3

Stated in

Status

Rem. F.21

Colour SU (3) and the content and chirality of one family; its
five hypercharged types, Ngen and the derived q1 = 3 fix
kmax = 60
The bracket coefficient c = 1/13 behind the last 21 ppm of α
Identification of the hypercharge-trace coefficient with 1/e2
The weak-block count 158 in mχ ; the uniform count gives 171

Rem. F.3

Input, shared with the
Standard Model; the
same input as axiom
(V5) of the seven
vacuum axioms
Inputs, shared with the
Standard Model

Rem. X.4
§X I
Rem. FK.7

√
ΛQCD = MP α19/2 and its 4π scheme matching
App. Z
Seven charged-fermion prefactors, one for each species other
App. FM
than the top and the charm
Three CKM counts, λ, A and η̄ as 31, 108 and 49 times α
App. AO
Euler projection of the CP-even CKM apex
Postulate AO.23
Nothing is added to the spectral action for the gauge fields; the App. L
spectral action then fixes the bare Yang–Mills angle at θYM = 0
The neutrino branch, one of four equally natural fractions
App. X
2
The Dirac-neutrino Yukawa rung, yD
= Fν α
sin2 θ13 = 3α

Prop. AT.1
App. Z

The tilt ns = 1 − 2/60

Table CXXVI

One sign bit; the orientation of time is the smooth beginning’s App. CE
The Bose-law branch, SOURCE
App. BS

The relic edge ϑmax = 12 and the Sugawara shift K = k + 2
The screen domain size RD

nearly cancelling each other (App. χ-matter), with the
operator (Casimir) forced, the measure conditional on
P4′ (Table CXXVII), and the normalization (k(k + 2);
θmax = 12 from χ’s derived Z2 , K = k + 2 from Sugawara)
resting on two stated conventions (Table CXXVIII). The
lone non-DFD input is the standard cosmological relicredshift 1.62 (shared with all DM), not a fit to ΩDM . (iii)
The vacuum coefficient R = 0.6042 ± 0.0027 is measured,
with today’s early-clock value C = 0.87784 ± 0.00266
(five concordant CMB-anchored channels); the eightroute no-go proves it underivable from the printed corpus
(Rem. AP.6). (iv) The coefficient of ωb h2 is calibrated:
its α4 power and B−L bias are derived, the printed mechanism gives ηB of about (2–3) × 10−9 , three to five times
Planck’s value, and Axiom E-transport′ adds one priced
discrete bit. Net count: 4 calibrated continuous
cosmological coefficients, three DFD-internal and
one shared with ΛCDM, against ΛCDM’s 6 free

Prop. FK.5
App. ZT

Stated; provenance open
Stated; open
Stated input; the CMB
tests it
Selected
Chosen by data
Selected
Selected
Stated premise
Chosen by oscillation
data
Stated input
Stated conjecture;
JUNO tests it through
θ12
Stated number; the
mechanism is open
Priced
Chosen by data; a
single-temperature
mode gives the same
branch
Stated conventions
Free; environments test
it

parameters (R, As , the ωb coefficient and τ ; ωχ h2 is
forced-amplitude with stated conventions), with ns a
stated number (the mechanism is open), and the H0 exponent a theorem. No derivation of As , of R or of the
ωb coefficient is manufactured here; no obstruction is
manufactured either: each candidate received an explicit
arithmetic check.
Falsifier FK.10. Two independent decisive tests. (a) Primordial B-modes. DFD predicts none that any planned
experiment can see: r ≲ 10−7 (App. SM, Cor. SM.4).
A confirmed primordial tensor signal (a LiteBIRD, BICEP Array or Simons Observatory detection) falsifies
the χ relic, and with it the dark sector; a firm upper
bound such as r < 5 × 10−3 is the predicted outcome,
not a falsifier. The ratio is not tied to the As normalization: 4α/π is the dictionary’s 16εW , not a B-mode
ratio. (b) Dark matter. The χ abundance is predicted :
the finite SU (2)60 CS-vacuum amplitude with the flux-

378
Table CXXVIII, continued.
Input
Continuous calibrations and the unit
Vacuum coefficient R = ΩΛ C = 0.6042 ± 0.0027; today’s
early-clock value is C = 0.87784 ± 0.00266
Primordial amplitude As , which 32πα5 matches
The ωb coefficient at the conventional point

§FK 3

Reionization optical depth τ

Rem. FK.8

Higgs prefactor

√
√
2π in v = MP α8 2π

Stated in

Status

Rem. AP.6

Measured

Prop. FK.4

Calibrated; no printed
source
Calibrated; the printed
mechanism gives ηB of
about (2–3) × 10−9 ,
three to five times
Planck’s value
Astrophysics, as in
ΛCDM
Fitted; no-go on
deriving it
Conjectured 1/8 at a
matching scale that is
not derived, so one
continuous number (mH
−0.40% at mt )
Imported; DFD’s own
running gives 4.21
Measured unit

Thm. Z.3

Higgs quartic λH

Thm. Z.3

Light-quark QCD enhancement ηq

Rem. Y.4

One dimensionful scale, G, with ℏ and c as units
§I E
Imported frameworks
Quantum mechanics: the unit ℏ, the interacting dynamics and App. QM
the measurement postulate; the Dirac β is derived
The standard thermal history: relic redshift, nucleosynthesis, Prop. FK.5
recombination
Baryonic astrophysics: star formation, feedback and the sources App. PK
of reionization
Lattice ratios of quantum chromodynamics
App. YM
The vacuum energy of matter cancels; uncancelled, it would
App. AU
gravitate as Λ does
A homogeneous, isotropic background

period identification (Lemma AV.6), the stated count
158 and the asserted premise P4′ (Lemma AV.27) gives
Ωχ h2 = 0.11928 (−0.60σ from Planck 0.1200 ± 0.0012).
(b1) A precision ΩDM h2 that moves firmly outside the
stated-convention band (across K, times the O(1) relic
prefactor) falsifies the derivation. (b2) The 32.005 eV
mass (= 2π × 5.0937 eV) is sharp, and χ has no photon
coupling (E = 0; App. AV): it neither decays nor converts,
so a laboratory or astrophysical null is the prediction, and
any dark-matter photon line at 16.0 eV (77.5 nm, twophoton decay) or axion-like signal at 32.005 eV (38.74 nm),
at any strength, falsifies χ.
a. Early-universe tests. DFD states the tilt ns =
1 − 2/60 = 0.9667 with no running (a stated number;
the mechanism is open), and it predicts no primordial Bmodes (r ≲ 10−7 ), no Gaussian dark-matter isocurvature
and no baryon isocurvature: the E-transport baryogenesis
step, A5 = Q d ln A, fires on the local e-fold count, so
ηB follows the local expansion and SB = 0 exactly for
an adiabatic onset of the radiation era (computed). The
CMB alone agrees: ns = 0.9682 ± 0.0032, r < 0.034 and

§XIX

Shared with every
quantum field theory
Imported; shared
Shared with ΛCDM
Inherited
Assumed, as in every
quantum field theory
with gravity
Assumed; shared

βiso < 0.038. The combination of Planck, ACT DR6,
CMB lensing and DESI gives ns = 0.974 ± 0.003, about
2.4σ above 0.9667, so the Simons Observatory, with
σ(ns ) ≃ 0.002, decides it: a tilt outside 0.957 to 0.977 at
that precision falsifies 1−2/60. A confirmed primordial Bmode signal from BICEP Array, the Simons Observatory
or LiteBIRD falsifies the χ relic. A detected Gaussian
dark-matter isocurvature mode falsifies the topological
origin of the relic (Rem. AV.16), and a detected running
|dns /d ln k| ≥ 0.003 falsifies the constant tilt.
Appendix SM: The “Scalaron Mass” is Not a DFD
Observable
1.

Statement of the (mis-framed) problem

A DFD scalaron mass can be estimated in two ways,
which disagree by eleven orders of magnitude:
• an As -route, which fits a Starobinsky scalaron mass
to the measured scalar amplitude through the R+R2

379
1
2 2
amplitude relation As = 24π
2 (M/M̄P ) N , giving
13
MAs ≃ 2.85 × 10 GeV; and

• a curvature / a4 -route, which reads a Starobinsky
mass from a dimensionless R2 heat-kernel coefficient
through M 2 = MP2 /(6 cR2 ), which with any natural
cR2 = O(1)/16π 2 returns M ∼ MP (Planckian).
The discrepancy (MP /MAs )2 ≈ 1.8 × 1011 was treated as
an unresolved tension between two estimates of a single
quantity. We show it is no such thing. The two numbers
measure two different objects, and the quantity they were
supposed to bracket — a Starobinsky scalaron mass —
does not exist in DFD.

2.

No native scalaron exists

Proposition SM.1 (No-native-scalaron ⇒ scalaron mass
undefined). The DFD gravitational action contains no
curvature-scalar (R2 ) term; its only four-derivative term
beyond the Gauss–Bonnet total derivative, the a4 Weylsquared term, reaches the scalar sector only as the helicity0 part of the spin-2 pole at 3.6 M̄P for f0 = 1, above the
spectral cutoff, and therefore the action propagates no
Stelle-type scalar degree of freedom. Consequently there
is no field in DFD whose mass can be identified with a
Starobinsky scalaron mass, and “the DFD scalaron mass”
is undefined-because-absent rather than undetermined.
Proof. Three independent facts close this.
(a) No R2 term in the action. The DFD gravitational
sector is the two-derivative Einstein–Hilbert term arising from the a2 Seeley–DeWitt coefficient of the spectral
action; the propagation analysis records explicitly that
DFD carries no R2 term: the a4 term of the Dirac trace
adds only the Weyl-squared and Gauss–Bonnet invariants
(Proposition WS.5); the Weyl-squared term is a spin-2
correction above the cutoff, not a scalar (Sec. V, with
αT = 0 identically). A Starobinsky scalaron is the additional propagating scalar polarisation of an R +R2 theory;
with no R2 term there is no such polarisation.
(b) The a4 coefficient carries no R2 term. In the Connes
spectral action the a4 heat-kernel coefficient yields the
Yang–Mills/Maxwell kinetic term, and for gravity only
the Weyl-squared and Gauss–Bonnet invariants (Proposition WS.5) (Ext. Deriv., App. AH; App. O), not an
Einstein-frame R2 term. The historical “curvature/a4 route gives a Planckian scalaron” read the gauge coefficient as a gravitational R2 coefficient; that identification
is incorrect, and with it the Planckian estimate loses any
claim to be “the scalaron mass.”
(c) No scalar mode is slow-roll-compatible. By Theorem FK.2 (proof in Ext. Deriv., App. AH series IV–VI;
six independent failure modes), no DFD scalar admits an
inflaton role with potential mass m2 ≲ 10−11 MP2 : ψ is
FRW-frozen and carries no potential term (exactly massless: the action displayed in Thm. AE.1 is purely kinetic

in ψ plus a linear matter coupling); the truncated microsector carries no size or shape
√ modulus (App. O); the
57 Kaluza–Klein modes sit at α MP ; complex-structure
and harmonic-one-form deformations vanish (h2,0 = 0,
b1 = 0); and kmax , the Chern–Simons levels are integerquantised. There is no continuous, light scalar to carry a
scalaron mass.
Since a scalaron mass is by definition the mass of the
R + R2 scalar, and no such field exists in DFD by (a)–(c),
the symbol Mscalaron has no DFD referent. □
3.

Why the two routes disagree by ∼ 1011

The disagreement is not numerical noise; it is the
exact signature of fitting an absent field.
Both
numbers are reproduced to printed precision in
scalaron mass analysis.py.
a. The As -route is the dictionary rung in disguise.
The de Sitter dictionary’s scale H⋆ is fixed by a rung
coincidence, not a potential. With fχ = M̄P α3 and the
3
3
Majorana
√ rung MR = MP α the same α rung, and
MP = 8π M̄P , the de Sitter dictionary’s rung is
√
H⋆ = 8πfχ = 8π MR = 2.38 × 1013 GeV
(SM1)
(Thm. AV.13, App. AV),
√
verified as an identity (8πfχ / 8πMR = 1 to ten digits).
Through the de Sitter dictionary As = H⋆2 /(8π 2 εW M̄P2 )
with εW = α/(4π), the amplitude
collapses to As = 32πα5
√
(App. AT); equivalently As ∝ H⋆ . Feeding this As
back through the Starobinsky amplitude relation therefore
returns a number MAs = 1.20 H⋆ , i.e. O(1) times the
rung-locked Hubble (SM1). The “As -route scalaron mass”
is simply H⋆ re-read — the de Sitter dictionary’s rung, a
DFD quantity, but not a particle mass.
b. The a4 -route defaults to Planckian. An R + R2
scalaron obeys M 2 = MP2 /(6 cR2 ) with cR2 the dimensionless R2 coefficient. To force Ma4 → MAs would require
cR2 = MP2 /(6MA2 s ) ≈ 3.1 × 1010 , larger than the natural heat-kernel value 1/16π 2 by a factor ∼ 4.8 × 1012 .
The single DFD power closest to bridging the gap is
α−5 = 4.8 × 1010 — and it still misses, and there is no
R2 term to host any cR2 in the first place. No DFD
heat-kernel number carries α−5 ; the route defaults to
M ∼ MP .
c. The gap is the missing α5 . The two routes differ
by

2
MP
= 1.84 × 1011 = 3.8 α−5 ,
MA s
(SM2)


2
logα (MP /MAs ) = −5.27.
The As -route carries the factor As = 32πα5 ; the a4 -route
carries α0 . The eleven-order gap (SM2) is that α5 power.
It is not a tension between two measurements of one mass;
it is the distance between two different objects — the
de Sitter dictionary’s rung H⋆ and the Planck scale —
artificially compared through a parameter (the scalaron

380
mass) that neither realises.

the dark relic (Theorem AV.23, App. AV).

4.

The decoupling is reinforced quantitatively: even granting the slow-roll dictionary, the hybrid is self-excluding.

The actual DFD primordial sector (no scalaron)

DFD states its primordial numbers with topological
integers and the α-tower, never a scalaron mass. In
R + R2 the scalaron mass is the fundamental input and
(ns , r, As ) are derived from it; DFD has no mechanism
that produces the primordial perturbations (App. FK),
and its primordial numbers are these:
H⋆ = 8πfχ = 2.38 × 1013 GeV,
2
ns = 1 −
= 0.9667,
60
r ≲ 10−7 (B-mode ratio; Cor. SM.4),

(SM3)

As = 32πα5 = 2.08 × 10−9 .
Here H⋆ is a rung of the α-tower, not the expansion rate
of any DFD stage: a de Sitter stage at H⋆ would wash
out the χ relic (Theorem SM.3). The tilt is a stated
number; the mechanism is open. As is one calibrated
amplitude, which 32πα5 matches numerically (App. FK).
The dictionary ratio 16εW = 4α/π = 9.29 × 10−3 is not
an observable; the tensor entry is the bound of Cor. SM.4.
Theorem SM.2 (Primordial optical decoupling). In
nativep DFD the slow-roll energy read-off Hinf =
π M̄P As r/2 is undefined. It presupposes three objects
DFD lacks: a slow-roll inflaton potential V (so that
ε = 12 M̄P2 (V ′ /V )2 exists; Theorem FK.2), a sustained
quasi-de Sitter clock at H⋆ (the optical engine can supply
one only far below it, Eq. (Q13)), and a Bunch–Davies
vacuum on that clock. The fourth ingredient, the tensor
normalization Pt = (2/π 2 )(H/M̄P )2 , does hold: any stage
that freezes the tensors at the rate H gives it (Eq. (SM6),
with the horizon-exit rate of App. Q 6). With no V there
is no ε and no consistency relation r = 16ε. Therefore
As and the dictionary ratio are not inflaton observables.
Written as optical-response amplitudes of the n = eψ
medium, they read
As = Pψopt ,

opt
ropt = Pshear
/Pψopt ,

(SM4)

the scalar optical-density power and its transverse spin-2
(shear) ratio, the TT sector being an independent native
mode (□ hTT =source, on the optical cone; App. AN,
App. WS). A shear built from the scalar ψ is a pure gradient (E-mode) pattern at linear order, so this ratio sources
no B-modes. The map (As , r) → Hinf is a category error,
and the de Sitter dictionary As = H⋆2 /(8π 2 εW M̄P2 ) is a
removable bridge, not the origin of As . The identification
(SM4) is a relabelling, not a source: no DFD stage produces Pψopt at the observed level, and a stage that froze ψ
and the tensors at a common rate would give Pt = 16 Pψ
(Eq. (SM9) against Eq. (SM6)), that is r = 16. With
r ≲ 10−7 (Cor. SM.4), ψ’s vacuum fluctuations supply at
most 6 × 10−9 of As (computed), and As is one calibrated
amplitude (App. FK). In particular there is no inflaton
epoch to randomize the χ flux state, which is what protects

Theorem SM.3 (Inflation-dictionary exclusion). Suppose, counterfactually, that DFD’s As = 32πα5 and
the dictionary ratio r = 4α/π were read as standard
slow-roll observables.
Then the slow-roll energy relation
p
Hinf = π M̄P As r/2 yields, exactly,
4α
As r = 32πα5 ·
= 128 α6
π
(SM5)
√
3
6
=⇒ Hinf = π M̄P 64α = 8π M̄P α = 8πfχ ,
since fχ = M̄P α3 . The compact-angle stochastic kick
per e-fold is then δθ = Hinf /(2πfχ ) = 4 (this 2π is the
Gibbons–Hawking factor, identical in both normalizations
of the flux coordinate, Lemma AV.6, App. AV), which
saturates the compact period-1 flux coordinate in a single
e-fold, washing the modular vacuum to the uniform measure — at best the finite-space
ceiling ⟨Q⟩∞ = 0.0853,
√
hence Ωχ h2 = 1.62 κrad 2π × 0.0853 = 0.139 (the
exact relic map of Rem. AV.12), excluded by Planck
(0.1200 ± 0.0012, +16σ). Therefore the slow-roll interpretation of (As , r) and the finite-CS χ relic (Ωχ h2 = 0.11928,
Theorem AV.23) cannot both be held. Retaining the relic
forces rejecting the dictionary — which Theorem SM.2
shows is in any case undefined.
Proof. Eliminating ε from r = 16ε and As =
H 2 /(8π 2 εM̄P2 ) gives H 2 = 12 π 2 M̄P2 As r, i.e. Hinf =
p
π M̄P As r/2; (SM5) is then arithmetic, verified to machine precision (Hinf /8πfχ = 1 to sixteen digits). The
over-horizon variance of a light spectator on the compact coordinate accumulates as ⟨θ2 ⟩ ≃ N (Hinf /2πfχ )2
until it saturates at the uniform value of the period-1
1
flux coordinate ( 12
in the continuum, ⟨Q⟩∞ = 0.0853 on
the finite 61-state space); with per-e-fold step δθ = 4,
four times the full period, saturation is immediate. The
numerical identity Hinf = 8πfχ also exposes why the apparent “As -route = H⋆ ” coincidence (§SM 4) is circular :
the dictionary was run forward with this same H⋆ = 8πfχ
and εW = α/4π to produce As , r, so inverting it returns
its own input, not an independent measurement of an
inflaton scale.
Corollary SM.4 (True tensor ratio; reclassification of
4α/π). If the finite-CS χ relic is retained, no de Sitter
stage runs at H⋆ : the primordial expansion is the ordinary
expansion read on the arena clock, and any accelerating
stage, such as the optical engine (App. Q 6), runs far
below H⋆ . Every species couples minimally to the optical
metric g̃ (App. AV), and the TT cone is the optical cone
(App. WS), so the tensor modes and the period-1 flux
coordinate θ = χ/fχ obey one mode equation and differ
only in their constant normalizations (M̄P2 /8 for hij , fχ2 /2
for θ). Whatever the history a(t), ψ0 (t), a stage that

381
freezes both leaves
fχ2
Pt (k) = 8 2 Pθ (k) = 8α6 Pθ (k),
M̄P

with the mode measure fixed a priori is
8α6 = 1.21 × 10−12 ,

(SM6)
and χ (mχ = 32 eV) is light at that freeze-out whenever
the tensors matter, since r ≳ 10−8 needsR a freeze-out rate
≳ 1010 GeV. The frozen variance
σθ2 = Pθ d ln k adds to
√
2
the relic, Ωχ h = 1.62 κrad 2π (QCS + σθ2 ) (Eq. (AV22)),
so Planck’s Ωc h2 = 0.1200±0.0012 allows σθ2 < 1.9×10−3
at two sigma. The modes that count lie between today’s
horizon and the horizon at χ’s oscillation onset (Tosc ≃
8.7 × 104 GeV), 36 e-folds of scale, which any powerlaw tilt reduces to no fewer than Neff = 14.7 effective
e-folds (10.7 if the engine froze only the observed scales,
k ≲ 10 Mpc−1 ). Hence
r=

2
8α6 σθ,max
Pt
<
As
Neff As

(3 × 10−8 if scale-invariant).
(SM7)
The looser criterion σθ2 < QCS with N = 60 gives the
same closed form,
α QCS
= 7.1 × 10−7
(N ≃ 60),
(SM8)
r<
4πN
but it tolerates a doubled abundance and an order-unity
χ density contrast. The super-horizon part of σθ2 enters
the density quadratically, a non-Gaussian isocurvature
∆2iso ≃ 4 ln(k/k0 ) (Pθ /QCS )2 (k0 today’s horizon scale);
at the value (SM8) it would exceed As three-million-fold,
and Planck’s βiso < 0.038 applied to it gives r ≲ 10−10
(estimate). Hence the number 4α/π = 9.3 × 10−3 is not
the observable primordial tensor-to-scalar (B-mode) ratio. It is the de Sitter dictionary’s 16εW . The optical
opt
shear/scalar response ratio Pshear
/Pψopt (SM4) is a property of the n = eψ medium, and a shear built from a scalar
is a pure gradient (E-mode) pattern at linear order, with
no B-mode. The gravitational-wave background the engine
can leave is r ≲ 10−7 , far below any planned experiment.
=⇒

r ≲ 10−7

Falsifier SM.5 (Tensor B-modes). The finite-CS branch
predicts r ≲ 10−7 (Cor. SM.4). A robust detection of
primordial B-modes at any r ≳ 10−7 falsifies the finiteCS χ relic, whether at r ≃ 4α/π ≈ 9 × 10−3 , at the
r ≈ 4 × 10−3 of an appended inflaton, or at any level
LiteBIRD or the ground-based programmes can reach
(≳ 10−3 ): by Eq. (SM6) the stage that froze those tensors
pushed Ωχ h2 past Planck’s two-sigma ceiling, and for
r ≳ 10−3 it kicked χ by more than a full period per e-fold,
washing it to the 0.139 ceiling (+16σ). A null result is
the prediction.
Remark SM.6 (Anatomy of 32π inside the de Sitter dictionary). A quantization of the canonical optical scalar
shows which part of 32π the dictionary fixes. The highgradient branch of Sψ reduces to a canonical massless
scalar with 1/(8πG) = M̄P2 (exact), whose frozen power

H⋆2
= 8 α6 = 1.21 × 10−12 ,
(SM9)
8π 2 M̄P2
the power such a scalar would freeze at the rate H⋆ ,
with the numerator 8 = 23 fixed and no εW imported.
It is not an amplitude DFD produces: H⋆ is not the
expansion rate of any DFD stage (Theorem SM.3), and at
the relic’s cap on a stage that freezes perturbations, H ≃
4 × 1010 GeV (Eq. (Q13)), the same formula gives 4.0 ×
10−18 , a factor 5.3 × 108 below As (computed); reaching
As would need H ≃ 9.9 × 1014 GeV. The entire residual
to As = 32πα5 is then exactly 32πα5 /(8α6 ) = 4π/α =
1/εW = 1722: one inverse power of the coupling α, times
4π. No solid-angle / horizon-cell integral can produce 1/α
(geometric measures yield O(1) transcendentals, never a
coupling), so the reading 32π = (4π)(23 ) is numerology
— the 23 is the derived Bunch–Davies numerator but the
4π enters only through the one-loop measure εW = α/4π.
Inside the dictionary the content is thus As = 8α6 ε−1
W
with the 8 fixed; the missing primitive there is a native
mechanism supplying ε−1
W = 4π/α
√(one power of the
coupling), equivalently H⋆ /MR = 8π. This sharpens
— and does not beat — the architecture-only grade; and
since no DFD stage freezes perturbations at H⋆ , the
power α5 is a numerical match as well, and As is one
calibrated amplitude (App. FK). (A parallel bookkeeping
note: the internally reduced-mass tensor At = 64α6 gives
r = 2α/π, while the corpus’s non-reduced At = 128α6
gives r = 4α/π; since 4α/π is the dictionary’s 16εW and
not the B-mode ratio (Cor. SM.4), this factor-2 convention
does not affect the observable bound r ≲ 10−7 .)
Pψbare =

Here H⋆ is a rung of the α-tower, not the expansion rate
of any DFD stage (Theorem SM.3); ns = 1 − 2/60 is a
stated number (the mechanism is open); r is bounded
by the χ relic independently of εW (r ≲ 10−7 ); and As
is one calibrated amplitude, which the formula 32πα5
matches without a derivation of its power or its coefficient
(App. AT; App. FK). A Starobinsky scalaron mass would
be a redundant fifth quantity over-determined by these
four, and it is precisely that redundancy that lets the “two
routes” return inconsistent values. The reframe therefore
dissolves the open entry: there is no fifth number to
compute.

5.

Status

Status SM.7. The “DFD scalaron mass” is undefined
because DFD has no such field. DFD has no native
Starobinsky R + R2 scalaron (Prop. SM.1, Thm. FK.2);
the symbol Mscalaron has no DFD referent. The ∼ 1011
“disagreement” is explained (SM2) as the missing α5 between two unrelated scales (the α-tower rung H⋆ and
MP ), not as a tension between two estimates of one mass.
We make no positive claim of a forced scalaron mass
(none exists to force), and we manufacture no obstruction:
every massive DFD scalar was checked against the slow-

382
roll ceiling m ≤ 3.86 × 1013 GeV with explicit arithmetic
√
(ψ massless with no potential; KK modes ∼ MP – αMP ;
χ = 32.005 eV cold relic), and none qualifies, in agreement
with Thm. FK.2.
Residual. This reframe resolves only the scalaron-mass
mislabel. It does not resolve the fitted item: the As coefficient 32π — equivalently√εW = α/(4π), equivalently
the unfixed ratio H⋆ /MR = 8π — remains asserted, not
derived (App. AT, App. FK). With no inflaton potential there is no V ′ /V from which to compute εW , so As
rests on an un-derived normalization; that WdW-closure
freedom stands. The tensor ratio does not rest on it:
the χ relic bounds r ≲ 10−7 directly (Cor. SM.4). What
the optical-response reframe (Theorem SM.2) settles is
narrower: it shows the de Sitter dictionary does not apply,
so the “inflation randomizes χ” objection to the dark relic
does not apply (Theorem AV.23). Writing As = Pψopt is a
relabelling, not a source; As is one calibrated amplitude
and 32πα5 a numerical match (App. FK), and 4α/π, the
dictionary’s 16εW , predicts no observable. The present
appendix neither adds nor removes a DFD result: it converts a mislabelled open/fitted scalaron-mass tension
into a correctly labelled non-question.
Falsifier SM.8. This appendix is falsified, and a scalaron
mass becomes a real DFD observable, if any of the following is established:
(i) a curvature-scalar (R2 ) term is shown to be required by the DFD spectral action, giving a propagating Stelle scalar with a computable mass (the
Weyl-squared term of Proposition WS.5 is a spin-2
correction above the cutoff and gives no scalar); or
(ii) a DFD scalar mode is exhibited with potential
mass m2 ≲ 10−11 MP2 and a flat (V ′′ /V ≪ 1) direction, contradicting one of the six failure modes of
Thm. FK.2; or
√
(iii) a first-principles derivation fixes H⋆ /MR = 8π
(equivalently the 32π in As ) and that same construction supplies an independent R2 scalaron mass
that lands on H⋆ — which would make the two
routes coincide for a real field rather than by the
present O(1) coincidence.
Absent (i)–(iii), the scalaron mass remains undefinedbecause-absent.

Appendix CE: The Improbability of the Pile: A
Combined-Evidence (No-Coincidence) Bound

DFD derives, parameter-free, a set of fundamental numbers that the Standard Model and ΛCDM treat as free
inputs. This appendix quantifies the combined evidence
after a full trials-factor audit (Sec. CE 2): every discrete
menu the construction could have scanned is enumerated
and charged against the hits. Worst case first: under a
full across-observable selection charge (the most hostile
admissible null) the combined evidence is not statistically significant (p ≈ 0.03–0.5); under hostile menu
accounting with no selection charge the four surviving
independent hits (sin2 θW , θ12 -TM1, the H0 /Λ grammar
block, Ωχ h2 at the flux-period value 0.1193, with one
binary period-convention bit charged — dimension 16)
give ≈ 4.0–4.4σ (conservative windows) to ≈ 5.1σ (moderate); conditional on DFD’s forcing theorems as stated
— the corpus’s pre-registered framing, on the corrected
row basis — the conditional tier is ≈ 5.1–5.4σ under the
conjunction-product convention; under referee-standard
calibrated combination (Fisher/Stouffer) the same rows
give 3.6–4.3σ, and 2.5–3.0σ under fully hostile menu pricing — the result is a ∼4σ closed-list no-coincidence signal,
not ≥ 5σ, a conditional figure that is always to be read
against the unconditional tiers just quoted. Separately,
and given the stated bracket coefficient c = 1/13 (Table CXXVIII), the α-ladder infeasibility theorem (Theorem CE.1) stands: landscape scanning cannot produce
the α−1 = 137.036 match by chance. DFD does not claim
5.5–7.1σ (Sec. CE 2, paragraph on that figure). The combined evidence is a preponderance / parameter-counting
argument — the same logic by which the Standard Model
and BBN were established — not a single falsification-test
discriminator; each tier states exactly what it grants; and
it is leading-order agreement, not precision. The α match
itself is excluded from the pile rows (α is the selector
datum and must not be double-counted); its evidence is
booked separately in Theorem CE.1.

1.

The surviving forced parameter-free rows

Each row is a quantity the SM/ΛCDM leave as a free
parameter (or do not predict at all), derived by DFD
from {α, M7 = CP2 × S 3 topology} with no fitted continuous coefficient (TM1 is a structural postulate and the
Ωχ h2 row rests on stated inputs; App. FK), and priced
against the menu of alternatives the construction could
have scanned (Sec. CE 2). Two candidate rows are excluded (details below): the CKM sin 2β row carries zero
information (Postulate E.1 is an independent normalization postulate and the CKM anti-theorem blocks apex
forcing, App. AO; the companion Wolfenstein integers
miss: λ at ∼1.8σ, A at ∼2.5–3.2σ), and the fine a0 row
is excluded because a price of p = 0.005 would be 40–
70× finer than the ∼10% empirical uncertainty on a0
permits. Included: the H0 /Λ grammar block (3 hits in

383
the declared 135-cell scan; look-elsewhere paragraph of
App. AU 5) and the Ωχ h2 row (the finite SU (2)60 CSvacuum Casimir abundance, App. AV, Thm. AV.13). The

quantity (DFD parameter-free formula) DFD

PMNS θ12 row is included at its pre-registered fair-menu
price 1/14: of the 14 standard mixing ansätze in the audit
menu, only TM1 hits.

measured agree

p (hostile)

p (cond.)

sin2 θW = 3/13 (weak mixing, tree)
0.2308
0.23129 0.23% 1/12–1/56
0.0045
PMNS sin2 θ12 (TM1; 1 of 14 ansätze) 0.3184
0.307
3.7%
1/14
0.074
3 57
H0 /Λ grammar block (ρdep = 8π
α ρP ) 2.55 meV 2.24 meV in band 3/135 (×0.074) 3/135
2
Ωχ h (SU (2)60 CS Casimir, flux period) 0.1193
0.1200
0.59% 0.047
0.0117
ns = 1 − 2/60 (spectral index)†

0.9667

†

Kept for the record but excluded from every tier: ns =
1 − 2/60 is a stated number; the mechanism is open (App. FK).
The companion’s 1 − 28.7α/2π (App. AH, Extended Derivations) is the same number, since 28.7 α/2π = 2/60 to the
printed digits. The combination of Planck, ACT DR6, CMB
lensing and DESI gives 0.974 ± 0.003, about 2.4σ above 0.9667.
The Ωχ /Ωb = 16/3 clue (“a clue, not a derivation,” App. J,
App. TP) enters no tier; the direct Ωχ h2 Casimir row above
carries the dark sector. The excluded sin 2β and fine-a0 rows
are documented in the paragraph on the 5.5–7.1σ figure in
Sec. CE 2.

2.

Trials-Factor Audit and Conditional
No-Coincidence Bound

The trials-factor audit (its scripts and outputs are in
the reproducibility package) asked the hostile question
directly: if DFD were numerology, how many numbers
could it have scanned, and how surprising are the hits
once every scan is charged?
a. The address space. The first instrument is an explicit enumeration of every discrete choice (“address coordinate”) the construction could have made. Sixteen
dimensions are charged: ten native coordinates from the
corpus’s own ledger, five hostile enlargements and the
period-convention bit. Table CXXIX lists them. The
core (numeric) address space is |A|core = 64 (baseline
reading) up to 8.01 × 109 (maximal-hostile: the conservative core 185,472 times the convention multiplier 43,200);
with sign/axiom-adoption bits, 7.0 × 1022 . The adoption
bits pick no value from a numeric menu, so they add
no hit-capable ladder, and the bound of Theorem CE.1
does not depend on them. Continuous inputs (α itself, ℏ,
one scale, the fitted knobs of the fitted-knob ledger) are
excluded from |A| and booked as fits.
b. The α-ladder infeasibility theorem. Within this
space, the number of hit-capable α ladders — distinct
(ladder, convention) assignments that could have produced
an α−1 match at all — is 10,800.

0.9649

0.18%

—

—

Theorem CE.1 (α-ladder infeasibility of landscape scanning). Let a false theory scan the fully enlarged address
space of Table CXXIX, containing n = 10,800 hit-capable
ladders. The observed match, with the stated bracket
coefficient c = 1/13, is α−1 (kmax =60) = 137.03599985
against CODATA 2022 137.035999177(21), a residual of
4.9 × 10−9 fractional (0.0049 ppm). The per-ladder probability of a false match at least as good as the observed
4.9×10−9 fractional match is p = 5.8×10−7 (rung spacing
≃ 1.7 × 104 ppm of α at k = 60), so the expected number
of false matches at least as good as observed, across the
whole space, is
E[false matches at least as good as observed] = n p ≈ 6.3 × 10−3

(0.006–0.04 under fair family inflation; ≤ 0.23 under maximal hostile family and convention inflation — never ≥ 1).
The infeasibility margin is ×160 (central), ≥ ×4 (maximal hostile), and ×7.8 under the pre-registered ±0.1 ppm
falsifier band. Landscape scanning cannot produce the
α−1 = 137.036 match by chance.
Proof sketch. (1) α−1 (k) is strictly monotone on k =
1, . . . , 2000 (computed), so any sub-percent window contains at most one integer rung: enlarging any address
cap adds trials but no new hit-capable addresses beyond
those counted (hit-capability lemma). (2) Enumerating
the space of Table CXXIX yields n = 10,800 hit-capable
ladders. (3) The per-ladder false-hit probability is the
two-sided window-to-step ratio, p = 2 × 137.036 × 4.9 ×
10−9 /2.318 = 5.8 × 10−7 . (4) Linearity of expectation
gives E = np ≈ 6.3 × 10−3 (exactly 6.257 × 10−3 ); hostile
family/bit inflation multiplies E by at most ∼36, topping
out at 0.23 < 1. The wording of the match criterion is
essential: under the weaker criterion “within 1 ppm” the
expectation is 1.28 ≥ 1 and no theorem survives — the
theorem is true only for matches at least as good as the
observed 4.9 × 10−9 fractional match, and it is stated only
in that form.
□
c. The four-tier null ladder (worst case first). Each
tier is a complete, self-consistent null; each grants strictly
more than the one above it.

384
TABLE CXXIX. The 16-dimension address space A charged by the trials audit (its artifacts are in the reproducibility package).
“Cons.” = maximal-hostile count. The two convention factors marked ,→ enlarge existing rows (CKM Wolfenstein basis ×3; AU
prefactor list 9 → 30, i.e. ×10/3) and are folded there, so the convention total multiplies to 43,200 = 10 · 12 · 3 · 6 · 2 · 3 · (10/3).
Core totals: 64 (baseline) → 185,472 (cons.) → 8.01 × 109 (cons. × conventions); with convention/axiom bits 7.0 × 1022 .
# / dimension

baseline

cons.

provenance (one line)

1. det-line charge q1
2. gauge partition

{3, 6}
2 (SM, PS)

{1, 3, 6}
7

3. generation count Ngen

{1, . . . , 6} at
q1 =3
{5, 6}

box to cap
kmax =440
{5, 6}
+7 alternates

6. CKM apex branch

1 (det-line
chain)
2

8

7. closure grammar

1 (forced)

135

8. sign bits

2

23

9. axiom-adoption bits

1

240

10. χ primordial state

1

4

11. surface/topology menu 1
12. lens-space quotients
1

10
12

13. dim X
14. formula grammar
15. 57-subtraction
convention
16. χ period convention
(flux vs rest-mass)

1
1
1

3
6
2

menu 3Z proven (App. F); q1 ≥ 9 tail α-hit-dead
anomaly audit; dim G ≤ 28 kills SO(10), E6 ;
SM-embedding rivals α-hit-degenerate
discrete input (App. F); interval saturation proven;
cap = evaluated ladder
B−L spin-Z4 six-species list, exactly one Y =0 entry
(App. F)
dichotomy lemma; every alternate ≥ 30% off α
(hit-dead)
anti-theorem names both branches (App. AO);
×unit convention, ×η̄; ,→ ×3 Wolfenstein basis
(convention)
declared 9 × 15 scan (App. AU 5); ,→ prefactors
9 → 30 (convention)
rank-2 GF(2) theorem; the second bit, the
orientation of time, is the smooth beginning,
counted in row 9; non-numeric, generate no rival hits
the forty adopted statements of Table CXXVII,
among them Y.10 and E-transport′ ; adopt no value
from a menu
natural-preparation menu; Ωχ h2 ≤ 0.140 hard
finite-k spectral cap for any state
hostile enlargement
hostile enlargement; menu removed by proof
(Rem. AB.7: every member has π1 ̸= 1, excluded by
axiom (V7)) — the ×12 is retained here only as a
hostile over-count; the repaired axiom bits (V7),
(V10), (V4′ ), (V2) fall under row 9
hostile enlargement
hostile enlargement
hostile enlargement

1

2

4. padding n
5. assembly rule

null (most hostile first)

period-convention correction (Lemma AV.6);
charged against the Ωχ h2 row in both tiers; struck
only Rif a printed three-form sector derives
P = S 3 C3

what it grants

p

Z

full selection charge (M =12–20 booked comparisons, best-k pick)
nothing
≈ 0.03–0.5
n.s.
hostile menus, conservative windows, no selection charge
row independence (0.6–3.7) × 10−5 4.0–4.4σ
hostile menus, moderate windows, no selection charge
row independence (1.0–2.0) × 10−7 5.1σ
conditional : forcing theorems granted (pre-registered framing, corrected rows) the derivations
(0.3–1.7) × 10−7 5.1–5.4σ

Under the full selection charge the conservative product
(2.5 × 10−4 ) is unremarkable: the median product of the
three smallest of 20 uniform draws is 3.2 × 10−4 , i.e. a
hostile skeptic picking the three best of twenty booked
comparisons expects to do this well half the time.

Adopted combined-evidence claim (worst case first)
Under a full across-observable selection charge (the most
hostile admissible null), the combined evidence is not
statistically significant (p ≈ 0.03–0.5). Under hostile
menu accounting with no selection charge, the four
surviving independent hits (sin2 θW , θ12 -TM1, the H0 /Λ
grammar block, Ωχ h2 repriced at the flux-period value

385
0.1193, δ = 0.59%, with one binary period-convention bit
charged — address dimension 16, struck when the flux
period is derived) give ≈ 4.0–4.4σ (conservative windows)
to ≈ 5.1σ (moderate). Conditional on DFD’s forcing
theorems as stated — the corpus’s pre-registered framing,
on the corrected row basis — ≈ 5.1–5.4σ under the
conjunction-product convention, 3.6–4.3σ calibrated
(Fisher/Stouffer, the referee-standard reading). DFD does
not claim 5.5–7.1σ: its sin 2β row is zero-information
(Postulate E.1 is an independent normalization postulate
and the CKM anti-theorem blocks apex forcing; λ and A
miss at ∼1.8σ and ∼2.5σ), and its a0 row was priced
40–70× finer than the empirical uncertainty permits.
Separately, and given the stated bracket coefficient
c = 1/13: the α-ladder infeasibility theorem stands —
across the fully enlarged address space (∼10,800
hit-capable ladders), the expected number of false α
matches at least as good as the observed 0.0049 ppm is
∼0.006 (≲ 0.04–0.23 under maximal hostile family and
convention inflation; margin ×160 central, ≥ ×4 hostile,
×7.8 under the ±0.1 ppm falsifier band) — landscape
scanning cannot produce the α−1 = 137.036 match by
chance. Strong evidence, not proof.

Remark CE.2 (Scope: what the audit decides, and what
it cannot). Given the stated c = 1/13, the audit closes
the landscape-scanning charge against the α match: no
admissible enlargement of the address space makes a false
α−1 hit likely (Theorem CE.1), and every surviving pile
row carries its menu price. What the audit does not —
and cannot — adjudicate is the grammar/framing choice
itself: a skeptic who rejects DFD’s forcing theorems is
charged no trials and simply reads the unconditional tiers,
down to non-significance. What refutes or confirms the
framing is the pre-registered falsifier set (App. RM 2):
Belle-II |Vcb | at 0.5% (a confirmed inclusive 42.2 × 10−3
refutes A = 108α outright), FLAG ξ firming at 1.206
(refutes the locked-CKM ∆Ms /∆Md ), the Simons Observatory σ8 determination, the a0 (z) sign (contested:
within the survey the rise in low-mass discs at z < 1.44
is about 1σ counted galaxy by galaxy and depends on
how the molecular gas is added; anchored to SPARC’s
z = 0 value, the galaxy-level bins still exclude a frozen a0
at about 3σ, and DFD’s zero-parameter law fits them on
the bound-matter clock; massive discs to z ≃ 2.5 show no
rise at face value), and mβ = 9.15 meV.
d. The 5.5–7.1σ figure. DFD does not claim 5.5–7.1σ
(that figure gave P6 = 1.4 × 10−12 ⇒ 7.1σ and a “refereeproof floor” 3 × 10−8 ⇒ 5.5σ; exact recompute 1.65 ×
10−12 ⇒ 6.96σ and 4.59 × 10−8 ⇒ 5.34σ). Both rest on
two invalid rows and are not claimed, nor are the related
central/maximal 6.5/6.8σ figures (which retain the invalid
a0 window): (a) the sin 2β row carries zero information —
Postulate E.1 (App. AO) is an independent normalization
postulate, the CKM anti-theorem blocks apex forcing,
and the companion Wolfenstein integers miss (λ at ∼1.8σ,
A at ∼2.5–3.2σ); (b) the a0 row was priced at p = 0.005,
40–70× finer than the ∼10% empirical uncertainty on a0
permits. A re-audit with the current apex slot strengthens

the zeroing rather than reopening it: under the corpus’s
own PDG 2024 centrals (App. K) neither apex integer
is even the nearest point of its own grid — ρ̄ measures
21.80α (nearest half-integer 22, not 43
2 ) and η̄ measures
48.28α (nearest integer 48, not 49) — so the apex pair
carries menu price pmenu = 1.0 outright; under the earlier
PDG-generation snapshot (ρ̄ = 21.67α, η̄ = 48.62α) the
per-channel prices 0.66 × 0.76 with the ×2 half-integer
charge again give a pair price of 1.0. The four-Wolfensteininteger input block scores χ2 /4 dof = 9.7 (p = 0.046) resp.
9.4 (p = 0.053) identically under both apex slots; the
current slot improves five downstream observables but
touches none of the zeroing legs, so the four-tier ladder
above does not depend on it. All audit artifacts and every
number in this subsection are reproducible from the audit
scripts and outputs in the reproducibility package.

3.

The conditional no-coincidence bound

Proposition CE.3 (Conditional combined-evidence
bound). Grant DFD’s forcing theorems as stated (the
corpus’s pre-registered framing; this is the conditional tier
of Sec. CE 2, and it is read with the unconditional tiers
there). Treat each surviving row as a blind parameter-free
formula that had to land within its achieved fractional
agreement δi of the measured value, in an O(1) prior
range (pi ≃ min(1, 2δi )), except the H0 /Λ block, which is
priced by its declared-grammar count:


p sin2 θW ≈ 0.0045, p(θ12 ) ≈ 0.074,
p(H0 /Λ) = 3/135, p(Ωχ h2 ) ≈ 0.0117.
The four being independent (distinct sectors: electroweak,
neutrino mixing, cosmological clock/vacuum, dark sector),
the combined conditional probability is
Y
Pcond =
pi ≈ 8.7 × 10−8 =⇒ Z ≈ 5.23σ
i

(5.10σ with the binary flux-vs-rest-mass period-convention
bit of address dimension 16 charged, the recommended booking), rising to 5.44σ if a deliberately weak,
measurement-limited a0 row (p = 0.3) is appended (the
two-stage-H0 subline is not used: the early clock’s coefficient is measured and the local value rests on the stated
two-clock postulate, App. ZS): the conditional band is 5.1–
5.4σ, read together with the unconditional tiers (4.0–4.4σ
conservative-hostile, 5.1σ moderate-hostile, not significant
under the full selection charge).
Why a χ2 “consistency” statistic is the wrong tool
here. The tree-level sin2 θW = 3/13 matches to 0.23%,
but the measurement is precise to 0.02%, so a naive pull is
∼ −13σ. That is an artifact of comparing a leading-order
parameter-free value to a precision measurement: 3/13 is
the bare angle, and its radiative relation to the value at
MZ is not computed. The figure of merit is the fractional
agreement a blind formula achieved (the table), not the
measurement error.

386
4.

Scope of the bound

Remark CE.4.
• It is a preponderance / nocoincidence argument, stated as a four-tier null
ladder with the worst case first (Sec. CE 2): not significant under the full selection charge → 4.0–4.4σ
(hostile-conservative) → 5.1σ (hostile-moderate) →
5.1–5.4σ (conditional on the forcing theorems). It
is the same kind of evidence that established the
Standard Model and BBN, a wall of independent
numbers, but with every menu priced.
• It is not statistically significant under the most hostile admissible null (full across-observable selection),
and it is not a single falsification-test discriminator
(DFD̸=standard, confirmed in isolation). The SN–
lensing sign test has been decided against DFD’s
optical-distance dressing (App. SL); the clock LPI,
under systematic audit, awaits a clean verdict.
• Conditional means conditional: the 5.1–5.4σ
tier holds only if each forcing theorem is granted as
stated; a skeptic who rejects the derivations reads
the unconditional tiers, down to non-significance.
• Leading-order: agreement, not precision; the last
digits need higher-order terms.
• The α match is booked separately (it is the
selector datum, never multiplied into the pile): given
the stated c = 1/13, landscape scanning cannot
fake it (Theorem CE.1). The full CKM matrix
contributes zero information here (Postulate E.1)
and is excluded.
One-line claim:
Fully hostile: not significant.
Hostile menus: 4.0–4.4σ to 5.1σ.
Conditional on the forcing theorems
(read with those tiers): 5.1–5.4σ.
Given c = 1/13: landscape scanning
cannot fake α−1 = 137.036.
Strong evidence, not proof.

Appendix RM: Falsifiable-Predictions Roadmap:
How DFD Is Confirmed or Killed

DFD is a frozen-shape theory: no continuous parameter is fitted to the shape of any observable it predicts,
except the baryon density. Its constants are derived
or discretely selected, with the discrete inputs enumerated and priced (App. CE, row 9; App. F remarks) and
four continuous cosmological calibration constants carried openly: the vacuum coefficient R, the amplitude As ,
the coefficient of ωb h2 and ΛCDM’s shared astrophysical τ (App. FK, Rem. FK.8). Each prediction below is
therefore a sharp target. This appendix consolidates the
corpus’s scattered falsifiers into one prioritized roadmap.
A systematic extraction and critical triage of the full
corpus yields 15 measurable bets (where DFD differs
from GR/ΛCDM/SM, or from their rivals, at a testable
level: the six live Tier-1 entries, the seven bets of Tier 2
(its three nulls are one-sided targets), and the Belle-II
|Vcb | and MAQRO coherence tests of §RM 2), resting on
∼17 already-passed numbers at the grades of Tier 0;
further consistency-checks (where DFD reproduces the
standard value exactly, or the signal sits below any foreseeable sensitivity) are listed separately (§RM 6) and are
not counted as discriminators.
Retired: this test was run and DFD’s sign
failed
DES measured the opposite (ordinary focusing) sign at
6.0σ (Shah et al., MNRAS 532, 932, 2024;
arXiv:2406.05047 [191]), which by the pre-registered
falsifier retires DFD’s optical-distance dressing. DFD’s
core metric is PPN γ = 1 at first order and predicts the
standard −5/ ln 10, consistent with that measurement;
the surviving discriminator in this channel is the
amplitude and scale dependence of the convergence
mapping, not the sign. The failed claim is recorded below
as it was made, and in App. SL: Supernova–lensing
correlation sign. DFD’s optical-distance dressing
DL → DL e∆ψLOS forced standardized SN Ia Hubble
residuals to correlate positively with foreground
weak-lensing convergence, d(δµ)/dκfg ≃ +11 mag/κ
(band [+7, +28]), whereas GR/ΛCDM lensing forces a
negative −2.17 mag/κ — a definite ∼12 mag/κ sign flip,
not an amplitude argument. The four usable existing
datasets (Pantheon+ × DES-Y3, BOSS, DES-SN5YR,
Planck CMB-lensing; the DESI subset is too small) lean
DFD-positive, joining at +0.7σ under a calibrated
rotation null (+0.2σ with the SDSS counter-lean;
App. SL) — map-noise limited, not SN-count limited;
LSST Y1 (∼650 well-mapped SNe) or Roman (∼160)
reach a clean ≥ 3σ either-way verdict within ∼5 years
(App. SL). The sign cannot be faked by ΛCDM; the one
confound is correlated foreground dust (a
detection-cleanliness requirement, not a sign error).

387
1.

The complete bet list — master table

DFD is a frozen-shape theory awaiting adjudication: its
constants are frozen, so every row below is a number the
theory cannot move if the data land elsewhere. Grades
are the recorded grades; conditional stays conditional. Established postdictions and exclusions are listed alongside
the live bets for completeness.
• νR / mβ = 9.1514 meV — prediction of the adopted
neutrino branch: νR is required on the spin-Z4 B−L
completion (Cor. F.23, App. F) and carried by the
neutrino sector on either completion; seesaw MR =
α3 MP . Experiment: KATRIN-II / Project-8. Kills:
inverted ordering; any mβ off the comb.
> 0, a0 (z) =
• a√
0 (z) running sign — ∂z a0
2 α cH(z). Theorem, conditional on Axiom WS.1
(epoch-consistency/anchor-fork, Thm. XIX.3): the nomemory argument makes the anchored branch structurally unavailable, so tracking is forced, and the
conditional magnitude bracket [1.79, 2.83] at z=1 collapses to the H(z)/H0 line (1.78 at the measured clock,
Ωm ≃ 0.312); unique vs static MOND. Data, contested: over 0.33 < z < 1.44 the MUSE–DARK III
bins [109], in the analysis published in MNRAS 551(4),
2026 September 7 [110], give a0 (z) = A(1 + z)γ with
γ = 0.78 ± 0.15 (0.83 ± 0.13 with the SPARC z ≃ 0
anchor), excluding a frozen a0 at 5.3σ (6.6σ anchored)
√
and a0 ∝ ρm at 4.8σ (on the exponent’s error; 4.7σ
by likelihood ratio) on the published bin errors; at the
measured clock H(z)-tracking fits with χ2 = 5.36/3 and
A = (1.394 ± 0.032) × 10−10 m s−2 , and the measured
exponent sits 2.1σ below DFD’s γE = 1.098. Refitted
galaxy by galaxy from the survey’s public products, the
rise reproduces (γ = 0.77 ± 0.21), but the galaxy-level
uncertainty is several times the published bin errors, so
within the survey a frozen a0 lies about 1σ away, while
anchored to SPARC’s z = 0 value the galaxy-level bins
still exclude it at about 3σ, with DFD’s zero-parameter
law fitting on the bound-matter clock; the trend depends on how the molecular gas is added, keeping the
rise when tied to the survey’s disc masses, about flat
when tied to SED-based masses (γ = 0.06 ± 0.21; −1.03
to 0.96 at galaxy level), and falling with SED stellar
masses (γ = −1.04 ± 0.22; −2.32 to −0.32), which exceed the kinematic 95% limit in about a third of the
high-z galaxies; above z ≃ 0.9 the fitted disc masses sit
at the edge of the halo model’s prior because the kinematics put them low; DFD’s zero-parameter amplitude
lies 0.094–0.103 dex below the tracking amplitude on
the published census, on either clock, and within 0.02–
0.09 dex of it with either fuller census (0.18 dex with
both changes together; computed; App. A0Z). Massive
discs at 0.6 < z < 2.5 with molecular gas counted [111]
give a0 = 1.9, 1.4 and 1.2 × 10−10 m s−2 at z ≃ 0.8, 1.5
and 2.2 against DFD’s 1.8, 2.6 and 3.7 on the early clock
(1.89, 2.82 and 3.95 on bound-matter clocks): DFD’s
value is met at z ≃ 0.8, and the rise beyond z ≃ 1.5 is

not seen on that census (within-sample γ = −0.86+0.46
−0.48
on statistical errors). DFD’s domain-local screen lowers
a0 in overdense domains, by about 10% for massive
z ≃ 2 discs at the average environment of their haloes
and by 20–34% in the densest known structures, so it
would match those discs only in 10 h−1 Mpc domains
about 6 to 11 standard deviations denser than average, which their measured environments do not show
(computed on published values; App. ZT). Two further
caveats on the survey bins: under a 95% reading of
the source’s unlabelled bars the exclusions strengthen
(≳ 10σ and 9.5σ on the exponent’s error, 9.2σ by likelihood ratio, exponent family γ = 0.78 ± 0.08) but the
tracking consistency does not survive (χ2 = 19.9/3,
p = 1.8 × 10−4 ); and the fitted amplitude is +17% on
the canonical SPARC value (0.8σ systematic, +6.3σ
statistical, +5.3σ including the SPARC statistical bar).
Experiment: direct molecular-gas masses (CO, [C i] or
dust), to 0.1 dex for the z ≃ 2 discs; kinematic masses
not limited by a halo-model prior, with one pressuresupport correction for all samples; per-galaxy analyses
with galaxy-level errors, of the MUSE–DARK public
products and of the massive discs with a0 (z) imposed,
the latter with an independent bound on the χ halo or
rotation curves beyond 3Re ; JWST and ALMA rotation curves at z ≳ 3, ultimately SKA H i kinematics,
where tracking and the survey’s linear form diverge
(on this corpus’s evaluation at the survey’s fitted amplitude, a0 ≃ 4.20 against 4.18 at z = 2, and 6.32
against 5.77 at z = 3, with the measured exponent
family at 3.38 and 4.23). Kills, on a baryon census that
counts molecular gas, with galaxy-level errors and a
stated pressure-support correction: measured ∂z a0 ≤ 0;
high-z frozen a0 ; γ more than 3σ from γE = 1.10. On
published calibrations the massive discs meet the second
at face value and lean toward the first.
• Stellar M/L ceiling Υ[3.6] ≤ 0.58 — theoremgrade, a0 -free, and testable with data that already exist
(App. AT 7, Falsifier AT.22). DFD’s occupation reading
carries ν ≥ 1 as a theorem; raising the 3.6 µm disk
mass-to-light ratio pushes SPARC’s innermost gbar bin
below that floor at Υ[3.6] = 0.5775 M⊙ /L⊙ , +0.063 dex
above the stellar-population value, which the data currently clear by ≃ 2.5σ. Read backwards, the same
zero-parameter relation returns Υ[3.6] = 0.475, −0.30σ
from the independent SPS value (post-hoc). No acceleration scale, no dark matter and no cosmology enter.
Experiment: an improved SPS calibration of the 3.6 µm
disk ratio, or a systematics-controlled re-analysis of
SPARC photometry and H i masses. Kills: a converged
Υ[3.6] > 0.58 M⊙ /L⊙ .
• Macroscopic coherence — sustained fringes, no objective collapse (forced: no-noise variational action;
operator-completion linearity, App. ZU — the meanfield Schrödinger–Newton mechanism is not DFD’s, and
its frequency shift is the falsifier of the completion).
Experiment: MAQRO-class levitated optomechanics,

388
109–10 amu. Kills: mass-scaled collapse at the Diósi–
Penrose rate.
• GW memory — consistency test: DFD’s tensor memory equals GR’s, linear plus nonlinear (Christodoulou),
because the TT action is the full a2 Einstein–Hilbert
term (Derived; Prop. AT.9). Experiment: LISA
105–6 M⊙ at z=1, single-event SNR 11–35; ET/CE stacking. Kills: a memory measurement consistent with zero
where GR predicts a ∼20–27% permanent strain offset
(one-sided: a detection does not favour DFD over GR).
• Neff = 3.044 exactly — integer 3 forced (no dark radiation); decimal inherited from SM physics; one-sided
falsifier (the value itself is SM-degenerate). Experiment:
the Simons Observatory, with a forecast σ(Neff ) ≃ 0.05–
0.07. Kills: an Neff that departs from 3.044 by 3σ or
more, about 0.15 to 0.2 at that precision.
• Proton decay: exactly zero — if baryon number is
violated only by the electroweak anomaly, which changes
it only in steps of three; the sharpest DFD-vs-GUT
adjudicator. Experiment: Hyper-K / DUNE. Kills: a
single-nucleon decay with τp < 1040 yr.
• Millicharged particles: none, at any charge and
any mass — forbidden at theorem level given the
adopted gauge group (BSM-exclusion theorem, Extended Derivations, App. AH): a kinetic-mixing millicharge requires a dark-photon U (1)′ , excluded because
a sixth chiral multiplet violates the saturated Spinc index χ(CP 2 , E) = 60; a bare millicharge is itself a new
charged internal mode, excluded by the same saturation;
hypercharge-twist integrality (App. F) corroborates conditionally — every admissible twist in the printed menu
quantizes electric charge coarsely, leaving no millicharge
window. Experiment: accumulator-augmented shieldedCavendish Gauss-law tests [192, 193], which reach the
irreducible cosmic-ray-produced millicharge population
on a few-year timescale, beyond accelerator reach at
sub-GeV masses; the bet is one-sided (a null is SMdegenerate). Kills: any confirmed millicharged particle,
at any ε and any mass (a bare Gauss-law anomaly alone
does not — a photon mass is the source paper’s stated
alternative reading).
• |Vcb | (A = 108α) — Aλ2 = 0.04033; DFD survives only
on |Vcb | ≲ 41 × 10−3 ; parked on the inclusive/exclusive
puzzle; nothing can move (A ≈ 113α forbidden by nA =
108). Experiment: Belle-II at 0.5%. Kills: inclusive
42.2 × 10−3 confirmed.
• ∆Ms /∆Md — 33.3 (−5.1%, ∼1.9σ at FLAG ξ =
1.206(17)); locked CKM, no rescue knob by construction.
Experiment: lattice ξ at 0.5%. Kills: firm ξ = 1.206
(⇒ 4σ + ; DFD needs ξ ≈ 1.24).
• σ8 ≃ 0.8063 (S8 ≃ 0.822) — current measured-clock
forward normalization (one inherited As coefficient),
with R8 = 11.848 Mpc and Ωm ≃ 0.312 recomputed
rather than frozen; KiDS-Legacy consistency is 0.45–

0.74σ depending on the printed primary/joint anchor;
DES Year 6 lies 2.8σ below (survey errors only). CMB
lensing gives Aϕϕ = 0.9887 (−0.80σ). Experiment:
Simons Observatory and SPT-3G lensing + shear. Kills:
a converged clustering amplitude incompatible with the
frozen-input forward calculation after stated theory
uncertainties.
a. Established
postdictions (already adjudicated).
√
H0early = C α57/2 /tP = 67.52 ± 0.09 (exponent derived, coefficient measured; five concordant CMBanchored channels; the C = 1 value 72.09 is excluded
as the early clock and is the bound-matter clock H0loc =
α57/2 /tP of Postulate ZS.11, 0.2σ from the Cepheid-free
local mean 71.87 ± 0.96). The dependency-aware current
result is S8 ≃ 0.822, consistent with KiDS-Legacy and
2.8σ above DES Year 6 on survey errors only (App. GR).
The geometric-vacuum exponent 57, the sin2 θW = 3/13
relation, and α−1 = 137.036 retain their printed grades;
no measured coefficient is promoted to a derivation.
b. Exclusions (not bets). No Born-statistics deviation is predicted (mean-field sourcing is exactly Bornconsistent — a null Born test does not discriminate).

2.

The four cleanest near-term decisive tests

1. Belle-II |Vcb | at 0.5%: a confirmed inclusive 42.2 ×
10−3 kills A = 108α outright — the integer cannot
move.
2. MAQRO coherence: DFD sustains fringes exactly
where Diósi–Penrose collapses; a mass-scaled collapse
signal kills.
3. Simons Observatory Neff : a departure from 3.044 by
3σ or more, about 0.15 to 0.2 at the forecast σ(Neff ) ≃
0.05–0.07, kills DFD (one-sided: it can only falsify,
since the value is SM-degenerate).
4. Hyper-K proton decay: a single-nucleon decay with
τp < 1040 yr kills DFD while confirming GUTs.
3.

Tier 1 — near-term decisive (≲5–10 yr, pass/fail)

Nine entries follow. Six are near-term decisive discriminators (each differs measurably from the standard
prediction or from its rivals and is resolvable by a named
experiment within roughly a decade); entries 1, 3 and
4 are retired as discriminators, and each names the test
that remains in its channel:
1. SN–lensing correlation sign: retired — the predicted +11 mag/κ was adjudicated against DFD by the
DES 6.0σ detection of the opposite sign (MNRAS 532,
932, 2024). DFD’s core predicts the standard −2.17.
What remains testable here is the amplitude and scale
of the convergence mapping (LSST Y1 / Roman), not
the sign.

389
2. PMNS solar angle sin2 θ12 = 0.3184 (TM1) vs measured 0.307 (+0.95σ; JUNO’s first result 0.3092 ±
0.0087, 1.1σ) — JUNO reactor/solar; ∼2027–30.
3. ISW amplitude: retired as a discriminator — a
∼30% optical-bias figure has no derivation and is not a
DFD prediction; under the adopted Rest-Mass Channel
closure (Qχ = 1, App. GR; no slip, Cor. NL.17) and the
real α57 vacuum term (App. AU) DFD gives AISW =
1.00, ΛCDM-identical, vs Planck 2015 1.00 ± 0.25 [92];
CMB × galaxy cross-correlation with AISW < 0.6 or
> 1.5 at 3σ would still kill the closure.
4. α(z) drift: retired as a Tier-1 discriminator —
the value +2.3 × 10−6 at z=1 comes from the retired
ψ-screen (∆ψ(1) = 0.27) and is not a DFD prediction;
res
the screen-free route (Eq. (253)) gives ξLPI
× (+2.0 ×
−6
res
10 ) with ξLPI undetermined, so no definite value
is predicted — QSO absorption (ESPRESSO/ELT)
remains the test.
5. CKM apex: γ=66.31◦ , sin 2β=0.719, an εK deficit
∼13% vs SM free parameters (tension box: companion
volume, App. AH) — LHCb / Belle II; ∼2027–30.
6. Th-229 nuclear-clock modulation (extended
strong channel; null under the minimal action,
Thm. XI.2): a perihelion-locked annual signal, 26 Hz–
O(1 kHz) vs GR/SM null — dedicated Th-229 clock;
near-term.
7. RAR intrinsic-scatter floor: ≲0.037 dex (haloshape floor alone 0.013–0.025; a0 a fixed constant; if
the χ halos inside galaxies are cored, as the rotation
curves require under the floor law, App. AV) vs untuned 0.06–0.08 — SPARC-class rotation curves.
8. Lepton-flavor universality: RK = 1 up to StandardModel corrections vs BSM RK ̸= 1 — LHCb Run 4;
∼2027–30.
9. Local wide binaries: γ = 1 exactly near the Sun,
with no orientation term of either parity (environmental
confinement, App. NL), vs MOND’s γ ≃ 1.4 and 6.9%
even modulation at 104 AU; Gaia DR4; near-term.
Kills: a confirmed boost of the Chae type in local
pairs.
4.

Tier 2 — future decisive (next-generation
instruments)

The following 10 await more capable facilities; the χ
photon line, the primordial B-modes and the linear-growth
statement are nulls that a detection would falsify, and the
other seven are bets: Neff = 3.044 with structurally forbidden dark radiation (Simons Observatory: a ∆Neff > 0
detected at 3σ, about 0.15 to 0.2 at its forecast precision,
kills DFD while ΛCDM absorbs it); no photon line from
χ: χ has no photon coupling (E = 0; App. AV), so a
dark-matter photon line at 16.0 eV (77.5 nm) or axion-like
signal at 32.005 eV (38.74 nm), at any strength, falsifies

it; maximal θ23 = π/4, δCP = −π/2 (DUNE/Hyper-K);
normal neutrino ordering + Σmν = 61.5 meV (Simons
Observatory+DESI; inverted ordering kills DFD); zero
single-nucleon proton decay (if baryon number is violated only by the electroweak anomaly); a step in the
dynamics of wide binaries across R ≃ 13–16 kpc (environmental confinement; halo pairs; the outer-halo globular
clusters already
contradict the boost beyond it, App. NL);
√
a0 (z) = 2 α cH(z) epoch evolution; one universe, two
clocks (Postulate ZS.11; Falsifier ZS.14): bright sirens and
mass-sheet-robust time delays at 72.1, not 67.5; early-ruler
and local ladders that do not converge with distance; the
absolute a0 scale at 1.1966×10−10 m s−2 on distances computed with H0loc ; no star established older than 12.92 Gyr
in laboratory units at better than 3σ (one programme’s
globular-cluster ages of 13.39 and 13.61 Gyr sit 1.8–2.0σ
above it, and the ages of the universe of 13.5–13.8 Gyr
that the same programme infers with a ΛCDM formation
delay sit 2.1–2.6σ above it, with statistical and systematic
errors in quadrature); no primordial B-modes, r ≲ 10−7
(LiteBIRD and the ground-based B-mode programmes:
any detected primordial tensor signal washes out the χ
relic, App. SM); plus the cosmological-growth statement
of the adopted Rest-Mass Channel: with the polarization
confined to bound systems, linear growth follows Qχ = 1
and σ8 rises by only 0.1–0.5% (App. GR), so kSZ and
f σ8 carry no DFD excess or suppression on linear scales
(§RM 6).

5.

Tier 0 — the already-passed bedrock (would
have killed DFD; did not)

Numbers DFD already matches, each at the grade
stated in its entry and in App. FK (absolute light-fermion
masses are fit-assisted and excluded):
• α−1 = 137.0360 (microsector closed form, exact under
two stated choices): 21 ppm from the integers alone;
the bracket coefficient is open.
• sin2 θW = 3/13 = 0.2308: 0.23%.
√
• Higgs vev v = MP α8 2π = 246.09 GeV: 0.05%, with
the exponent derived and the prefactor fitted.
√
• mt √
= (1−α)v/ 2 = 172.74 GeV (0.52σ); mc =
αv/ 2 = 1.270 GeV (within 0.25%).
√
• Forced ratio mτ /mc = 2: +1.1%.
p
√
• MOND scale a0 = 2 α cH0loc = 2α29 c7 /Gℏ =
1.1966 × 10−10 on bound-matter clocks (Postulate ZS.11): within SPARC’s systematic-dominated
error.
√
• H0early = C α57/2 /tP (67.54 at C = 0.87784; 67.52 ±
0.09 across channels): exponent forced, vacuum coefficient R measured (today’s early-clock value C); on
bound-matter clocks C = 1 and H0loc = α57/2 /tP =
72.09, 0.2σ from the Cepheid-free local mean 71.87 ±
0.96, conditional on Postulate ZS.11.

390
• sin2 θ13 = 3α = 0.0219: 0.1σ. Neutrino splittings:
χ2 = 0.025 vs NuFIT.
• CKM sin 2β = 0.719 (1.0σ), λ = 31α = 0.2262 (0.54%);
Bs → µµ = 3.40 × 10−9 (0.2σ).
• Strong-CP θ̄ = 0 at tree level with θYM = 0 fixed by
the spectral action given that nothing is added to it for
the gauge fields, loop-bounded below 10−16 (no axion);
nEDM bound satisfied.
• CMB-lensing Aϕϕ = 0.9887 at the current background
(−0.80σ); the AL = 1.0542, σ8 = 0.820 row belongs to
the one-clock C = 1 background.
• cT = c (GW170817); proton τp > 1034 yr (Super-K).
• Kennedy–Thorndike null: cavity/atom ratio boostindependent at fixed ψ (Cor. XI.3); remainders ≲ 10−22
against the measured βRMS − αRMS − 1 = −1.7(4.0) ×
10−8 [61].

6.

Separation and open cracks

Consistency-checks (not discriminators). The following reproduce the standard value exactly, or lie below
foreseeable sensitivity, and a null result does not distinguish DFD: PPN γ = β = 1 and all classic 1PN tests (GRidentical by construction); BH entropy S = A/4 = SGR
and TDFD = TH , from the event horizon’s surface gravity
κ = c4 /4GM (a magnified-area entropy is not a DFD
prediction); with the parent strain’s σ kept, the strongfield sector: black-hole horizons, shadows and ringdowns,
neutron-star masses, the second post-Newtonian dynamics and the absence of echoes, all general relativity’s
(App. WS; for spinning holes argued beyond first order
in the spin); GW speed cT = cγ exactly (kills TeVeS,
does not favour DFD over GR); zero spatial curvature,
with no dark curvature either (App. WS, Rem. WS.7),
flat ΛCDM’s value, which a detected curvature of either
kind falsifies; the NDW = 1 wall relic and βiso ≃ 0 (onesided/structural, observationally null); the 0νββ effectivemass comb (all teeth 1–2 orders below LEGEND/nEXO);
charged LFV (null, ≥31 orders below limits); the neutron
EDM (coincides with the SM CKM background); the Oort
spike (a Newtonian Sun under environmental confinement;
GR-identical, but excluding MOND); the Local Group’s
matter barycentre, which moves with the cold local Hubble flow (ΛCDM-identical; a drift of 15 km/s or more
along the Milky Way–M31 line falsifies the momentumconserving energy); and the clock-normalized hierarchy
ρΛ /ρPl = ΩΛ (3/8π)Cα57 ≃ 1.14 × 10−123 (DFD derives the 10−123 exponent parameter-free, microsectorscoped with the O(1) prefactor via the clock dictionary
(Rems. AU.3, AP.6) — a parsimony gain, but the value
matches the observed Λ, so it is not a measurable difference). The Ġ/G channel is resolved by the coupling/statefunction distinction (Thm. XIX.3, Cor. XIX.4, both conditional on Axiom WS.1): G is an epoch-independent

action coupling, Ġ = 0 exactly — GR-degenerate, passing lunar-laser-ranging trivially, hence still not a discriminator —
√ while the discriminating partner prediction ȧ0 = 2 α c Ḣ ̸= 0 lives in the a0 (z) entry above.
Gravitational-wave memory is also a consistency-check:
DFD’s tensor memory equals GR’s (Prop. AT.9), so a
LISA/ET detection at the GR level does not favour DFD
over GR, while a memory null where GR predicts ∼27%
of the edge-on strain peak falsifies DFD.
Open cracks, and two resolved by axiom. The
χ-abundance is forced-amplitude/stated-convention (production; App. AV Step 5b): the finite SU (2)60 CS-vacuum
Casimir expectation gives Ωχ h2 = 0.1192806 (−0.60σ);
the amplitude is forced (Casimir; measure conditional on
the asserted premise P4′ (Lemma AV.27); k(k+2) from χ’s
derived Z2 + Sugawara), the only non-DFD input being
the standard cosmological relic-redshift. The operator is
proven and the density is not fitted, but the Z2 /Sugawara
edge, the relic-map normalization, the count 158 and the
flux-period identification remain stated premises. The
Q = Geff /G = 1 linear-clustering (third-peak, lensing) is
provably not derivable from the single-W action (“structured ⇔ gradient ⇔ screened” is a theorem; App. GR),
so DFD adopts the minimal Rest-Mass Channel postulate (a second µ=1 rest-mass Poisson operator; cost: +1
gravitational axiom; matter moves under a momentumconserving energy, and χ falls like the baryons wherever
it fills the µ-law’s demand), and given it Qχ = 1 holds
and the clustering is consistent. That postulate carries
a testable cluster-vs-galaxy prediction: clusters need χ,
at 0.84 ± 0.12 of the cosmic mix when it adds its Newtonian pull (under the adopted floor composition the need
is 1.28–1.51, and the derived abundance under ΛCDMlike assembly gives 1.21 ± 0.05, at estimate grade), while
baryon-dominated disks show little dark mass beyond the
µ-law (App. GR), distinct from both ΛCDM and pure
MOND. The MOND response of a subsystem inside a
strong host field is closed by a second adopted axiom, environmental confinement (+1 gravitational axiom), which
the Bose statistics provably cannot supply (App. NL).
One further crack is open. χ composes with the µ-law by
the floor composition, adopted in energy form; under that
energy χ halos assemble as in ΛCDM (estimate), and the
rotation curves then need cored inner halos, whose origin
is open, and in discs above about 1011 M⊙ of baryons
lighter halos than abundance matching gives; in spherical
simulations of dwarf and low-surface-brightness galaxies,
feedback from stars does not make these cores under that
energy (App. AV). The photon coefficient of χ is zero,
E = 0 (App. AV). The growth statement of Tier 2 follows
from the same structure. The abundance crack and the
results above stand independently.
DFD makes 15 sharp, falsifiable bets (the principal ones,
with the numbers that kill, are consolidated in App. RM 1)
that differ from the standard models, or from their rivals,
in ways telescopes and labs can check. One further bet has
been settled against DFD: the supernova–lensing sign flip
was adjudicated by DES at 6.0σ in favour of the ordinary

391
focusing sign (MNRAS 532, 932, 2024), retiring DFD’s
optical-distance dressing while leaving its core metric —
which predicts that same standard sign — intact. The
remaining bets stand as printed.

Appendix WS: Wave-Sector Closure: Elliptic Scalar,
Derived Gravitomagnetism, δrad = 0, and Ringdown

This appendix derives three elements of the wave sector: (i) the radiative status of the scalar sector, δrad = 0;
(ii) the gravitomagnetic sector g0i , beyond PPN-template
matching; (iii) the relation between DFD’s physical coordinates and the standard PPN gauge. Two named inputs
carry the result, Axiom WS.1 and Proposition WS.5.
It then states the ringdown and echo items: with the
scalar-longitudinal piece σ of the parent strain kept, the
remnant of a merger is a Kerr hole (argued beyond first
order in the spin), it rings as Kerr, and there are no echoes
(Subsec. WS 5).

1.

Axiom E (Elliptic Constraint Reading)

Axiom WS.1 (E: the scalar sector is an instantaneous
constraint). In the preferred foliation (absolute time t),
the scalar sector obeys the elliptic AQUAL constraint at
each instant,


16πG
∇· µ(|∇ψ|/a⋆ ) ∇ψ = − 4 S[matter; ψ](t), (WS1)
c
with no independent temporal kinetic term in the local
(non-cosmological) sector; through first post-Newtonian
order the source S is the matter source of Lemma WS.3.
The absence of a temporal kinetic term, which makes the
action differ from general relativity’s by the (K − K̄)2
term of Proposition WS.6, holds, once baryons have decoupled from the photons, on the whole of every slice
on which the µ-law acts somewhere; while no baryons
are bound anywhere, and at least until baryons decouple from the photons, the scalar sector carries instead
the Einstein–Hilbert term’s own temporal term, and the
axiom selects its slicing (Remark WS.7). The one-field
form holds for systems without χ; inside bound systems
that contain χ the main text’s adopted energy of the two
Newtonian fields governs. Wherever the corpus writes
a c-speed d’Alembertian for ψ, that form is the numerical hyperbolic-relaxation of Eq. (WS1), not a claim that
ψ radiates (the channel-scope remark of App. AT). The
temporal functional K(∆) of App. Q remains the cosmological sector: ∆ is defined against the FRW screen flow
and governs homogeneous-deviation (dust-branch) physics
only.
For static matter in the µ → 1 band the source is
the main text’s optical-source law to all orders: on the
static matter metric, with the divergence taken in the
unimodular spatial part (eh )ij , that law is the time-time
component of the trace-reversed Einstein equation (computed), and with the parent strain’s σ kept, σ obeys the
traceless part of the Einstein equation and the Hamiltonian constraint follows (Proposition WS.6), so static
stars obey the equations of general relativity. Beyond
first order the source of moving or rotating matter also
contains the energy of the gravitational field, among it the

392
gravitomagnetic term ∂i (g0j Fij ) with Fij = ∂i g0j − ∂j g0i ,
and terms in ∂t ψ (computed); with the parent strain’s σ
kept the source is that of the maximal-slicing lapse equation and the slicing is maximal (Proposition WS.6(b),(e)).
Loop-induced higher-derivative terms of the matter fields
enter the constraint only through the renormalized source
⟨ε̂ + 3P̂ ⟩ren of the main text’s optical-source theorem, as
a relative correction of order (k/M̄P )2 ; written as terms
in ψ itself, their R2 part would supply, wherever the
(K − K̄)2 term acts, the only time derivatives and give
Eq. (WS1) a growing root at a rate of order (M̄P k)1/2
(computed).
Remark WS.2 (Why the elliptic reading is forced). A
no-go is proved: if the scalar sector is hyperbolic, the
far-zone breathing flux of a compact binary is δrad =
0.193 for PSR B1913+16 (monopole 0.015, cross 0.010,
quadrupole+trace 0.168; pipeline validated against Peters–
Mathews to 5 digits), excluded at ∼100σ; and no referenceflow choice in the K(∆) sector repairs this without shifting
the 1PN periastron advance by δ ω̇/ω̇ ≃ +0.064 (direct orbit integration), which breaks the measured mass triangle
at ∼90σ. The elliptic reading is the only structure specified in the corpus that survives both tests. Instantaneity
in the preferred frame is thereby a prediction carrier : it
is exactly why the scalar channel is radiation-free, while
all radiative physics lives in the (hyperbolic, luminal) TT
sector. Laplace-aberration safety is supplied by Theorem WS.9 below: matching EIH dynamics through 1PN
is precisely the statement that no anomalous aberration
effects appear at current sensitivity.

2.

The verified 1PN source and the Standard-Gauge
Coincidence Lemma

Lemma
WS.3
(1PN source,
point-particle
form;
machine-verified).
Direct
variation
of
R p
Spp = −mc2 dt e−ψ − eψ v 2 /c2 gives the conventionfree 1PN source
X ma c2 h
v2 i
S =
1 − 21 ψa + 32 2a δ 3 (x − xa ), (WS2)
2
c
a
(ψa : external field at a; self-terms renormalized). The
fluid form is S = 12 ρc2 [1+Π/c2 +2v 2 /c2 +2U/c2 +3p/ρc2 ]
(sympy-verified); the two agree through the proper-volume
factor e−3ψ/2 .
Lemma WS.4 (Standard-Gauge Coincidence). Solving
Eq. (WS1) with source (WS2) at 1PN (µ → 1) gives
ψ = 2U/c2 + ψ2 /c4 with
ψ2 = 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4
(no χ̈ term: the solve is instantaneous),

(WS3)

so that
2U
c2
4Φ1 + 4Φ2 + 2Φ3 + 6Φ4 − 2U 2
+
,
c4
(WS4)


2U
(WS5)
gij = e+ψ δij = 1 + 2 δij ,
c
which is exactly the standard-PPN-gauge 1PN metric of
general relativity at γ = β = 1 with all other parameters
zero (no A term, no Whitehead term). DFD’s physical coordinates — the diagonal optical dictionary
plus absolute time — coincide with the 1PN standard PPN gauge, with no transformation and no
residue.
g00 = −e−ψ = −1 +

Proof. Green-function solution of ∇2 ψ2 = −8πGρ[Π +
2v 2 + 2U + 3p/ρ] term by term against ∇2 Φk =
−4πG (densities). The χ̈ term of the hyperbolic treatment
arises only from iterating c−2 ∂t2 ψ; Axiom E removes it at
the root. The Whitehead potential cannot arise: the field
equation’s only nonlinearity is the local, multiplicative eψ
in the source, and a local source under the flat Green
function never generates the non-local double integral
ΦW (Whitehead Locality Lemma). β = 1 is exact because ψ2 contains no U 2 piece; −2U 2 comes solely from
e−ψ = 1 − ψ + ψ 2 /2 − · · · .

3.

Vector-Sector Completion: g0i derived, not
matched

The Einstein–Hilbert term that the TT sector carries:
Proposition WS.5 (Spectral Einstein–Hilbert reduction). Let F be the truncated microsector, with Hilbert
space of dimension N (N = 60) and internal Dirac
operator DF , and let D = DM ⊗ 1 + γ5 ⊗ DF on
M4 × F . For a plateau cutoff f , constant near the origin, with every internal mass inside the plateau
(with G
√
fixed by Eq. (WS7) the top internal mass α MP sits at
m2 /Λ2 = αN f2 /(3π +f0 TrDF2 /MP2 ), at most 0.046 f2 , so
f2 /u0 ≲ 21.5 suffices for a plateau on [0, u0 ]; for a smooth
cutoff with f2 ≃ 3f0 and u0 = 1 that gives f0 ≲ 7, and
the internal masses of the near-arithmetic lattice relax it
to f0 ≲ 8.6; computed),
 D2 
Tr f 2
Λ
Z
N f4 Λ4 − f2 Λ2 TrDF2 + 21 f0 TrDF4
√ 4
=
gd x
4π 2
Z
N f2 Λ2 − f0 TrDF2
√
−
g R d4 x
48π 2
Z


f0 N
√
+
g 11E4 − 18 Cµνρσ C µνρσ d4 x
2
5760π
+ Sgauge + Sscalar ,
(WS6)

393
R∞
R∞
with f4 = 0 uf du, f2 = 0 f du, f0 = f (0) and E4 the
Gauss–Bonnet density. No local term beyond a4 appears;
this is an asymptotic statement, with a remainder that
falls faster than any power of Λ for curvatures and momenta well below Λ. The a2 term is the Einstein–Hilbert
functional of the four-dimensional metric, with
1
N f2 Λ2 − f0 TrDF2
.
(WS7)
=
16πG
48π 2
The result rests on two premises. (G1) The Einstein–
Hilbert term acts on the scalar field only through the
rest-mass channel of App. GR, that is, on the uniform
background and on the rest-mass perturbations, where it
replaces the printed Newtonian term rather than adding
to it; the µ-law acts outside it. The printed scalar action
is therefore the µ-bent scalar part of the Einstein–Hilbert
term: one term, not the Einstein–Hilbert term plus a
second one. G1 is not independent: its one term is the
static-energy identity of Proposition WS.6 with one G,
and its partition is the Rest-Mass Channel. (G2) Gravity’s
spectral trace uses the four-dimensional Dirac operator
D above, while the closed form for α uses its own trace
on the internal space (the regularization rule of the main
text’s convention-locked α); these are two different traces.
2
Proof sketch. Since D2 = DM
⊗ 1 + 1 ⊗ DF2 , the trace is a
sum over the internal eigenvalues λ2i of four-dimensional
heat-kernel expansions with shifted moments; on the
plateau f2 (x) = f2 − f0 x, f4 (x) = f4 − xf2 + 12 f0 x2 and
f0 (x) = f0 exactly, and every higher coefficient carries a
derivative of f at the origin, which vanishes. The truncation is exact in the four-dimensional sense on M4 × F ;
on a continuous internal space of odd dimension it would
not be. The Lichnerowicz formula gives E = −R/4, so
tr(E + R/6) = −R/3 per internal mode, which fixes the
attractive sign of G (an operator with E = 0 would give
the opposite sign), and the a4 coefficient of the Dirac operator contains no R2 term. Equation (WS6) reproduces
the exact Dirac spectral sums on the round four-sphere
and on S 2 × S 2 times a 60-mode internal space, with a remainder that falls faster than any power of Λ (computed);
its normalization agrees with that of Chamseddine and
Connes, the factor between 48π 2 and their 24π 2 being
the moment convention, with their mass trace equal to
half of TrDF2 .

The massless zero mode hµν (x) is the four-dimensional
metric itself, so its quadratic part is the linearized
Einstein–Hilbert action used at 1PN below and its cubic
vertex carries the tensor memory (App. AT). In strong
fields, where µ → 1, the Einstein–Hilbert term governs
the whole metric: with the parent strain’s σ kept, matter,
light and the tensor field share it, and the static exterior
is the Schwarzschild metric in DFD’s coordinates. The
tensor and vector sectors, and the scalar sector at µ → 1
(Lemma WS.4), share one G. Its value is not predicted:
it fixes N f2 Λ2 − f0 TrDF2 in Planck units, one scale measurement, as in
√ general relativity, and the internal masses,
57 modes at α MP in the printed spectrum, shift 1/G

by about 4%. The Weyl-squared
term implies an extra
p
spin-2 pole at m2 = Λ 10f2 /3f0 ≃ 3.6 M̄P for N = 60
and f0 = 1, above the cutoff, a tachyonic ghost under the
continuation that gives G > 0; it moves no observable
((k/m2 )2 ∼ 10−81 in the LIGO band), and the action
stops at a4 only below the cutoff. The a4 gravitational
term does not enter the Friedmann law: the Weyl term
vanishes on the Friedmann background and the Gauss–
Bonnet term is a total derivative. The displayed a0 term
is Planckian: on the near-arithmetic lattice of the internal spectrum, m2n = (n − 3) αMP2 /57 (n = 4, . . . , 60), its
4
term −f2 Λ2 TrDF2 /4π 2 alone is −(0.532 + 0.012f
√0 ) M̄P ,
and −(1.045 + 0.046f0 ) M̄P4 with all 57 modes at α MP
(computed). The dark-energy density is the α57 microsector term of App. AU with its coefficient R measured, and
reading it as the α57 -suppressed a0 term is a conjecture,
which needs the whole DF -dependent vacuum suppressed
with it. On the truncated microsector there are no Kaluza–
Klein modes; on a continuous K the Lichnerowicz count
(b1 (CP 2 ) = b1 (S 3 ) = 0, both factors Einstein-rigid) leaves
no extra massless spin-2 mode.
The spectral action and its Einstein–Hilbert term are
the construction of Chamseddine and Connes (Phys. Rev.
Lett. 77, 4868 (1996); Commun. Math. Phys. 186, 731
(1997)), and the idea that gravity is induced by the quantum fields of matter is Sakharov’s (Dokl. Akad. Nauk
SSSR 177, 70 (1967); Sov. Phys. Dokl. 12, 1040 (1968)).
What DFD adds is this: its own 60-mode finite space, the
structure that fixes α, supplies the finite geometry of that
construction; the Friedmann law and the nucleosynthesis
check follow (the optical-time Friedmann law of the main
text); the scalar sector and the Einstein–Hilbert term
are one term (premise G1); and the clock constant is
independent of G (App. AP).
Proposition WS.6 (Einstein–Hilbert completion of the
parent strain). Let the physical metric carry the full parent
strain of the main text’s gravitational-wave section: lapse
N = e−ψ/2 , a shift, and spatial metric gij = eψ (eh )ij
with hij the traceless part of the parent strain (hTT
ij , the
vector piece and the scalar-longitudinal piece σ), so that
N 2 (det gij )1/3 = 1 (the lock); let its dynamics be the
Einstein–Hilbert term of Proposition WS.5, with no temporal kinetic term for ψ (Axiom WS.1), in DFD’s absolute
time, in the rest-mass sector, where µ → 1. Then, for
isolated systems:
(a) the lock leaves only the spatial diffeomorphisms that
preserve the arena’s volume form; they remove the
vector piece but not σ, so σ is physical, and its
equation, from the traceless part of the Einstein
equation, is not a gauge condition;
(b) every solution has vanishing mean curvature of
DFD’s slices, K = 0: the absolute time is the maximal slicing;
(c) general relativity’s Hamiltonian constraint holds,
and the solutions are exactly those of general rela-

394
tivity in maximal slicing, written in the coordinates
of the lock;
(d) static stars obey the Tolman–Oppenheimer–Volkoff
equations; the exterior of a static star is the
Schwarzschild metric, e−ψ = 1 − 2GM/(c2 R)
at areal radius R, with ψ = 2GM/(c2 r) −
4
2 3
3 (GM/c r) + · · · in the arena radius r; a stationary rotating exterior is Kerr’s at first order in the
spin, and a spinning black hole’s is Kerr’s; through
second post-Newtonian order the two-body dynamics is general relativity’s; and for compact bodies
inertial and gravitational mass agree;
(e) the scalar equation is the maximal-slicing lapse equation, which for static matter is the main text’s
optical-source
divergence of (eh )ij :
 −h ij
 law with the
4
∂i (e ) ∂j ψ = −(8πG/c )(ε + 3P ) eψ ;
and on the Friedmann background:
(f ) linear perturbations keep the mean curvature of
DFD’s slices uniform, δK = 0 for k = 0 wherever no phantom field changes: the absolute time
is the uniform-expansion slicing, no scalar mode
propagates, the curvature perturbation ζ of adiabatic modes is conserved outside the horizon, and
for the curvature constant k = 0 of the first integral
the linear equations are general relativity’s.
The equivalence with general relativity in (b)–(d) and
(f ) is proved or computed for static configurations, for
linear perturbations of flat space and of the Friedmann
background, through second post-Newtonian order, and at
first order in the velocities to all orders in compactness;
for fully nonlinear time-dependent fields, for stationary
rotation beyond first order in the spin, and for maximal
slicing during generic rotating collapse, it rests on the
argument given for (b).
Proof sketch. (a) On the lock a spatial diffeomorphism


ξ at fixed slicing changes ln N (det gij )1/6 by 13 ∂i ξ i , so
only ∂i ξ i = 0 survives: two functions, which remove the
transverse vector piece; setting σ = 0 would take a third.
At first post-Newtonian order σ does not enter gij , so
the isotropic dictionary of Theorem WS.8 is exact there.
(b) Without a temporal kinetic term for ψ the action
differs from general relativity’s by a term quadratic in
K; general relativity’s action is invariant under normal
deformations of the slices, and the variation of the remaining term along them gives Di (N 2 Di K) = 0 on every
solution, so N > 0 and K → 0 at infinity force K = 0
(for actions whose momentum trace is proportional to
K, the Hamiltonian form of this identity is that of Henneaux, Kleinschmidt and Lucena Gómez, Phys. Rev. D
81, 064002 (2010), eq. (4.1)). This is computed for static
fields, for linear perturbations, and at first order in the
velocities to all orders in compactness, where the momentum constraint alone forces K = 0; for general fields
the identity, and the existence of the lock’s coordinates

on general data, are argued, not computed. (c) Every
term of the action is spatially covariant; with the matter, momentum, traceless and ψ equations imposed, the
spatial diffeomorphism identity
gives the Hamiltonian√
constraint density as c(t) det g N 2 , and asymptotic flatness sets c = 0; the term quadratic in K vanishes on
solutions, so the remaining equations are general relativity’s, and conversely general relativity’s maximal solutions,
solve them. For static fields

√ written with
√ the lock,
ˆ − 1 |∂ψ|2 , ĝ = eh , so
N det g (3)R = det ĝ R̂ − 2∆ψ
ĝ
2
the Einstein–Hilbert scalar energy is the printed Newtonian term with the same coefficient c4 /32πG: premise
G1’s “one term” (computed). (d) The exterior map is
r3 = R3 − 3GM R2 /c2 + C ′ , with C ′ set by the interior;
the reference Γ = 2, K = 100 polytrope has maximum
mass 1.637 M⊙ , the same computed in DFD’s coordinates
and from the TOV equations, and a moving neutron
star’s inertial mass equals its gravitational mass to better
than 10−8 up to the maximum mass (computed). At
second post-Newtonian order σ is sourced by the combination that sources general relativity’s conformal mode,
and its exchange cancels, at orders G, G2 and G3 , the
terms that the lock would otherwise add to the twobody Lagrangian (computed). (e) With N = e−ψ/2 the
lapse equation D2 N = N [Kij K ij + 4πG(ρ + S)] reads
∂i [(e−h )ij ∂j ψ] = −eψ [8πG(ρ + S) + 2Kij K ij ] (c = 1,
S the trace of the spatial stress), whose static form is
the stated law. (f) On the Friedmann background the
Einstein–Hilbert term keeps its Friedmann law, so the
term quadratic in K is quadratic in K − K̄, with K̄(t) the
uniform mean curvature of the slice itself, and the identity
of (b) becomes Di (N 2 Di K) = N K̄˙ (K − K̄); linearized
in conformal time for k = 0 it reads (Q − ∇2 ) δK = 0,
with ∇2 the comoving Laplacian, Q = 3(H2 − H′ ) and
H the conformal Hubble rate; Q > 0 for ordinary matter, so δK = 0. The three scalar equations, the lapse
equation, the momentum constraint, in which the shift
enters through the shear of σ, and general relativity’s
traceless Einstein equation, which determines σ, are then
Einstein’s in the uniform-expansion slicing; the trace, the
shift and σ are fixed by the matter at each instant, and
the constant c of (c) is proportional to the curvature
constant k (computed).
Black holes follow. The exterior’s event horizon is
Schwarzschild’s, at the areal radius 2GM/c2 , and Kerr’s
for a spinning hole, argued beyond first order in the spin.
For an eternal static hole it is the main text’s optical
horizon, the surface where n = eψ → ∞, which the lock
places at the arena’s origin; a hole formed by collapse
settles on the late-time maximal slicing, where the horizon
is a flow horizon, the inflow of the slices equalling the
local speed of light, with n = 256/27 on it, and n diverges
only on the slices’ limiting surface, R = 3GM/(2c2 ), at
the arena’s origin (computed).
The Boltzmann-code computations of the main text,
the χ third peak among them, solve general relativity’s
linear perturbation equations; by (f) these are DFD’s

395
own equations, in its uniform-expansion slicing, for the
curvature constant k = 0, which Remark WS.7 derives
and which the main text’s fits take; while no baryons
are bound, where the (K − K̄)2 term is absent, they are
the Einstein–Hilbert term’s own equations in the same
slicing. The temporal functional K(∆) of App. Q does
not enter them: it acts on homogeneous deviations only
(Axiom WS.1). Wherever the µ-law acts the equivalence
does not extend to it: its term lies outside the Einstein–
Hilbert term (premise G1).R Writtenpon the physical metric, that term is (a20 /8πG) d3 x N det gij [W (X) − X],
with X = c4 g ij ∂i ln N ∂j ln N/a20 the squared proper acceleration of the observers at rest √
in the slices of absolute
time in units of the epoch’s a0 = 2 α cH, with H read on
bound-matter clocks (App. ZS); it keeps the spatial diffeomorphisms that the lock leaves, at Newtonian order it
is the flat form of Axiom WS.1, and where µ ̸= 1 the two
differ at relative order (1−µ)ψ. It is the acceleration term
of the preferred-foliation theory of Blanchet and Marsat
(Phys. Rev. D 84, 044056 (2011); arXiv:1107.5264), written on DFD’s fixed slices.
Remark WS.7 (The curvature constant is zero). While no
baryons are bound anywhere the µ-law acts nowhere, so
under Axiom WS.1 the action is the Einstein–Hilbert term
alone. Its identity under a change of slicing then reads
ċ = 23 c ∂i β i , with c the constant of Proposition WS.6(c),
proportional to the curvature constant k of the first integral. For c ̸= 0 the identity fixes the longitudinal shift,
so no slicing keeps Eq. (WS1) elliptic, and ψ and σ carry
two modes of their own per comoving
√ wavenumber κ, one
of them growing at the rate κ/ 3 per unit conformal
time whatever the size of k, the matter content or the era,
with no upper limit in κ (computed). So k = 0 is the only
value that keeps the axiom and a well-posed evolution
before any baryons are bound. At k = 0 the equations
keep a2 δK constant before the (K − K̄)2 term acts, and
the momentum of ψ stays continuous where it switches on,
which sets δK = 0: the slicing is the uniform-expansion
slicing, K = K̄, throughout. Switched on there, with the
first phantom growing from zero, the term changes neither
the action nor the momenta, and c stays zero (computed;
after the switch at linear order). A detected spatial or
dark curvature falsifies this.
Theorem WS.8 (Vector-Sector Completion). Under
Proposition WS.5, the h0i components obey the linearized
EH momentum constraint. In the gauge fixed physically by
the DFD dictionary — (a) gij isotropic, = e+ψ δij through
first post-Newtonian order, (b) g00 = −e−ψ through the
same order, with ψ the elliptic solution of Lemma WS.4
— the unique 1PN solution is

1
g0i = − 3 72 Vi + 12 Wi
(WS8)
c
— the standard-gauge value of general relativity. The farzone coefficient −4, the Lense–Thirring precession, and
the leading spin terms of App. AX follow as consequences.
Beyond linear order the Einstein–Hilbert term couples the

shift to ψ: at O(JM ) it gives the stationary exterior the
general-relativistic factor 1 − GM/c2 r in gtϕ (App. AX).
Proof sketch. By Proposition WS.5 the zero-mode dynamics is GR, and at 1PN only its linearization enters: the
h0i equation is the momentum constraint sourced by ρvi .
Conditions (a)+(b) are precisely Will’s standard-gauge
conditions: (a) fixes the spatial gauge function; (b) — the
statement that all c−4 scalar content of g00 sits inside
e−ψ[elliptic] , i.e. no χ̈ term — fixes the temporal gauge
function (Lemma WS.4). With both functions spent, the
momentum constraint has the unique solution (WS8);
uniqueness is the standard 1PN elliptic uniqueness with
Vi , Wi → 0 at infinity. In DFD the “gauge” functions are
not free: coordinates are physical (preferred foliation),
and the dictionary conditions (a)+(b) are what define
those coordinates; the theorem states that the physical
frame the scalar dictionary selects is the same frame
in which the vector constraint takes the standard-gauge
form.
Condition (b) also fixes the slicing. For gij = eψ δij
the mean curvature of the constant-t slices is K =
3
− 2c
∂t ψ + ∂i g0i at leading order, and the standard-gauge
3
value (WS8) has ∂i g0i = 2c
∂t ψ, so K = 0: the preferred
foliation is the zero-mean-curvature (maximal) slicing at
this order. The longitudinal part of g0i is then set by
the slicing, and its transverse part obeys the Einstein–
Hilbert equation. At linear order the longitudinal part
of the momentum constraint, ∂t (∇2 ψ + 8πGρ/c2 ) = 0,
follows from the Poisson equation, so the two statements
agree there. Conditions (a) and (b) hold through first
post-Newtonian order; exactly, the lock fixes the lapse,
N = e−ψ/2 , and det gij = e3ψ , and with the parent
strain’s σ kept the slicing is maximal for isolated systems
(Proposition WS.6(b)).
Theorem WS.9 (EIH equivalence; binary-pulsar triangle). Lemma WS.4 + Theorem WS.8 reproduce the
full Einstein–Infeld–Hoffmann 1PN two-body Lagrangian.
Hence the periastron advance, Einstein delay, Shapiro
parameters, and the (ω̇, γE , Ṗb ) mass-triangle consistency
of PSR B1913+16 and J0737−3039 are GR-identical.
(The +6.4% periastron excess of the hyperbolic/K-branch
readings is absent by construction.)
Remark WS.10 (Data lock: J0737−3039 forces this result twice over). The published double-pulsar timing
(Kramer et al. 2021, PRX 11, 041050) gives Ṗb /ṖbGR =
0.999963(63) — a 1.3 × 10−4 test of the radiative sector — and measures ω̇ at 8 × 10−7 fractional precision.
Both pillars of this appendix are therefore independently
data-forced by the same system: (i) radiative: any scalar
breathing residual ≳ 10−4 of the GR flux is excluded, so
the elliptic reading’s δrad = 0 exactly (Theorem WS.11) is
the only DFD reading that survives — the K(∆)-branch
estimate (δrad ∼ 10−3 ) would fail this bound by ∼8×;
(ii) conservative: at 8 × 10−7 on ω̇, nothing short of the
exact EIH equivalence of Theorem WS.9 suffices. Both
bounds are met, with no tuning available or needed.

396
4.

δrad = 0 with a mechanism; scalar memory;
falsifiers

Theorem WS.11 (Radiative budget). Under Axiom E
and Proposition WS.5: the scalar sector, being an instantaneous constraint, carries no radiative flux in any
multipole (monopole, dipole, quadrupole breathing all identically zero as radiation); the TT sector carries the GR
quadrupole; therefore
δrad = 0

exactly at leading order,

(WS9)

with corrections only from µ-function effects where
|∇ψ|/a⋆ is not asymptotically large (a consequence of
the mechanism above, not an assertion).
Remark WS.12 (Scope: energy flux, not remnant spectroscopy). Theorem WS.11 governs the energy-flux channel — inspiral phasing and orbital decay (Ṗb ), where it
is confirmed at 1.3 × 10−4 (Remark WS.10). It does not
by itself fix the quasinormal-mode spectrum of a merger
remnant, which is set by the remnant’s metric; with the
parent strain’s σ kept that metric is Kerr’s, argued beyond
first order in the spin (Subsec. WS 5).
Corollary WS.13 (Scalar memory closes). The open
case of the App. AT channel-scope remark (head-on/burst
scalar memory) closes: a constraint field has no far-zone
radiative degree of freedom, so scalar memory vanishes
for all source classes; total memory is the tensor (TT)
memory, which equals GR’s, linear (ordinary) plus nonlinear (Christodoulou), because the TT sector is the full
a2 Einstein–Hilbert term (Proposition WS.5; App. AT).
Remark WS.14 (Consistency panel). (i) GW170817: cT =
c untouched (TT hyperbolic, luminal); on the optical-cone
reading of the TT sector the constraint is satisfied identically (GWs and photons ride the same cone). (ii) LVK
polarizations: no scalar/vector far-zone modes — consistent with the corpus falsifier clause; a confirmed far-zone
breathing mode falsifies Axiom E outright. (iii) Preferredframe causality: instantaneous constraint in a preferred
foliation generates no closed causal loops (absolute time
orders all events). (iv) Energy balance: Ṗb is TT-only;
radiation reaction enters the matter EOM through the
TT interaction (Burke–Thorne analog in the TT channel),
with the scalar sector conservative. (v) Frame-dragging
phenomenology at leading order (LAGEOS, GP-B, the
leading QPO spin terms) rests on Eq. (WS8); with the
parent strain’s σ kept the strong-field shift is Kerr’s, so
the spin-ISCO and the shadow displacement of App. AX
are Kerr’s, and the e−2u closed form of the first-order
shift gives way at O(JM ) to general relativity’s 1 − 3u.

5.

Ringdown and echoes: the optical and tensor
channels

a. Ringdown. Where µ → 1, with the scalarlongitudinal piece σ of the parent strain kept, light, matter

and the tensor field share one metric, whose dynamics is
the Einstein–Hilbert term (Proposition WS.5): the remnant of a merger is a Kerr hole in DFD’s coordinates and
its quasinormal modes are Kerr’s, beyond first order in the
spin by the argument of Proposition WS.6. Against the
ˆ
published constraints — GW250114: δf
220 = 0.02 ± 0.02
(90% C.L.); GWTC-4.0 hierarchical: 0.00 ± 0.06; damping
ˆ 220 = −0.01+0.10 — DFD’s δf220 = δτ220 = 0 is consisδτ
−0.09
tent, and the ringdown does not distinguish DFD from
GR. The eikonal
Ωc = c/bcrit is Schwarzschild’s,
√ frequency
2
since bcrit = 3 3 GM/c : the optical and tensor channels
ring together.
b. Echoes. The remnant’s Kerr horizon, argued beyond first order in the spin, has no reflecting surface, so
there is no echo train in either band and no third
√ polarization; the echo-polarization claim h3 /h+ ≃ 1/ 20 ≈ 0.22
(App. AH) does not apply.
c. The flux channel. Theorem WS.11 and Remark WS.10 govern the energy-flux channel (Remark WS.12) and are unchanged.

6.

Consistency notes on other sections

1. PPN vector sector. The pure-V solution Ni =
−4Vi /c3 of the shift equation is not transverse; the
transverse-gauge solution is Ni⊥ = −2(Vi + Wi )/c3
(since Vi⊥ = (Vi + Wi )/2 by the continuity identity; Vi and Wi already contain G). The two differ
by a longitudinal gradient, so no physical quantity
changes; Theorem WS.8 gives the vector sector,
and the PPN section states the transverse-gauge
solution.
2. TT coupling normalization appears as 16πG/c4
at every print (§formalism, §gravitational waves,
App. AT). The canonical value (fixed by the
Sh Rnormalization c4 /64πG and the interaction
ij
4
4
+ 12 hTT
ij T ) is 16πG/c ; a c /32πG prefactor or
a − 12 coupling is inconsistent with it (App. GR,
remark on the TT action normalization, item (ii)).
3. Echo polarization (App. AH vs. the
gravitational-wave section). Under Axiom E the
far-zone scalar mode is absent, and with the parent
strain’s σ kept the h3 /h+ ≃ 0.22 echo-polarization
claim does not apply: there is no third polarization
and no gravitational-wave echo (Subsec. WS 5).
4. EH attribution. a2 carries the Einstein–Hilbert
term (gravitational-wave section, App. O, App. SM);
a4 carries the gauge kinetic term and α (App. O)
— two different operators, not a convention choice.
This appendix uses a2 for the Einstein–Hilbert term
throughout (Proposition WS.5).
5. Ringdown and echoes: the statements of
App. AT, App. AH, App. RM and the strong-field

397
section follow Subsec. WS 5: Kerr ringdowns, argued beyond first order in the spin, and no echoes.

7.

What remains open in this sector

(1) A full nonlinear well-posedness statement for the
coupled elliptic-constraint + hyperbolic-TT system (an
open program item of the corpus; constraint-hyperbolic
mixed systems are standard in NR as “elliptic-hyperbolic”
formulations).

Appendix ZS: The Early-Clock CMB Fit: Conformal
Invariance Theorems, the Measured Clock
Coefficient, and the Shape Inputs

This appendix sets DFD’s position on the Hubble tension. Three exact theorems (Sec. ZS 1) close the “Planck is
ψ-screen-biased” interpretation; an enumeration-complete
no-go (Sec. ZS 2) establishes that the coefficient C in
(H0 tP )2 = C α57 is a free continuous input of the printed
theory — the exponent 57 remains a theorem (App. AP)
— and it is measured; a C = 1 axiom is one the CMBanchored data exclude at ∼46σ as today’s early-clock
value, while bound matter’s clock has C = 1 (Postulate ZS.11). With C measured once, every remaining
spectrum-shape input of the CMB but three is carried by
derived or adopted-and-priced structure: the ωb coefficient
and As are calibrations, and the tilt is a stated number;
the mechanism is open (per-input grades in Sec. ZS 3).
DFD fits Planck plik-lite with two free calibrations, three
counting the ωb coefficient, against ΛCDM’s five effective
parameters at essentially equal χ2 (Sec. ZS 3). Its position on the tension is one universe read on two clocks
(Postulate ZS.11): the CMB-anchored channels read the
early clock, H0early = 67.52, and local probes read the
bound-matter clock, H0loc = 72.09; the local evidence and
its decider are set out in Sec. ZS 4.

1.

Three theorems: what a ψ screen cannot do

Conventions as in App. AN: M1 denotes the optical
metric ds̃2 = −c2 dt2 /n2 + dx2 with n = eψ , and M2 the
matter-coupling metric ds̃2 = −c2 e−ψ dt2 + e+ψ δij dxi dxj ;
the corpus already records that the two are conformally
related with identical light cones (App. AN).
Theorem ZS.1 (Conformal θ∗ -invariance). M2 = eψ M1
as an exact matrix identity (conformal factor Ω = eψ/2 ),
so the two metrics share their null sets and unparametrised
null geodesics. Under a conformal rescaling g̃ = Ω2 g the
(M )
(M )
angular-diameter distance obeys DA 2 = Ωs DA 1 with
ψ(z∗ )/2
Ωs = e
, while the proper transverse sound-horizon
scale measured by the plasma’s own (M2 ) rods carries the
same factor Ωs . Hence
(M )

(M )

θ∗ 2 − θ∗ 1 = 0

identically.

(ZS1)

Computing DA (z∗ ) “on the optical metric” cannot move
θ∗ at all: the shift is exactly zero.
Corollary ZS.2 (Etherington reciprocity is structural;
no probe selectivity). Distance duality dL = (1 + z)2 DA
holds exactly in DFD because it is conformal invariance,
not a coincidence [84]; DFD cannot break the duality ratio
η = 1. Consequently a ψ screen is not probe-selective:
CMB θ∗ , transverse and radial BAO, SNe Ia luminosity
distances, and strong-lensing time-delay distances all ride
the same null congruence on the same conformal class
and must move together. A split in which time-delay H0

398
probes the topological value directly while Planck is screenbiased is refuted as stated. A viable screen could only be
redshift-selective, which the corpus’s own low-z-weighted
screen kernel cannot satisfy.
Theorem ZS.3 (The split of A = a eψ̄/2 is pure gauge).
For DFD–FRW on M2 with arena scale factor a(t) and
background ψ̄(t), define dτ = e−ψ̄/2 dt and A = a e+ψ̄/2 .
Then ds2 = −c2 dτ 2 + A2 dx2 exactly, and every CMB
observable
is a functional of (τ, A) alone: 1 + z = A0 /A,
R
χ = c dτ /A, θ∗ = χs /χLSS . At fixed (τ, A) the background field cancels identically, dθ∗ /dψ̄ = 0: with an
arena factor, ψ̄ is a field redefinition of (a, t), not a physical screen. This carries the App. AE flat-direction theorem from the action to the observables for the split: the
split of the scale factor between the arena factor and ψ̄
is unobservable; in the static arena, where a = 1, the
homogeneous field is the scale factor itself, ψ̄ = 2 ln A,
and carries no screen beyond the expansion it describes.
Theorem ZS.4 (Skewness sign theorem for the mean
line-of-sight screen). The only channel Theorems ZS.1–
ZS.3 leave open
R is the inhomogeneous mean ⟨δψ⟩LOS entering ⟨χ⟩ = (c/H) ⟨e−δψ ⟩ dz. In GR this vanishes identically (linear Poisson operator, zero-mean source). In
DFD the √
deep-MOND solution operator is odd and sublinear (ψ ∼ GM a0 ), so ⟨δψ⟩ ̸= 0 requires skewness of the
density field, and for any monotone concave µ on a positively skewed field the sign is forced: ⟨δψ⟩LOS < 0 — the
sign a θ∗ resolution would require. The magnitude, however, is bounded: the pre-registered lognormal computation updated to the measured-clock forward normalization
σ8 = 0.8063 gives a best path-weighted |∆| ≈ 3.7 × 10−6
against the required 1.32×10−2 — short by a factor ∼3500
(3.55 orders; generous upper bound still 29× short). The
GR control run returns machine zero. The mean-vs-RMS
distinction also corrects App. LS: an RMS is positivedefinite and cannot be the coefficient of a systematic distance shift.
Status ZS.5 (The optical route is closed). The optical
route to a DFD-favourable reinterpretation of Planck’s H0
is closed by two exact theorems, and the inhomogeneous
route is 3.5 orders of magnitude too small. The reading
that “the Planck-vs-TD tension is structurally resolved
by interpreting Planck as ψ-screen-biased” is therefore
excluded.
2.

The clock coefficient C: an eight-route no-go, and
its measurement

Write the clock identity of App. AP as
(H0 tP )2 = C α57 ,

(ZS2)

with the exponent 57 = kmax − Ngen a theorem. Taking
C = 1 (equivalently H0 = 72.09 km/s/Mpc) would be an
axiom. An eight-route derivation analysis shows that C
is not derivable, at Derived grade (enumeration-complete
over the printed corpus):

Proposition ZS.6 (No-go: C is not derivable from the
printed corpus). All 128 printed corpus equations connecting the α57 deposit to an expansion rate are exponentonly, convention-importing, or C-covariant. The spectral
vacuum-matching route requires a regulator shape moment outside the independently forced Riesz class p ≥ 3;
the homogeneous ψ-sector Lagrangian, momentum, and
Hamiltonian vanish identically, and the Friedmann law
of the Einstein–Hilbert term converts the vacuum density
into the clock rate without fixing either; epoch-selection,
deposit-repartition, horizon-entropy, and dictionary-audit
routes all fail quantitatively. C is a free continuous input. Its time-independent form is the vacuum coefficient
R ≡ ρΛ,obs /[(3/8π)α57 ρP ] = 0.6042 ± 0.0027, and C is
today’s value of the early clock, C(z) = R + 1.924 [ωm (1 +
z)3 + ωr (1 + z)4 ]; any derivation must fix R, not an epochdependent C. A ninth route, fixing the whole shape of
the spectral cutoff from a printed principle, fails as well
(Remark AP.6). Any future closed-form claim must beat
a measured false-positive-rate bar: at least 952 grammar
candidates fall inside the ±1σ window.
Status ZS.7 (C measured; C = 1 excluded as the early
clock). Measured once against 13 pre-DESI BAO points
(6dFGS, SDSS MGS, BOSS DR12, eBOSS DR16) plus
the Planck θ∗ [58], with the other inputs of Sec. ZS 3:
C = 0.87784 ± 0.00266 (χ2 = 10.52/13); the plik-lite
conditional channel independently gives C = 0.8772; the
time-independent input is R = ΩΛ C = 0.6042 ± 0.0027.
As today’s value of the early clock C = 1 is excluded at
∼46σ by the CMB-anchored channels, and it is the clock
of bound matter (Postulate ZS.11). The resulting clock is
H0early = 67.52 ± 0.09 km/s/Mpc, concordant across five
CMB-anchored channels (plik-lite 67.520±0.094; BAO+θ∗
67.543±0.103; ACT DR6 67.481±0.094; SPT-3G 67.417±
0.232; θ∗ -only 67.52 ± 0.10). On DESI DR2 BAO [194]
the BAO+θ∗ channel gives C = 0.8801 ± 0.0026, while
BAO alone gives H0 = 68.71 ± 0.33 km/s/Mpc, 3.4σ
above the θ∗ -only value. The split is 3.4σ here because
ωχ is computed, not fitted; ΛCDM shows 2.3–2.5σ with
ωc free and 4.2–4.4σ with ωc held at Planck’s value. No
printed input moves it (the printed range of ωχ moves
it the wrong way). On the early clock the Hubble time
1/H0 = 14.48 Gyr and the age t0 ≈ 13.8 Gyr (12.92 Gyr in
laboratory units, Sec. ZS 5). This is exactly the epistemic
status of H0 in ΛCDM: one measured normalization, with
the structure (here the exponent theorem) supplied by
the theory.

399
3.

The CMB fit on the measured clock

With C measured, the spectrum-shape inputs are supplied upstream of the data, with per-input grades printed:
ωb = 0.0222730 (conventional-point value, a calibration;
α4 power derived, coefficient calibrated;
Axiom E-transport′ , +1 priced discrete bit —
Ext. Deriv. App. AH),
ωχ = 0.1192806 (flux-period relic,
Lemma AV.6, App. AV),
P

Neff = 3.044,

mν = 61.46 meV (normal-ordering
spectrum),

ns = 1 − 2/60 = 0.96667 (a stated number;
the mechanism is open),
As (calibrated; amplitude-profiled).
(ZS3)
Two calibrations are free in the fit, the clock coefficient C
of Sec. ZS 2 and one overall amplitude absorbing As e−2τ ;
a third, the ωb coefficient, is fixed at its conventionalpoint value. Against Planck 2018 plik-lite native (613
bandpowers, TT+TE+EE, ℓ = 30–2508) [58]:
Out-of-sample: SPT-3G outright prefers DFD (∆χ2 =
−0.77); ACT DR6 disfavours it by +5.94 at the fluxperiod abundance (Lemma AV.6) (∼1.2–2.4σ, below 3σ,
and partly the known ACT–Planck ΛCDM tension).
CMB lensing at this background with full conventions
gives Aϕϕ = 0.9887 (−0.80σ). The flux-period abundance agrees with three Planck-based channels (Planck
fit −0.6σ; BAO+θ∗ −0.18σ; plik-lite −0.98σ), and the
neutrino spectrum is fit-neutral once H0 is re-profiled in
both arms (−1.2 χ2 plik-lite, −0.8 CamSpec PR4 in favor
of massless — negligible; the spectrum thus enters as mild
anti-tuning).
Remark ZS.8 (Anti-tuning facts). The inputs are not riding luck: DFD’s stated tilt ns = 0.96667 (a stated number;
the mechanism is open) fits plik-lite better than Planck’s
own fitted 0.9649 by 0.11 χ2 (the combination of Planck,
ACT DR6, CMB lensing and DESI gives 0.974 ± 0.003,
about 2.4σ above it), and DFD’s conventional-point ωb
costs DFD 1.25 χ2 relative to Planck’s fitted value. Swapping both to Planck’s values would improve DFD to
∆χ2 = +1.88. No continuous input in Eq. (ZS3) was
tuned to the CMB: the flux-period relic (with its two
stated conventions) and the neutrino spectrum were fixed
for unrelated reasons, and they jointly resolve a five-way
abundance inconsistency (χ2 = 120.4/4, extremes 10.6σ)
without a single CMB-facing continuous adjustment. The
ωb input is the conventional-point value of the baryogenesis chain (Axiom E-transport′ , a priced discrete bit
adopted after Planck’s ηB was known; App. FK and
App. CE); its α4 power is derived and its coefficient is a
calibration (Ext. Deriv. App. AH).
Status ZS.9 (Standing claim). DFD with the tilt ns
(a stated number; the mechanism is open), ωb at its

conventional-point value (a calibration; per-input grades
in Eq. (ZS3)), and two free calibrations (the clock coefficient plus one amplitude) fits Planck plik-lite at
∆χ2 = +3.2 for 2 fewer effective parameters with the
ωb coefficient counted (AIC −0.8, BIC −9.6; 3 fewer,
AIC −2.8 and BIC −16, if it is not counted; computed),
needs no screen, and is sustained out-of-sample by SPT3G (preferred) and ACT DR6 (below 3σ). The CMB
model-comparison figures of App. CL, priced at the C = 1
background (e.g. the plik-lite ∆χ2 = +23.3/ + 8.0 there),
are not current: they use neither the flux-period abundance nor the measured clock. Under the adopted Axiom
E-transport′ (Ext. Deriv. App. AH) the shape input ωb
has its α4 power and its B−L bias derived, but not its
coefficient (with the stated carrier, the mechanism gives
ηB of about (2–3) × 10−9 , three to five times Planck’s
value), and the fit uses its conventional-point value. The
enumeration, with per-input grades: ωb derived power, calibrated coefficient (+1 priced discrete bit); ωχ conditionalderived, on two stated conventions, the stated count 158,
′
the flux-period identification and
P the asserted premise P4
(App. FK, Lemma AV.27);
mν + hierarchy derived,
given the stated inputs and the data-selected branch of
App. X; Neff standard; ns a stated number, the mechanism open; As calibrated and amplitude-profiled — with
H0 (the vacuum coefficient R) measured and one overall
amplitude calibration absorbing As e−2τ (τ astrophysical).
4.

The local arm and the acoustic arm, and the
decider

Two arms of the theory measure the clock coefficient
C at different values. The local arm (galactic and local)
is the SPARC absolute a0 scale together with the local
distance-ladder H0 , whose SH0ES value of 2022 reads
C = 1.027±0.029; the acoustic arm is the CMB and BAO
geometry of this appendix, which gives C = 0.87784 ±
0.00266. Read as one clock, this would be the sharpest
conflict in the corpus; under the two-clock postulate of
Sec. ZS 5 both readings are right, the local arm reading
the bound-matter clock (C = 1) and the acoustic arm the
early clock. The domain-local screen clock of App. ZT is
the structural extension of the acoustic side — a single
conditional axiom under which the local kinematic flow
raises the effective local clock and a0 by about 1% — and
its theorems are conditional on that axiom; the a0 -SPARC
/ SH0ES data question below is not settled by it.
Falsifier ZS.10 (The absolute a0 scale tests the two
clocks). If the Euclid/DESI-era lensing RAR pins the
absolute
the bound-matter clock (H0loc = 72.09:
√ a0 at
loc
a0 = 2 α cH0 = 1.1966 × 10−10 m s−2 ), the SH0ES-side
value, the outcome is the one the two-clock postulate
expects (Falsifier ZS.14, item iv). If it pins a0 at the
early clock (H0early = 67.52: a0 = 1.1208 × 10−10 m s−2 ),
bound systems do not read the bound-matter clock, the
postulate fails, and the local arm’s statistical excursion
resolves as systematics of the local measurements. Ei-

400
TABLE CXXX. The fit numbers (effective parameter counts: ΛCDM 5 vs DFD 2 free in the fit, 3 with the calibrated ωb
coefficient; gap 3 or 2).
χ2 (DFD, 2 calibrations)
586.77
χ2 (ΛCDM, Planck best fit; refit shave 0.23) 583.80
fair ∆χ2 (for 3 fewer effective parameters) +3.20
AIC / BIC
−2.8 / −16.1 in DFD’s favour (−0.8 / −9.6 with ωb counted)
100 θ∗ (no screen of any kind)
1.04110
vs measured 1.04109 ± 0.00031
+0.03σ
plik-lite conditional H0
67.520 ± 0.094
vs BAO+θ∗ channel
0.16σ

TABLE CXXXI. The local and acoustic arms and their falsifiers. The SPARC split on the early clock is 3.9σ(stat) but only
∼0.3σ against the a0 -SPARC systematic band; the decider is the absolute a0 scale, when baryonic masses are known to about
2%.
Arm

Evidence

Coefficient reading

Status

acoustic (this
appendix)
local (dynamical)

plik-lite, BAO+θ∗ , ACT, SPT,
θ∗ -only
absolute a0 from SPARC

C = 0.87784 ± 0.00266

five concordant
CMB-anchored channels
within the systematic error on
bound-matter clocks

SH0ES distance ladder [106]
App. AH tether channel (Λ
theorem)

within the systematic
√
error of 2 α cH0loc ;
−3.9σ(stat) / −0.3σ(sys)
on the early clock
6.4σ high vs 67.52, 1.5σ read on the bound-matter
vs 72.09
clock
C = 1.027 ± 0.029
SH0ES 2022 re-expressed,
(73.04/72.09)2 ; not a separate
arm; 0.9σ from C = 1
Cepheid-free H0 0.2σ
both arms hold; size λ
measured

two clocks
(Postulate ZS.11)

bound matter reads C = 1, the
early plasma the measured C

Decider

absolute a0 scale: Euclid/DESI
baryonic masses to ∼2% pre-registered discriminator
weak-lensing RAR
stellar/globular-cluster ages; f σ8 (z) matter-clock integrals on untouched by any screen
growth; BBN
M2
(Thm. ZS.1)

Screen-immune
arbiters

ther outcome is a sharp result. The two arms differ by
6.8% in a0 , and an a0 read from a lensing RAR scales
inversely with the adopted baryonic mass, so a 3σ decision
needs baryonic masses to about 2%, against stellar-mass
systematics of about 0.1 dex today; and each arm must
be read on distances computed with its own H0 : an a0
inferred from redshift distances with photometric baryonic masses scales as h2assumed /htrue (with masses fitted to
rotation curves it scales as hassumed alone, and such data
cannot tell the arms apart), so at an assumed h = 0.70
the early-clock arm gives the larger value, 1.205 against
1.128 × 10−10 m s−2 (computed). Because Corollary ZS.2
locks all null-path distance probes together, the optical
escape routes are closed.

the difference to an unidentified error in one of them.
DFD reads both as right, through one postulate.
Postulate ZS.11 (One universe, two clocks). Matter and
the expanding universe keep clocks offset by one constant
factor λ, set when matter first binds and falls behind
the expansion. In the cosmological field equations gravity
enters as Gcos = λ2 Glab at every epoch, and before a
switch redshift zsw every non-gravitational microphysical
rate (Thomson scattering, the atomic recombination rates,
Compton heating, the nuclear rates, the oscillation rate of
the χ field) runs faster by λ than the same rate computed
with laboratory constants, both read on the local light
geometry; λ2 = 1/C.
The two readings of the one expansion are labelled

5.

One universe, two clocks

Both measurements of the present expansion rate are
robust. The early-plasma yardsticks read 67.52 and the
locally calibrated yardsticks read about 72, about 5σ
apart (4.5σ for the Cepheid-free local mean below, 6.4σ
for SH0ES’s full 2025 sample [106]). ΛCDM has to assign

H0loc = 72.09 km/s/Mpc,
H0early = 67.52 km/s/Mpc,

(ZS4)

λ = H0loc /H0early = 1.0677.
H0loc is the Hubble rate read on bound-matter clocks
and rulers (after the switch every matter clock, bound or
diffuse, runs at laboratory rates), the C = 1 value α28.5 /tP

401
of the clock identity; H0early is the same expansion read
through early-plasma yardsticks (the CMB, BAO, and
supernovae anchored on the BAO ruler), the measuredclock value of Sec. ZS 2. They are one universe read on
two clocks, not two expansions.
The postulate is the rate-rescaling symmetry of the
CMB [195], with its size fixed by the measured coefficient.
Under it:
1. Local clock. Gravity and light geometry read with laboratory constants obey the clock identity with C = 1,
which contains no measured coefficient, so the local
expansion rate is H0loc = α28.5 /tP = 72.09 km/s/Mpc,
from the measured G with no parameter (Sec.
√ XIX).
Because the CMB-anchored rate scales as C and
λ2 = 1/C, the measured value of C cancels from
the local value: setting λ2 = 1/C is the statement that the local coefficient is 1, and 72.09 is
the content of the postulate, not an output of a fit.
Seven Cepheid-free local measurements (CCHP TRGB
and JAGB [108], SH0ES JWST TRGB [105], megamasers [196], TRGB-calibrated surface-brightness fluctuations [197], tailored-EPM type II supernovae [198],
and GW170817 with its VLBI jet [199]) give a weighted
mean of 71.87 ± 0.96, which is −0.2σ from 72.09 and
+4.5σ above 67.52 (the type II supernova systematic
error, not quoted by its authors, taken equal to its statistical error; errors treated as independent; the TRGB
analyses share hosts).
2. Bound systems. Bound systems read the same
clock, so the acceleration pscale of Eq. (488) is
√
a0 = 2 α cH0loc = 2α29 c7 /(Gℏ) = 1.1966 ×
10−10 m s−2 , with no measured coefficient, consistent
with the SPARC extraction (1.20 ± 0.02stat ± 0.24sys ) ×
10−10 m s−2 [10] within its systematic error.
3. Early clock. Every dimensionless CMB and BAO
observable (θ∗ , z∗ , zd , zeq and every D/rd ) is unchanged to 10−4 , so the five CMB-anchored channels
still return H0early = 67.52 when read with laboratory
rates. A forward Boltzmann run of the postulate at
λ = 72.09/67.52, inside the error of C −1/2 , on plik-lite
plus DESI DR2 BAO [194], without refitting, changes
χ2 by −0.06 (RECFAST) and −0.17 (HyRec) against
the ΛCDM fit with H0 held at 67.52, for zsw of 9.5, 30
and 100 (HyRec at 9.5 and 100), with the amplitude
shift As → As λ1−ns that the symmetry implies (+0.18
without it). Raising Gcos without the rate factor costs
∆χ2 = +156, so the rate factor carries the fit. The
sound horizon is rd = 137.8 Mpc in laboratory units
and 147.1 Mpc in the units of the early plasma.
4. Mirror symmetry. The postulate is homogeneous,
so an observer anywhere reads its own neighbourhood
at 72.09 and the early plasma at 67.52, and the two
readings do not change when observers are exchanged.
The coefficient is therefore even under exchange.
5. Size. λ2 = 1/C = 1.1392 ± 0.0035, and through
R = ΩΛ C it carries the same information as the vacuum
coefficient R, so the postulate adds no parameter. Its
switch redshift is not located, and Proposition ZS.6

applies to its size unchanged.
a. The structure behind the postulate. Three relations taken from the printed equations give the form that
Postulate ZS.11 takes, though not its size. (a) The gravitating contrast of a region is its accumulated expansion
deficit. With θ the local volume-expansion rate of pressureless matter and 3H that of the uniform-expansion
R
slices, mass continuity gives 1 + δ = exp (3H − θ) dt;
since the expansion is carried by ψ itself (Theorem ZS.3),
the contrast that sources ψ is the matter’s failure to expand with the slices. This is the continuity equation,
not a new law. (b) The deficit rate of bound matter
is fixed. Inside a bound region θ = 0 while the slices
expand at 3H, which on the lock of Proposition WS.6
is a ψ-rate 2H; measured√ in units of the epoch’s a0 /c
it is ∆b = 2cH/a
√ 0 = 1/ α, the same at every epoch
because a0 = 2 α cH (Eq. (488)). This is a rate, not
the constant gap of (c): integrated between epochs it is
the slices’ own change of ψ, the expansion itself (Theorem ZS.3). (c) The clock exponents are fixed. Matter
clocks carry the lapse e−ψ/2 and light the one-way optical
index n = eψ , so a constant gap ∆ψ > 0, with bound
matter at the larger ψ, changes matter rates by λ = e∆ψ/2
and the one-way light index by λ2 = e∆ψ , a ratio between
frames and not a screen on the light path (Theorem ZS.3);
then 1/λ2 = e−∆ψ , and the measured C corresponds to
∆ψ = 0.130. A uniform ψ cancels from every co-located
ratio (Theorem XI.2), so the gap appears only in comparisons between two frames. None of (a)–(c) makes the gap
constant, fixes its size or locates the switch redshift: the
printed equations give bound matter only its potential
depth above the slices, so a constant gap, set when matter
first binds, is what the postulate adds, and its size is the
measured coefficient.
b. A stated rule for the size. A second postulate,
separate from Postulate ZS.11, states a value for the size.
It is a rule to be tested, not derived, and no other result
in the corpus uses it.
Postulate ZS.12 (Size of the clock gap). (i) When
matter first binds, its ψ gap above the expansion slices
is the microsector response of Proposition N.4 with the
shifted level of Lemma N.10 responding, keff + 2 = (k0 +
2)(1 + s) in place of Assumption N.3, so that ∆ψ =
3
2 ln(1 + s), with argument s = a0 /(c |ψ̇sl |), where |ψ̇sl | =
2H is the slices’ ψ-rate. (ii) Both rates in that ratio are
read along one worldline, as in Eq. (Q4), and in the same
era, with a0 the epoch’s value of Eq. (488).
√
Because
a0 = 2 α cH (Eq. (488)), clause (ii) gives
√
s = α at every epoch. With λ = e∆ψ/2 from (c),
√
λ = (1 + α)3/4 ,
(ZS5)
so λ = 1.06341, ∆ψ = 0.12296, C = 1/λ2 = 0.88430, and
the rule predicts H0early = H0loc /λ = 67.79 km/s/Mpc,
against the measured (pre-DESI BAO+θ∗ ) C = 0.87784±
0.00266 (+2.4σ), the DESI DR2 BAO+θ∗ value 0.8801 ±
0.0026 (+1.6σ) and the CMB-anchored H0 = 67.52 ± 0.09
(+3.0σ); at 67.79, with ωb and ωχ held at the val-

402
ues of Eq. (ZS3), the acoustic angle moves by 0.08%
(+2.6σ against θ∗ ). It goes beyond the printed equations.
Clause (i) reads App. N’s single-environment response
as a gap between two frames, replaces the printed level
response with the shifted one, and applies a temporal
response inside bound matter, which Axiom WS.1 does
not provide (it confines App. Q’s K(∆), whose closure
K ′ (∆) = µ(∆) is the minimal natural choice rather than
a forced one, to the homogeneous sector); its argument
is the inverse of the printed one, and with s = ∆b the
same response gives C = 0.022. Reading instead a0 on
the bound-matter
clock√and ψ̇sl on the early clock gives
√
s = α λ, λ4/3 = 1 + α λ and C = 0.87729 (−0.21σ;
−1.1σ on DESI DR2), closer to the data but at the cost
of comparing rates across the two clocks. The inverse
argument, the reading of clause (ii) and the choice of response and level were made with the measured C known,
so neither reading meets the false-positive bar of Proposition ZS.6, which concerns derivation from the printed
equations and stands unchanged, and the rule is not a
derivation of C. On the bare level of Assumption N.3
with k0 near 60, the O(1/k0 ) term moves C by +0.003
(to 0.8877; +3.7σ, +2.9σ on DESI DR2); on the shifted
level Eq. (ZS5) holds up to the O(k −2 ) term of the lemma.
The rule fails if the BAO+θ∗ channel places C more than
3σ from 0.88430; on the CMB-anchored H0early it already
sits at 3.0σ. The predicted 67.79 lies between the two ingredients of the BAO+θ∗ channel, which differ by 3.4σ on
DESI DR2 (Status ZS.7): θ∗ alone gives 67.52 ± 0.10 and
the DESI DR2 BAO points alone 68.71 ± 0.33 km/s/Mpc,
and 67.79 sits 2.7σ above the first and 2.8σ below the
second.
In the 3D BAO and supernova data the split is fixed
by the anchor, not by distance. Eighteen 3D BAO measurements at twelve redshifts from z = 0.106 to 2.33,
read with the early-plasma ruler rd = 147.09 Mpc on
the measured-clock shape (Ωm = 0.312), give a flat
H0 = 68.22 ± 0.19 (χ2 = 15.2/17), whose straight-line
value at z = 0 is 68.65 ± 0.37. The same ruler anchoring
the Pantheon+ supernovae [88] (1,086 flow supernovae
with 0.01 < z < 0.3, SH0ES calibrator hosts excluded)
gives 68.35 ± 0.36, with no rise toward low redshift across
seven bins, while the Cepheid anchor reads the same supernovae about 5.5 km/s/Mpc higher, one 0.17 mag shift
of their zero point. A value that slides from 72.09 locally
to 67.52 at large distance fits worse (∆χ2 = +8.9 on the
BAO points). The stretch 2.33 < z < 1090 is itself measured to 0.96% in ruler units by θ∗ , with the calculated
ratio r∗ /rd of the sound horizons at last scattering and at
the drag epoch, and the Lyman-α BAO point at z = 2.33;
holding H0 = 72.09 at fixed ωm and restoring θ∗ with
a uniform change of H there worsens the fit to θ∗ and
the BAO by ∆χ2 ≈ +70; the joint rescaling of the ruler
and the rates in Postulate ZS.11 leaves both unchanged.
Angular (2D) BAO in thin redshift shells reads 72.7 ± 0.9
with the same ruler, the known difference between the 2D
and 3D methods; it is not used here.
Status ZS.13 (Postulate: form derived, size measured).

Under Postulate ZS.11, DFD returns H0early = 67.52 in the
CMB-anchored channels and the C = 1 value of the clock
identity, H0loc = 72.09, in local probes. The form and
structure of the postulate are derived from the printed
equations (the paragraph on the structure behind the
postulate), and what it adds is that the gap is constant;
its size λ, equivalently C, is the one measured input, as
C already was, and the derivation of the size is open.
The postulate lies outside the printed premise that one
G governs every epoch and every sector (Sec. XIX), the
premise on which the CMB-anchored channels exclude
C = 1 as today’s value at ∼46σ; under the postulate that
exclusion states that the early clock is not at C = 1. It
does not say where gravity passes from the cosmological
to the laboratory value (if matter perturbations grew with
the laboratory value, σ8 would fall well below 0.81); its
switch redshift is not located (the CMB and BAO are
blind to it between 9.5 and 100), but a sharp switch would
move the sky-averaged 21-cm signal 6.2 ± 0.1% toward
zero in one step at the frequency 1420.4 MHz/(1 + zsw ),
for any switch redshift, which would locate it. It costs
the age of the universe in laboratory units, 12.92 Gyr,
against ages of the universe of 13.5–13.8 Gyr inferred
from globular clusters with a ΛCDM formation delay (2.1–
2.6σ with statistical and systematic errors in quadrature,
one programme), a cost the postulate does not remove,
since faster rates before the switch add at most 0.03 Gyr
to a star’s age; late stellar clocks prefer no switch by
∆χ2 ≈ 7–10 (one globular-cluster programme; the cosmicchronometer reading moves by about 7% between stellarpopulation models). Every dimensionless tension of the
measured-clock fit, including the DESI BAO-alone split
above, is unchanged. The runs labelled elsewhere as on
the one-clock C = 1 background use H0 = 72.09 together
with Ωm ≃ 0.274 at every epoch; the postulate does not
reproduce that background and keeps Ωm ≃ 0.312.
Falsifier ZS.14 (The two-clock postulate). The postulate
fails if any of the following holds. (i) The early-plasmaanchored and locally anchored distance ladders converge
as redshift grows: the postulate requires one constant
ratio λ at every distance. (ii) Bright gravitational-wave
sirens, which use no calculated rate, settle at 67.5 rather
than 72.1. (iii) Time-delay lenses with mass-sheet-robust
modelling settle at 67.5. (iv) The absolute a0 scale settles at the early-clock value 1.1208 × 10−10 m s−2 on
distances
with H0early : bound systems read
√ computed
loc
a0 = 2 α cH0 = 1.1966 × 10−10 m s−2 on distances
computed with H0loc , the SH0ES-side outcome of Falsifier ZS.10, which is the expected one. (v) Stars older than
12.92 Gyr in laboratory units are established at better
than 3σ, with statistical and systematic errors in quadrature. (vi) Any co-located ratio of clocks or masses moves:
clock-null (Sec. XI) must hold.

403
Appendix ZT: The Domain-Local Screen Clock: A
One-Axiom Spatial Completion of the a0 Theorem

√
The corpus derives a0 = 2 α cH0 from the crossover
invariant Ξ (App. N 4) and, conditional on Axiom WS.1,
promotes the normalizer
to the instantaneous cosmological
√
rate, a0 (z) = 2 α cH(z) (Thm. XIX.3): the temporal
half of locality — the clock is read now, with no stored
scale — is already a theorem (the no-memory argument).
As printed, however, the corpus hard-codes the global
average rate cH(t) into Ξ. This appendix books the
spatial completion: a single new axiom under which the
normalizer is the domain’s own screen flow, together with
everything that follows from it at its verified grades. All
results below are conditional on the axiom (price list:
Sec. ZT 3); this appendix works on the early clock, and its
Hubble posture (Sec. ZT 6) is a statement about that clock.
The two-clock postulate of Sec. ZS 5 is separate: under it
local probes read the bound-matter clock H0loc = 72.09,
bound systems read a0 = 1.1966 × 10−10 , about 6% above
this appendix’s kinematic local value, and time delays
read 72.1, not the global-clock value given below.

1.

The axiom

Definition ZT.1 (Domain). A domain D is a bounded,
mass-conserving comoving region, homogeneous to leading
order on its own scale. Its screen flow is
Hscr (D) = − 12 uµ ∇µ ψopt D = − 21 ψ̇opt,D ,

(ZT1)

the bridge H = − 12 ψ̇opt (Eq. (484)) applied to the domain’s matter volume, with ψopt,D = −2 ln aD and aD
the scale factor of that volume; DFD’s own slices keep
the background rate there.
Axiom ZT.2 (DLSC: the crossover normalizer is domain-local). The cosmological rate normalizing the constitutive crossover µ(|∇ψ|/a⋆ ) of the elliptic scalar sector
(Axiom WS.1, App. WS) is the domain’s own screen flow:
the cH entering the crossover invariant Ξ (Eq. (N10),
App. N 4) is c Hscr (x, t), not the global average c H(t).
The axiom asserts
only this normalizer substitution.
√
The coefficient 2 α is not part of the axiom — it is derived
(Theorem ZT.3). The temporal half (instantaneous flow,
no stored scale) is already a corpus theorem (no-memory,
Thm. XIX.3); the spatial half is the new content: the
corpus as printed hard-codes cH0 .
2.

The theorems

Theorem ZT.3 (A: domain-local a0 ; conditional on
Axioms WS.1 and ZT.2).
√
a0 (x, t) = 2 α c Hscr (x, t),
(ZT2)
with the same parameter-free coefficient as the global theorem. The proof is the App. N 4 chain run verbatim on a

region Ω ⊆ D: the microsector coefficient ka = 3/(8α) is
environment-independent; the crossover value Ξ∗ = 3/2
is normalizer-independent (λ-stationarity; no circularity); solving Ξ∗ = ka (a0 /cHscr )2 yields Eq. (ZT2). The
global theorem (Thm. N.14) is the special case
√ D =
FRW background; the epoch theorem a0 (z) = 2 α cH(z)
(Thm. XIX.3) is the spatially averaged case.
Theorem ZT.4 (B: exact kinematics of the screen flow;
no dynamical input). From mass conservation alone
1/3
(VD ∝ a3 /(1 + δ), aD = VD ),
Hscr = H −

δ̇
,
3(1 + δ)

(ZT3)

identified with − 21 ψ̇opt,D via eψopt,D = a−2
D , giving the
split
ψopt,D = ψopt + 23 ln(1 + δ).

(ZT4)

Corollary ZT.5 (Environment ratio).
a0
δ̇
Hscr
= 1−
;
=
ā0
H
3H(1 + δ)
(ZT5)
∆a0
f
0.55
= − δ, f = Ωm = 0.527.
a0 lin
3
A growing underdensity raises the local H and the local
a0 ; an overdensity lowers both.
Proposition ZT.6 (Scale rule: the
√ minimal domain
Dmin ). For a system S, a0 |S = 2 α c Hscr (Dmin (S)),
where Dmin is the smallest domain satisfying Definition ZT.1 that contains S (inner edge = the zero-peculiarvelocity surface, ∼2× turnaround; outer edge = leadingorder homogeneity, RD ≳ 10–20 h−1 Mpc). The virialized
reading — evaluating Hscr inside the collapsed system —
is triply excluded: Cor. N.13 fails inside rvir ; App. Q fixes
uµ to the background congruence; and exact stable clustering gives Hscr = 0, which violates the SPARC BTFR
at −7σ under a uniform virial rule.
Status ZT.7 (Compatibility). Growth, σ8 , and the CMB
are untouched: a0 never enters linear growth, and the
coupling is strictly one-way (growth → δ̇ → Hscr → a0 ).
The homogeneous-mode theorem (Thm. ZS.3) survives:
Hscr is a functional of the invariant data (τ, A, δ), and it
never enters null geodesics — θ∗ is protected.
3.

Price list and the declared δ̇ closure

The package costs 2 discrete bits and 0 fitted continuous parameters:
1. spatial locality of the normalizer (Axiom ZT.2 itself);
2. the minimal-domain clause of the scale rule
(Prop. ZT.6; the spatial completion of no-memory
— natural, not a theorem).

404
Declared residual freedoms: the outer patch scale RD (the
environment-spread prediction varies 4× over RD = 5–
30 h−1 Mpc — the package’s largest soft spot) and the
cluster-patch density denv ∈ [0, 1].
Status ZT.8 (Declared δ̇ closure for the cluster band).
Mapping denv to Hscr requires a δ̇ model. The quasilinear closure δ̇ = f Hδ is invalid by construction at δ ∼ 1:
it understates the a0 suppression by 1.377× at denv = 0.5
and 1.725× at denv = 1.0 against the exact top-hat closure,
so the band it gives (“−2.8% to +2.2%, ≤ 0.72σ”) must
not be used. The band under the declared exact top-hat
closure: 16-cluster mean coherent shift (on the five-factorcorrected ratios of item 5 of §ZT 5, which size the package,
not a cluster closure) −2.8% to +6.4% over denv ∈ [0, 1]
(exact ODE; the harsher Bernardeau approximant gives
+6.6% — conservative). At denv = 1 the coherent shift
is +6.36% = +1.65σ (Bernardeau +6.57%/ + 1.71σ)
and the√relaxed class lands at 1.094 = +1.9σ (SER =
15.4%/ 10 = 4.9%; Bernardeau 1.096 = +2.0σ); mergers
improve 0.908 → 0.951 (Bernardeau 0.953). The band is
absorbable for denv ≲ 0.5, strained at denv ∼ 1. denv
is a measurable per-cluster quantity (galaxy counts in the
10–20 h−1 Mpc shell) and is listed as such — a declared
band with a declared closure.
4.

Kinematic numbers

A strong-void construction of the local excess is not
used (it is excluded, and none of its predictions is used
in this appendix). The kinematic application below uses
only measured flow data on the measured clock.
5.

Predictions and pulls

1. Galactic sector at the a0 of Table CXXXII —
does not break: BTFR Υ-marginalised −0.85 to
−0.94σ naive, +0.17/ + 0.47σ with self-consistent
Hubble-flow distances; the dwarf-galaxy channel
−0.02σ; the Milky Way improves (+0.54 →
−0.13/ + 0.28). The SPARC stat-only pull (−3.4
to −4.0σ) is not the right meter (the a0 –Υ degeneracy).
2. RAR environment modulation (new, signed):
∆a0 /a0 = −(f /3) δenv ; spread 0.015–0.027 dex at
RD = 10–20 h−1 Mpc — inside the 0.057 dex intrinsic budget, and a candidate explanation for part of
it. a0 anti -correlates with environment density. The
current Chae-sample SPARC test is a null with the
right sign (ρ = −0.144, p = 0.26; sensitivity 17% vs
a 2–10% effect). The named path is conditional —
a forecast, not in hand : 2M++ densities [200] (exists) plus ∼300 TRGB-quality RAR galaxies — a
sample that does not yet exist (SPARC = 175 total
/ 64 matched; TRGB-distance rotation-curve galaxies today = a few dozen; future-feasible via JWST

TRGB reach + WALLABY/BIG-SPARC). Power:
with the package’s own intrinsic floor (∼0.096 dex
non-environmental at RD = 10), 300 galaxies give
∼1.4σ, not 3; the 3σ figure holds only on the optimistic branch where DLSC dominates the intrinsic
scatter — otherwise ∼2–4 × N is needed. Test on
SPARC × 2M++ (pre-registered, 7,000-mock calibrated; both field routes agree at corr = +1.000):
bound. At the patch band R = 10 h−1 Mpc the
slope on the prediction is s = +0.40 (recipe-faithful
profile, 95% [−0.55, +1.00]; full-nuisance hierarchical MCMC s = +0.16, 95% [−1.45, +1.11]) —
DLSC-signed at every scale, consistent with both
s = 1 and 0; the mocks certify a-priori nondecisiveness at N = 110 (detect power 30–38%,
driven by the measured 0.27 dex population scatter).
Established by the same execution: strong coupling
s ≳ 1.1–1.5 excluded at 95%; the linearized coupling
below RD ∼ 10 h−1 Mpc excluded at > 4σ (a δ ≃ 10
galaxy would need a0 < 0), so the coupling must
either live at RD ≥ 10 h−1 Mpc or saturate with a
δ/(1+δ)-class form — the shape of DFD’s own µlaw; and the hierarchical population normalization
√
lands
about 1.5σ of both 2 α cH0early and
√ within
2 α cH0loc (not yet independently verified). The
full analysis pipeline (simulation cubes, mocks, precommitted analysis plan) is archived with the reproducibility package.
3. The a0 (z) letter: the sign da0 /dz > 0 gains the
fixed-z environment lever. The screen lowers a0 in
the survey’s overdense domains (item 4), which adds
to the amplitude offset of App. A0Z (+0.095 dex on
either clock); the letter itself fits a free amplitude
on a Planck background (H0 = 67.7) and takes no
position on the DFD clock.
4. High-redshift discs at fixed z (signed; estimate):
the screen lowers a0 where a disc’s domain is overdense, and massive discs at z ≃ 1–2 sit in strongly
clustered haloes. At RD = 10 h−1 Mpc under the
declared exact closure the shift is about −10% at
the average environment of such haloes and −20 to
−34% in the densest known structures at z ≃ 1.6–
2.5, whose matter overdensity on that scale is about
0.8–1.3; at RD = 20 h−1 Mpc every shift is about
three times smaller. Low-mass discs like those of
MUSE–DARK are lowered by 6–7% at their average environment. The massive discs whose a0 is
1.2–1.4 × 10−10 m s−2 at z ≃ 1.5–2.2 (App. A0Z,
Stat. A0Z.12) would be matched only in domains
with δenv ≃ 2–3, about 6 to 11 standard deviations
denser than average, and their measured environments show no such trend, so the screen does not
reconcile them. Test: matter-density maps of the
deep fields set against per-galaxy a0 ; at fixed z the
a0 must fall as the domain density rises. The sign
alone does not separate the screen from a conventional external-field effect (Rem. ZT.9); the size of

405
TABLE CXXXII. The kinematic numbers. CF4 radii are in h−1 Mpc (150 h−1 Mpc ≈ 222 Mpc at h = 0.675).

√
Global clock
H0 = 67.52; aglobal
= 2 α cH0 = 1.1208 × 10−10 m/s2
0
−1
Kinematic deficit (CF4 monopole) δm (< 150 h Mpc) = −0.057 ± 0.009
Local H excess
+1.0–1.2% ⇒ Heff = 68.20 ± 0.11; a0 (< 150 h−1 Mpc) = 1.1321 × 10−10
Very-local (< 70 Mpc)
H = 70.3 ± 0.4; a0 = 1.1669 × 10−10 (carries the SH0ES Cepheid zero point — not DFD-independent)
Local a0 band
1.13 × 10−10 (1.17 on the SH0ES zero point) vs SPARC 1.20 ± 0.02(stat) ± 20%(sys):
unresolved at stat, absorbed by sys. Not an exact match.

the fall and its 10 h−1 Mpc scale are the screen’s
own.
5. Cluster sector (patch rule): the scatter and
window figures below are measured on five-factorcorrected cluster ratios, which App. I does not print
because that correction stack does not hold (clusters
need χ); they size the package, not a cluster closure.
Coherent shift −2.8% to +6.4% for denv ∈ [0, 1]
(exact ODE; Bernardeau +6.6%) under the declared
exact top-hat δ̇ closure (Status ZT.8); at denv = 1
the numbers of Status ZT.8 apply — absorbable
for denv ≲ 0.5, strained at denv ∼ 1, with denv
measurable per cluster. Both-sided scatter: the
relaxed class improves 4.6% → 3.9%, but the full
16 worsens 7.09% → 10.36% and the ±10% window
drops 13/16 → 12/16. Merger-class cost −1.2 to
−1.8σ. Epoch mismatch: App. I uses a0 (z = 0) at
z = 0.02–0.87; correcting the epoch alone moves the
16-mean 0.978 → 0.924, away from Obs/DFD = 1
(mergers 0.833), so a “toward 1” reading is wrongsigned; the DLSC patch-rule offset, not the epoch
correction alone, is the result of this appendix.
6. Time-delay lenses on this appendix’s early
clock: H0 (TD) = 67.6–67.8. Status: consistent, not yet discriminating — the current anchors are the collaboration’s own 2025 reanalysis [201]: TDCOSMO-2025 primary 71.6+3.9
−3.3
(+1.2σ) and TDCOSMO+SLACS+SL2S combined
74.3+3.1
−3.7 (+1.8σ); the collaboration’s 2026 update
with eleven lenses (arXiv:2610.14781) gives 71.1+2.4
−2.3
(+1.5σ).
The earlier TDCOSMO-IV+SLACS
67.4+4.1
−3.2 (−0.1σ) is superseded by the 2025 reanalysis, which attributes the old low value to resolved
methodology issues. The 2020 TDCOSMO-only hierarchical value is 74.5+5.6
−6.1 (+1.1σ); only the powerlaw H0LiCOW value 73.3 ± 1.75 is 3.2σ off; SN
Encore MACS J0138 (2026) 66.9+11.2
−8.1 supports. No
mass-sheet-robust ≥ 73 at 2σ exists. Under the twoclock postulate they read 72.1: TDCOSMO-2025
lies −0.1σ, TDCOSMO-2026 −0.4σ, the combined
value +0.6σ and H0LiCOW +0.7σ from it.
7. Growth / σ8 / θ∗ : the domain-local screen does
not alter the current baseline. The measured-clock
run gives σ8 = 0.8063, S8 ≃ 0.822, 100θ∗ = 1.04110,
and Aϕϕ = 0.9887 (−0.80σ; App. ZS).

Remark ZT.9 (The external-field caveat, stated both
ways). Ref. [110] carries a caveat that touches this appendix directly, and it is reproduced here at its own grade
rather than paraphrased into support. A per-galaxy RAR
fit requires an environment-correlated external-field term,
so the effective a0 inferred galaxy by galaxy depends on
the ambient field [166]; both the mean cosmic field and
typical environmental fields evolve with redshift; therefore
an evolving external-field effect can in principle mimic
an evolving apparent a0 . Ref. [109] excludes interacting
galaxies and cluster members and argues the impact is
minor for its sample. The letter’s own point about the
sign is the sharp one: weaker external fields at higher z
mean weaker apparent-a0 suppression there, which steepens the apparent evolution — i.e. acts in the direction
of the measured signal — and so deserves quantitative
treatment in a per-galaxy analysis.
(i) As a caveat on the cosmological reading. Some part
of the measured a0 (z) slope could be environmental rather
than cosmological. The epoch reading of Eq. (ZT2) is
therefore not the only reading those bins admit, and the
exponent γ measured on them is, at present, a sum of a
cosmological and an environmental contribution that no
published analysis has separated.
(ii) As independent motivation for the environment test,
not as support. This is the same environmental channel
that Ax. ZT.2 parametrizes, which is a reason to run the
per-galaxy environment test already specified in §ZT 5
— the reason arriving from outside this appendix rather
than from it. It is not evidence for the domainlocal screen. A mechanism that can mimic the signal
is a confound to be measured, not a confirmation: a
conventional external-field effect in standard modified
dynamics produces the same sign without Ax. ZT.2, so
the sign agreement discriminates nothing. Only a pergalaxy analysis that measures the environment coefficient
at fixed z — the test of §ZT 5, currently a bound, not a
detection — can separate the two, and until it does, the
environmental and cosmological contributions to γ stay
entangled.
(iii) The screen cannot make the rotation-curve form
of the effect. It sets one a0 per system, and the Bose
response u(ν) contains no a0 (App. AT, Prop. AT.20), so
every rotation curve stays on the isolated shape locus and
its outer part stays flat (theorem). In fits that hold a0
fixed the screen raises the fitted external field by about
0.007 at the median galaxy, against the 0.05 reported by
Chae et al. [166] (computed).
(iv) The local data do not show the environment-

406
correlated term the caveat assumes. In the rotation-curve
fits of Chae et al. the fitted external field does not correlate
with the environment density on 4–20 h−1 Mpc scales; it
follows its own error bar and the inclination, and weighted
by the per-galaxy errors it is 0.012 ± 0.009 (App. GR;
computed).

6.

The early-clock Hubble posture: SH0ES is its
named external falsifier

Status ZT.10 (Posture). SH0ES, at 73.49 ± 0.93 in its
2025 analysis of HST and JWST Cepheids [106],
is the named external falsifier of this early-clock
posture at 5.0–6.4σ against the kinematic prediction
in SH0ES’s own window (0.023 < z < 0.15: 67.6–68.8
when SH0ES’s own peculiar-velocity correction, which
raises its H0 by 0.8%, is applied to the flow the prediction
carries; pulls (73.49 − 68.80)/0.93 = 5.0σ and (73.49 −
67.56)/0.93 = 6.4σ; computed). SH0ES’s 2022 value,
73.04 ± 1.04 [202], sits 4.1–5.3σ above the same band, and
the community Local Distance Network’s 73.50±0.81 [107]
5.8–7.3σ. This is a bet against SH0ES specifically, not
against all ladders: the named systematic channel is calibrator selection (the live Freedman–Riess dispute); supernova standardization is disfavoured, since replacing the
supernovae with galaxy-based indicators moves the ladder’s H0 by less than 0.1 km/s/Mpc [107] — Cepheid photometric crowding is excluded at 8.2σ by JWST (Riess et
al. 2024, ApJL 962, L17) (> 1000 Cepheids, HST–JWST
difference −0.01 ± 0.03 mag), so generic “ladder systematics” is not defensible. The supporting side: CCHP’s
TRGB from HST and JWST, 70.39 ± 1.22 ± 1.33 ± 0.70,
its JWST-only TRGB 68.81 ± 1.79 ± 1.32 and JAGB
67.80 ± 2.17 ± 1.64 [108] lie 0.8–1.5σ, 0.0–0.6σ and within
0.4σ of the band (stat+sys in quadrature). Against it,
SBF 73.3 ± 2.5 (40–99 Mpc) [203], and SBF calibrated on JWST TRGB with neither Cepheids
nor supernovae, 73.8 ± 0.7 ± 2.3 [197], sit 1.8–2.3σ
and 2.1–2.6σ above the band. The “very-local 70.3”
figure is the favored small-void fit of Stiskalek et al. [204]
(2025) and carries the SH0ES Cepheid zero point; on the
zero point this posture implies it falls to about 65–66, below the global 67.52, so its sign is not a DFD prediction.
CF4 radii are in h−1 Mpc (150 h−1 Mpc ≈ 222 Mpc at
h = 0.675); corrected local excess +1.1–1.2%, the quoted
+1.0 ± 0.2% covers it. If SH0ES stands, this posture
is refuted; under the two-clock postulate (Sec. ZS 5)
SH0ES’s value is read on the bound-matter clock, 1.5σ
above H0loc = 72.09.
7.

Falsifiers

DLSC falsifiers. Item 4’s forecast path is conditional:
2M++ densities plus ∼300 TRGB-quality galaxies — a
sample not in hand; ∼1.4σ at N = 300 under the package’s own intrinsic floor, 3σ only if DLSC dominates the

intrinsic scatter.
1. SH0ES vs CCHP (Freedman–Riess dispute).
Dispute resolves for SH0ES (its calibrator selection vindicated) ⇒ the kinematic posture is refuted at 5.0–6.4σ
(5.8–7.3σ on the Local Distance Network). Resolves
for CCHP (68–70) ⇒ posture supported.
2. Mass-sheet-robust TD H0 . ≥ 73 at 2σ ⇒ refutes
this posture. 67–69 ⇒ confirms this posture and falsifies the two-clock postulate (Falsifier ZS.14). (Currently: consistent, not discriminating.)
3. a0 (z) sign. A measured frozen (da0 /dz ≤ 0) or ρm scaling a0 ⇒ refutes the carrier (the letter excludes
γ = 0 at 5.3σ by likelihood ratio and γ = 3/2 at 4.8σ
on the exponent’s error and 4.7σ by likelihood ratio,
both on the published bin errors; counted galaxy by
galaxy the exclusion of γ = 0 is about 1σ within the
survey and about 3σ anchored to SPARC’s z = 0 value,
and the trend depends on how the molecular gas is
added, and massive discs at 0.6 < z < 2.5 show no rise
at face value, App. A0Z).
4. RAR environment binning (conditional forecast
path; see above). Binned a0 shifts with the wrong
sign, or magnitude incompatible with −(f /3) δenv at
every RD ∈ 5–30 h−1 Mpc ⇒ refutes DLSC modulation.
5. High-redshift disc environments. At fixed z, the
a0 of high-redshift discs shifts with the wrong sign as
the density of their domain rises, or by a magnitude
incompatible with the screen at every RD ≥ 10 h−1 Mpc
⇒ refutes DLSC modulation at high z. The massive
discs with a0 ≃ 1.2–1.4 at z ≃ 1.5–2.2 sit in average
domains ⇒ the screen does not reconcile them, and
their tension with the epoch law stands (App. A0Z;
the screened values of Stat. A0Z.12 are an estimate).
6. Cluster booking. Coherent shift > 3σ under every
admissible denv ∈ [0, 1] ⇒ kills the cluster booking.
(Current, exact top-hat closure: ≤ +1.65σ coherent
(Bernardeau +1.71), relaxed class +1.9σ (Bernardeau
+2.0) at denv = 1 — strained, falsification threshold
not crossed; merger class −1.2 to −1.8σ carried at
denv = 0.)
8.

Named open items

1. The domain scale is not derived: the definition of
a domain is met by every region between the zerovelocity surface and the homogeneity scale, and
the minimal-domain clause needs a homogeneity
threshold the definition does not contain. It is an
empirical scale of this extension, bounded below by
the first environment test (RD ≥ 10 h−1 Mpc for
the linear form) and measured by the test of §ZT 5.
No result
√ outside this appendix uses the extension:
a0 = 2 α cH0 , the RAR and BTFR and the epoch
law hold without it.

407
2. The a0 (z)-letter amplitude worsens 0.024 dex under
DLSC (the survey’s bin environments are not yet
measured).
3. Merger-class deficit −1.2 to −1.8σ under the patch
rule (a physically sign-correct mitigation exists —
the densest nodes have the largest denv — but denv
is a declared band, not a fit).

Appendix ZU: The Gravity Completion: the Exact
Newton Identity of the Quantized ψ Sector, and the
Kohn Null

This appendix books the operator (second-quantized)
completion of DFD’s gravitational sector and its laboratory consequence. Two independent mean-field treatments of the corpus-forced four-fermion vertex — a
Wick/Hartree–Fock–Bogoliubov route and a disjoint
Hubbard–Stratonovich saddle route — agree exactly:
Newton’s law survives as an exact, zero-parameter identity of the quantized ψ action (Sec. ZU 2); the pairing gap
has a closed form but the condensate is not the carrier —
Hartree+Fock is (Sec. ZU 3); second-order corrections are
controlled (Sec. ZU 4); and, because every term of the derived completion is translation-invariant or proportional
to N̂ , the completion predicts exactly zero gravitational
self-shift of the center-of-mass mode of a trapped crystal
— the Kohn null (Sec. ZU 5) — where the semiclassical
reading of the corpus’s own Schrödinger–Newton equation predicts a nonzero sideband. The completion thereby
overturns the semiclassical commitment: the self-gravity
nonlinearity of Sec. XVIII B is not a DFD prediction but
the falsifier of this completion (Sec. ZU 6). The scope,
including what the null does not establish, is stated in
Sec. ZU 7.

1.

The forced four-fermion vertex

Every object in this appendix descends from two printed
anchors, with no freedom introduced anywhere in the
chain.
The action. The scalar energy of Def. AP.26 (App. AP)
carries the prefactor c4 /32πG,
Z
Z
c4
c2
2
3
Eψ =
κα F (Y ) d x −
ρ ψ d3 x, (ZU1)
32πG
2
whose Euler–Lagrange equation (Thm. AP.27) is the
AQUAL field equation; the Newtonian regime is F ′ = 1.
The matter coupling. The single dispersion relation of
App. AN, E 2 = c2 e−2ψ |p|2 + m2 c4 e−ψ , expands at weak
field to U = − 12 mc2 ψ.
√
Canonically normalizing, ϕ = ψ/ 16πG makes 12 (∂ϕ)2
equal to (1/32πG)(∂ψ)2 exactly, and the Yukawa coupling
is forced to
√
√ m
m
= 3.5449077018
,
y = 12 m 16πG = 2 π
MP
MP
y1 y2
Gm1 m2
V (r) = −
=−
exactly.
4πr
r
(ZU2)
Second-quantized (position space, forced — no channel
choice exists: the three-Wick-channel contraction set is

408
complete),
h ℏ2 ∇ 2
i
XZ
Ĥ =
fs† (x) −
+ Vext fs (x)
2m
s
Z
Z
1X
+
fs† (x)fs†′ (y) v(|x − y|) fs′ (y)fs (x),
2 ′
s,s

v(r) = −

Gm2
.
r

(ZU3)
On the finite sector the pairing kernel is derived (not
chosen) from the Berezin–Toeplitz matrix elements of the
ψ modes on H 0 (CP 2 , O(9)), and the exact mode sum
X 1
3665699
K=
=
exactly
(ZU4)
ωk2
1764
k

renders the self-energy finite with no counterterm.

2.

Newton survives as an exact zero-parameter
identity

Theorem ZU.1 (Exact Newton identity; zero parameters, forced normalization). Newton’s law is an exact identity/saddle of the quantized ψ action: the vertex (ZU2)
saturates V (r) = −Gm1 m2 /r with zero adjustable quantity, by two independent symbolic routes (canonical normalization + Yukawa exchange; direct Poisson solve),
and in the disjoint Hubbard–Stratonovich formalism Newton is the exact saddle of an exact Gaussian identity
whose weight is forced to the corpus c4 /32πG prefactor
— uniqueness proven, so no hidden normalization is possible. Consequences, all with the same G: the fifth-force
strength α5 = 0 identically, and, with the metric dictionary (Lemma WS.4), γPPN = βPPN = 1 — matter’s G
and light’s G are one number by construction. Numeric
closure at CODATA: Gout /Gin − 1 = −5.6 × 10−16 .
G itself is consumed, not derived: it enters through
the printed c4 /32πG prefactor and nothing in this chain
derives its value. What is parameter-free is the identity —
the coefficient ratio is exactly 1 — not a value of G.

3.

The gap in closed form; the carrier is Hartree,
not a condensate

On the exactly-diagonal pair kernel (diagonality forced
by the torus selection rule), the gap equation decouples
orbital-by-orbital and admits the closed form
q
y 2 Gii
y2
Ei =
, δ= P
, ∆i = Ei2 − δ 2 ,
2
2 shell 1/Gii
(ZU5)
nonzero on exactly 7 of 60 orbitals (indices 27–33, the degenerate Fermi shell), matching the reference Bogoliubov–
de Gennes solver to 4.99 × 10−18 . Numeric labels (electron): the quasiparticle-energy maximum is Emax =
9.366856 × 10−27 GeV; the true gap maximum is ∆max =

9.355917 × 10−27 GeV (0.117% lower, the δ 2 correction).
The condensate is not the carrier : under the complete
three-channel (Hartree/Fock/pairing) functional the BCS
minimizer runs to the κ = 0 boundary — confirmed
Wick-free by direct Fock-vector minimization — and the
mean-field completion of the vertex is Hartree+Fock.
The Hartree channel reproduces − 12 mc2 ψ[Ψ] coefficient
for coefficient: the c-number source ρ = m⟨Ψ|ρ̂|Ψ⟩ is the
Hartree factorization of the two-body operator (ZU3), so
the Schrödinger–Newton form needs no classical-ψ axiom.
4.

Second-order control

2
The beyond-mean-field residual reproduces O(αψ
) with
fitted slope → 2, and the one-loop/MP2 layer carries
99.7% of it; on DFD’s own finite sector the residual is
exactly zero. The finite self-energy (via the exact K of
Eq. (ZU4)) is proportional to N̂ and hence unobservable
(absorbed into the measured mass). Two layers are priced,
not computed, and are explicitly not claimed discharged:
3
the two-loop layer at αψ
and the ψ-loop layer at (m/MP )2 .

5.

The Kohn null

Theorem ZU.2 (The Kohn null). Every term of the
derived operator completion is translation-invariant or
proportional to N̂ . By the generalized Kohn theorem,
DFD’s operator completion therefore predicts exactly zero
gravitational self-gravity sideband on the center-of-mass
mode of a trapped crystal — no ωSN sideband, no twopeak structure — to all orders in αψ , with zero adjustable
parameters; the statement extends to unequal masses at
common trap frequency (real multi-species crystals).
The semiclassical reading of the corpus’s own
Schrödinger–Newton equation (Sec. XVIII B) predicts
instead
Gm
2
ωSN
= √ 3 > 0,
(ZU6)
6 πσ
with σ the measured Debye–Waller wave-packet width —
an apparatus property, not a DFD parameter. Citationfaithful magnitudes at the cryogenic (100 mK) parameters of Großardt et al. [205] (Table I): osmium ∆ω/ω =
1.2 × 10−13 (this chain) to 6.7 × 10−13 (Großardt et al.’s
own number — ours is the conservative one); silicon
1.1 × 10−15 to 6.2 × 10−15 . The discriminator statement stands on osmium at the ∼10−13 order; silicon
is secondary. The relevant protocols are the optomechanical Schrödinger–Newton tests of Yang, Miao, Lee
and Chen [206] and Großardt et al. [205]; no published
dataset reaches this discrimination yet — a live test, not
an archival one. An independent calculation additionally
bounds the maximum anharmonic/trap-internal leakage
into the c.o.m. sideband at 1.9 × 10−18 (Os) / 2.0 × 10−19
(Si) — a factor 5.5×103 –3.5×105 below the discriminator,
so anharmonic corrections can neither fake nor mask it.

409
Falsifier ZU.3 (The SN sideband falsifies the completion). A measured positive self-gravity sideband at the
predicted size (ZU6) falsifies the operator completion of
this appendix outright (sign-definite) and vindicates the
semiclassical reading. A null result confirms the operator completion but does not uniquely fingerprint DFD
(Sec. ZU 7).

6.

The Schrödinger–Newton term is the
completion’s falsifier

The operator completion of the same printed action
forces the linear side: the c-number source is the Hartree
factorization of a two-body operator (Sec. ZU 3), the
one-particle sector is exactly free, and the Kohn null
holds to all orders. The Schrödinger–Newton frequency
shift of Sec. XVIII B is therefore not a DFD prediction
but the falsifier of this completion. Novelty credit is
stated precisely: the single-outcome analysis (App. QM)
had already closed the classical-vs-quantized-ψ root in
the quantized direction (Page and Geilker [207]; and the
MAQRO sustained-fringe prediction); the present completion completes that line constructively — exact Newton
identity, controlled completion, and the Kohn-null laboratory discriminator.

7.

Scope and verification status

Status ZU.4 (What is and is not claimed). (i) The Kohn
null itself is shared with standard linear QM : the unique
DFD content is that a zero-parameter chain from the
printed action (corpus action → forced vertex → forced
channels) lands the theory on the linear side. (ii) The two3
loop (αψ
) layer is priced, not computed. A translationbreaking ψ-loop term cannot arise: the operator completion is invariant under spatial translations, so the totalmomentum Ward identity holds at every loop order, and
the loop dressing of the external trap is local and suppressed by (m/MP )2 ≃ 2 × 10−34 for osmium, more than
twenty orders below the discriminator. The Kohn null
therefore holds to all orders in ψ loops as well. (iii) G is
consumed, not derived. (iv) Two independent formalisms
(Wick/HFB; Hubbard–Stratonovich saddle) agree exactly.

Appendix NL: The Scale-Additive Field Equation
and Its Test Record
1.

Why a scale-resolved form is needed

The local operator of Eq. (21) evaluates µ at a point.
DFD’s derived response is, however, a mode statement:
ν = 1/µ is a single-mode partition function and τ is an optical depth (App. N), so the physical question is not “what
is the field here?” but “how many independent modes
does the field here consist of?”. Those two questions
have the same answer only when all the sourcing mass
lies at one distance from the field point. This appendix
constructs the operator that asks the second question
directly, records its forward evaluations, and proves five
no-go results that close the obvious alternatives.
a. The measurement that forces the question. Any
response functional whose argument is the vector sum
|∇ψint + ∇ψext | — which includes every local AQUAL
and QUMOND operator — generates a static anomalous
quadrupole in the planetary potential,


Q2
δΦ(x) = −
xi xj êi êj − 31 δij ,
(NL1)
2
with ê the direction of the Galactic external field. The
measured bounds are Q2 = (3 ± 3) × 10−27 s−2 from
nine years of Cassini range and Doppler tracking of Saturn [29], tightened to Q2 = (1.6 ± 1.8) × 10−27 s−2 on
the DE440 dataset [30]. Evaluating DFD’s own derived
response through the local reading at the bound-matter
value a0 = 1.1966 × 10−10 m s−2 (Postulate ZS.11) gives
Q2 = 30.6–31.8 × 10−27 s−2 across four independent evaluation methods, i.e. +9.2 to +9.6σ against Cassini and
+16.1 to +16.8σ against DE440. The same obstruction is
what Desmond, Hees & Famaey [179] report for MOND
itself: a 8.7σ tension between the SPARC RAR and the
Solar-System quadrupole under fiducial assumptions (relieved to 1.9σ only by discarding bulge galaxies). It is
structural, not numerical: Q2 = 0 requires the response
functional never to form the vector sum g⊙ + gext , and
no local operator in ∇ψ can avoid forming it.

2.

Statement of the scale-additive equation

Let the mode label be log-distance from the field point,
which is forced: DFD’s established mode reduced wavelength is λ̄ = s. Define the shell field density
Z ∞

I
γ(x, s) = −G

γ(x, s) ds = ∇ΦN (x),

dΩ ρ(x + sn̂) n̂,
0

(NL2)
which carries direction: a symmetric shell contributes
zero. Writing m(λ) = eλ γ(x, eλ ) for the field generated
per e-fold, the completion is a causal ODE in scale,
dgc
= m(λ) − w0 w(q) gc (λ),
q ≡ |m|/|gc |, (NL3)
dλ

410
closed by the two exact quadratures
Z
∇ΦN (x) = dλ w0 w(q) gc (λ),
Z
p
τ (x) = 12 dλ w0 w(q) |gc (λ)|/a0 ,

sition law that forces the exponential shell (App. N),
recovered exactly.
(NL4)

4. Subdivision: splitting any body into N pieces leaves
τ invariant to all orders, because γ is a linear functional of ρ.

with w(0) = 1 (in vacuum the mode forgets) and w(∞) =
0 (inside structure there is no Rforgetting). Both displayed
identities are exact because (dgc /dλ) dλ = 0 over the
whole
axis; the prefactor 12 is forced, not chosen — for
R
p
w gc dλ = g p the prefactor must be p, and p = 12 for
τ , p = 1 for ∇ΦN . The response is then the established
one, ν = 1/(1 − e−τ ), evaluated on this τ . For scaleseparated sources this τ composes additively, a reading
that weak-host-field data disfavour against the quadrature
form (§NL 10, item 2); a subsystem whose host mode is
typically empty is Newtonian (Axiom NL.14).
In words: gravity keeps a running memory of the field it
has passed, and forgets it only while crossing vacuum. A
galaxy is matter all the way through and stays one mode;
the Sun and the Galaxy, separated by 18.955 e-folds of
vacuum, stay two.
a. The window width
R is derived, not fitted. The window realisation gλ = ψ(ln s −Rλ)γ ds admits exactly
two normalisations and no more: ψ du
R √= 1 (partition of
unity, so Eq. (NL4) left is exact) and
ψ du = 1p
(a single point source must return
R τ = g/a0
√ the established
exactly).
Writing φ ≡ ψ these read φ = 1 and
R 2
φ = 1 — two conditions, which fix the width. Solving
them exactly: the top hat has support exactly one efold; the triangle 4/3; the raised cosine√3/2; the Gaussian
√
σ = 1/(2 π), i.e. equivalent width 1/ 2.
b. Closed form for point sources. Sorting sources
by distance from x, with λk = ln |x − yk | and gk =
Gmk (x − yk )/|x − yk |3 ,

Proof. (1) and (4) follow from Eq. (NL2) and the two
normalisations: a single Rsource contributes one jump in
∞
gc and the tail integral 0 w0 e−w0 v/2 dv/2 = 1 returns
p
g/a0 for every w0 . (2): all mass at one λ makes gc a
single jump equal to the vector sum, and the 12 prefactor
returns p = 12 of it. (3): for ∆ → ∞ the accumulator
has decayed to zero p
between contributions, so Eq. (NL5)
P
telescopes into
gk /a0 . Numerically, at machine
k
precision: residuals 0 (item 1, all w0 tested), 0 (item 2,
six random sources), 4.4 × 10−14 (item 3), and item 4
returns τN /τ1 = 1.000000000000000 for N up to 103 colocated pieces and exactly 1 thereafter by the contraction
in Eq. (NL5).

gc(k) = gc(k−1) e−w0 ∆k + gk ,
X q (k)


τ (x) =
|gc |/a0 1 − e−w0 ∆k+1 /2 ,

Corollary NL.3. σ ≡ τ − τN ≥ 0 on every configuration
tested (400/400 random 2–6-source configurations spanning four decades in distance and random directions; zero
negatives).
4.

Conservation

(NL5)

k

with ∆k = λk − λk−1 and ∆N +1 = ∞.
3.

Remark NL.2 (The Beer–Lambert limit needs a mass condition as well as distance). Scale separation alone is not
sufficient. The leaked field is g1 e−w0 ∆ = Gm1 /(d1 d2 ), so
the per-source limit requires (mnear /mfar )(dfar /dnear ) ≪ 1.
For the Sun in the Galaxy at 10 AU this ratio is 1.69×10−3
(with Menc (R0 ) = 1.01 × 1011 M⊙ from vc = 229 km s−1 ),
and the separation is ln(R0 /10 AU) = 18.955 e-folds. Two
comparable masses at a large distance ratio do not separate.

The exact limits

Theorem NL.1 (Limits of the partition of unity). For
any admissible vacuum detector w with w(0) = 1, w(∞) =
0, and any w0 > 0:
p
1. Single point source: τ = g/a0 , the established law,
exactly and independently of w0 .
2. Co-distant
sources (any masses, any angles): τ =
p P
| i gi |/a0 — the local, per-point (AQUAL) reading of Eq. (21), recovered exactly.
P
3. Scale-separated
sources: τ =
i τi , equivalently
Q
1 − µtot = i (1 − µi ) — the Beer–Lambert compo-

Theorem NL.4 (E, p, J conserved). The scale-additive
action
Z Z
n
S[Φ, ΦN ] = − dt d3 x ρΦ

o
1
+ 8πG
2∇Φ·∇ΦN − a20 Q + Smatter ,
4
Q(x) = ν(τ ) τN
− τ4




Z τ
+4

ν(v) v 3 dv,

0

τN =

p
|∇ΦN |/a0 ,

τ = τFB [ρ] (§N L 10, §N L 5),

has no explicit t-dependence at fixed a0 , is invariant under translations and rotations, and is differentiable in the
sourcePpositions. Noether therefore gives exactly
R conserved
E = a 12 ma ẋ2a + W with W = −(a20 /8πG) Q d3 x, together with p and J.
Proof. Time- and Euclid-invariance hold for every construction in this family, since every kernel is built from
|x − y| alone. Differentiability is the entire content: the

411
FB-OUT depth’s first-order response to a displaced point
source is explicit (Eq. (NL6)), and Q contains no subtraction, and is finite where τFB → 0 because τN ≤ τFB there,
so the integrand of W is finite on every positive-mass configuration; its volume integral grows as ln r at large distance, through a term fixed by the total mass, as the field
energy of any isolated deep-MOND mass does. Variational
residuals (symbolic): δS/δΦ ⇒ ∇2 ΦN = 4πGρ, residual
0; the constitutive step ∂(a20 Q)/∂(∇i ΦN ) − 2ν(τ )∇i ΦN ,
4
residual (0, 0, 0); Newtonian limit Q → τN
+ 4π 4 /15, an
inert additive constant, residual 0.
In this action ΦN is an auxiliary field: δΦ S = 0 makes
it the Newtonian potential of the matter, and it is not
a potential of the metric, which at first post-Newtonian
order carries the one function ψ = −2Φ/c2 (Cor. NL.17).
Remark NL.5 (Why a hard connectivity rule could not
have this). A hard partition of the mass into connected
components is a global operation and therefore jumps.
Measured at a merge surface, its |∆W |/|W | = 1.4005 ×
10−1 step-independently over four decades of step size, i.e.
a genuine 14% discontinuity, i.e. a δ-function force. The
scale-additive |∆W |/|W | is 8.29 × 10−5 , 10−7 , 10−9 at
the same three steps — exactly proportional to the step.
Status NL.6 (Numerical conservation). The theory’s own
central potential
is Φ(r) = −GM/r + a0 rM ln ν(rM /r)
p
with rM = GM/a0 . Symbolically (residual 0) dΦ/dr
=
√
ν(τ ) GM/r2 with τ = rM /r, and r dΦ/dr → GM a0
as r → ∞, i.e. the baryonic Tully–Fisher relation vf4 =
GM a0 holds exactly. Integrating a bound orbit in that
potential with a fixed-step fourth-order composition integrator at dt = T /8000 for 40 periods — 320 000
steps — gives max |∆E/E| = 1.5–2.6 × 10−12 (integratordependent), max |∆L/L| ≤ 5 × 10−14 , and a secular slope
per period of |·| < 2 × 10−15 , i.e. consistent with exactly
zero. The error falls under step refinement, so it is integrator truncation and P
not dynamics. On a random
−12
four-body
configuration,
|
a Fa |/ max |F| = 2.1 × 10
P
−11
and | a xa × Fa |/(max |x| max |F|) = 9.5 × 10 .
5.

Action completion of the scale-additive equation

The forward law ∇2 Φ = ∇· [ν(τ )∇ΦN ] on the FB-OUT
depth τ = τFB [ρ] (§NL 10, item 2; the depth of record for
this subsection) is the static limit of exactly one action
of the form Q(t, τ ) that keeps the isolated point-mass
law, and p
that action fixes the force onR matter. Write
τ
t ≡ τN = |∇ΦN |/a0 and QQU (τ ) = 4 0 ν(v) v 3 dv.
Theorem NL.7 (Unique static completion). With Q =
QM (t, τ ) ≡ ν(τ ) (t4 −τ 4 )+QQU (τ ) in Thm NL.4, δΦ S = 0
gives ∇2 ΦN = 4πGρ and δΦN S = 0 gives the forward
law, because ∂(a20 QM )/∂(∇i ΦN ) = 2ν(τ )∇i ΦN ; QM is
the unique Q(t, τ ) with this property whose isolated point
mass (τ ≡ t) carries the QUMOND energy. It contains no
subtraction, is finite on every positive-mass configuration
because τN ≤ τFB there, and a20 QM → |∇ΦN |2 +4π 4 a20 /15

(an inert constant) in the Newtonian limit. The force on
body a is
Fa = −ma ∇Φ(xa )
Z
a2
∂τFB (x)
+ 0
d3 x ν ′ (τ ) (t4 − τ 4 )
,
8πG
∂xa
i
Gm h
δ(a0 τ 2 ) = 2 cos ψ + (1 − ρ3 )(1 − cos ψ) Θ(s < r) ,
s
(NL6)
the second term being the first-order response of the depth
to a displaced source m with one other source present (the
Galaxy enters only through ge ; s = |x − y|, r the distance
from x to the other source, ρ = s/r, ψ the angle between
the two sources’ fields at x). For an isolated point mass
τFB ≡ t and the reciprocal term vanishes identically, so
the test-body law ν(t) gN is exact. For a disc, τFB = t at
the midplane (§NL 10, item 2) leaves the forward law there
unchanged; the reciprocal term on a midplane body samples
the off-plane depth, where non-reversal is not established.
For exponential discs of thickness 0.05–0.2 Rd , from MilkyWay-like to deep low-surface-brightness, it is below 3% on
a midplane star and cancels against the off-plane change
of the forward field to ≤ 0.002 dex between 0.5 and 8 Rd
(computed); for a spherical density that decreases outward
τFB = t at every point, so the reciprocal term vanishes and
the completed law is the local law. Uniqueness holds within
energies that depend on ρ through τFB alone; an energy
built from a second printed depth would add terms with
coefficients no printed principle fixes. In the frozen regime
of Axiom NL.14 the law is resolved by mode and has no
reciprocal term (§NL 6); the uniqueness above concerns
the law of weak hosts.
Theorem NL.8 (Cassini, exactly). For the Sun in
the Galactic field ge ê the depth a0 τ 2 = GM⊙ /r2 + ge
is spherical about the Sun (the vector sum is never
formed), while a20 t4 = |∇ΦN |2 is quadratic in that sum, so
t4 − τ 4 = −2ge GM⊙ (1 + cos θ)/(a20 r2 ) carries only l ≤ 1;
both kernels of Eq. (NL6) are rotationally covariant about
the Sun. Hence the test-body potential is l ≤ 1 and Q2 = 0
exactly at first order in the planet mass (−0.89σ against
DE440, −1.00σ against Cassini, from the central values).
The reason is sharper than the statement of §NL 1: an
energy linear in |∇ΦN |2 with a spherical coefficient has
no l = 2 term even though it contains the vector sum,
whereas AQUAL/QUMOND evaluate ν at |∇ΦN | itself.
Under Axiom NL.14 the Sun’s own field is Newtonian
inside the Galaxy, so Q2 = 0 at every order; the Galactic
tide itself is 8 × 10−31 s−2 (computed).
P
Theorem NL.9 (Newton’s third law).
a Fa = 0 by
translation invariance. The forward force alone does
not have this property: on a body of mass M in a
host field ge ê it exerts a net push −(ge /3)(νe − 1)M ê
(νe = ν(τext )), cancelled in closed form by the reciprocal
1
∞
term, 8π
(−2ge ) 4π
3 [ν]τext M = +(ge /3)(νe − 1)M ê (units
G = a0 = 1). At the Sun’s depth, with the Newtonian
Galactic field of 1.19–1.33 a0 as the argument, the push
would be 0.201–0.204 a0 M⊙ (computed); the Sun itself

412
is frozen by Axiom NL.14, the mode-resolved law has no
cross term there, and the
Ppush and the reciprocal term both
vanish. Cor. NL.18’s a Fa = 0 is therefore a theorem
of the action completion, external-field regime included.
Theorem NL.10 (Orientation: affine in the external
field). For bodies scale-separated from the Galaxy, τFB
depends on |ge | but not on ê, and t4 is affine in ê; hence
every Fa is affine in ê, and a pair’s relative acceleration at
angle θ to ê is exactly γ(θ) = γ0 +γ1 cos θ: no even (cos2 θ,
G∥ /G⊥ ) modulation at any separation or mass ratio, and
an odd term antisymmetric under m1 ↔ m2 — zero for
equal masses, ≤ 1% of γ0 for m2 /m1 = 0.5 at ≤ 3 ×
104 AU, and +1.5%, +3.0%, +2.8% of γ0 in the test-body
limit at 7274, 104 , 2 × 104 AU (larger attraction for the
companion on the Galactic-centre side). The statement is
about the force; the velocity observable vobs /vN inherits
an even term of order γ12 /2γ0 ≲ 7 × 10−4 at ≤ 3 × 104 AU
in the test-body limit, identically zero for equal masses.
At the Sun Axiom NL.14 makes local pairs Newtonian,
so both the even and the odd terms vanish there; the oddterm values above are the completed law’s for a host at
the Sun’s depth and apply to pairs in weak hosts.
Status NL.11 (Wide-binary force boost, external field
included). Orientation-averaged, ge = p
1.793 a0 , total mass 1 M⊙ (separations scale as
Mtot /M⊙ ).
Equal-mass pair (full three-source depth): γ =
1.26, 1.37, 1.50, 1.59, 1.69, 1.78 at 7274, 104 , 1.45 ×
104 , 2 × 104 , 3 × 104 , 5 × 104 AU (m2 /m1 = 0.5: within
0.012 of these). Test body (comets, planets): γ =
ν(τ0 ) + γrec = 1.37, 1.52, 1.74, 1.81 at 7274, 104 , 2 ×
104 , 3 × 104 AU, tending to 1.87 = νe + 0.511 for
rM ≪ d ≪ pc: the reciprocal weight falls as 1/r2 ,
so the boost does not saturate at νe = 1.355 as the
forward response does (ge = 1.838 a0 moves each testbody entry by ≤ 1%; with the Newtonian Galactic field
gN,e ≃ 1.2 a0 as the depth’s argument, the observed
1.79 a0 being the boosted total, every test-body entry
rises by 3–16%). These are the completed law’s values at
the Sun’s host field, and they are not realised there: the
Galaxy’s mode at the Sun is typically empty, so under
Axiom NL.14 local pairs are exactly Newtonian, γ = 1,
with no orientation term. On the Gaia DR3 analyses
this sides with the null camp, Banik et al. (2024) [52]
(−0.3σ on αgrav ) and Saad & Ting (2026) [208] (0.0σ),
against Chae (2023) [51] (+7.2σ), Chae et al. (2026) [209]
(+4.3σ), Yoon et al. (2025) [210] (+2.9σ on its combined
low-acceleration value) and Hernandez et al. (2024) [211];
Boufourou (2026) [212], γ = 1.045 [1.025, 1.068], lies 2.2σ
above it on that 68% statistical interval; its eccentricitymodel systematic band, 0.957–1.068, includes γ = 1. A
confirmed boost of the Chae type in local pairs falsifies
the axiom. Pairs beyond the edge of the frozen zone
keep the boost: on the forward law γ = 1.70–1.97 at
104 AU and 2.11–3.29 at 2 × 104 AU for R = 20–100 kpc,
and the reciprocal term adds to these (computed). The
orientation statements of Thm NL.10 hold for such pairs.

Status NL.12 (The Oort cloud excludes the external-field-regime boost as printed). Long-period comets are test
bodies of exactly the field of Status NL.11. Their original
1/a is the Newtonian specific energy inferred at ∼ 250 AU
(planets removed), and the CODE catalogue [213] shows
the Oort spike at 0 < 1/a ≲ 0.75 × 10−4 AU−1 (centroid
≃ 3–4 × 10−5 ; 277 of 369 long-period comets in [0, 10−4 ];
17 hyperbolic originals (4.6%); per-comet errors ≃ 10−6 ).
Every reading here is Newtonian to e−28 inside 250 AU
and conservative for the comet, soR for the test-body force
ra
−γ(r)GM⊙ /r2 , 1/aN = 2/ra − 2 250
(γ − 1) r−2 dr exAU
actly for a comet turning at ra . A comet tidally injected from atrue = 25 000 AU (1/a = 4 × 10−5 ) is catalogued at −2 × 10−4 (this completion), −1.0 × 10−4
(FB-OUT forward response), −3 × 10−5 (accumulator forward response, SOURCE branch) and +0.3 × 10−5 AU−1
(TOTAL branch), 30–230 per-comet errors and 3–23
non-gravitational shifts from the observed centroid; the
spike position alone bounds a boost saturating at rM to
γ∞ − 1 ≲ 0.02–0.04, against the printed νe − 1 = 0.36
(SOURCE), 0.20 (TOTAL) and 0.87 (this completion).
Independently, the odd l = 1 term of every reading is
102 –103 times the Galactic tide 4πGρ0 d at ≲ 104 AU,
collapsing the tidal injection radius from ≃ 2.3 × 104 AU
(Newton) to 3–4 × 103 AU in every reading, the completion included. A direct integration (2 × 104 test comets,
1 Gyr, Galactic tide) reproduces the observed spike under
Newton (median original 1/a = 4.0 × 10−5 AU−1 , 77%
within [0, 0.75] × 10−4 , no hyperbolic originals) and gives,
for the mildest printed reading (accumulator, TOTAL
branch, monopole), a median of −1.4 × 10−5 with 77%
hyperbolic originals; for the SOURCE branch 100% hyperbolic; for the completion a median of −1.9 × 10−4
with 75% hyperbolic; and, with the forward law’s own
l = 1 term, the stripping of two thirds of the cloud within
1 Gyr. Vokrouhlický, Nesvorný & Tremaine [214] exclude
AQUAL-MOND by this test for every transition function
(“a nearly uniform distribution in 1/a . . . far more LPCs
on hyperbolic orbits than the data”) and leave open theories “in which non-Newtonian effects are screened . . .
in external gravitational fields comparable in strength
to the critical acceleration”. DFD as printed does the
opposite: at the Sun the Galactic mode’s occupation
ne = 1/(eτext − 1) = 0.46–0.51 is passed to the Sun’s field
by the composition laws of §NL 2 (τext = 1.09–1.15, taken
on the Newtonian Galactic field for an observed field of
1.79–1.94 a0 ; the entries of Status NL.11, which enter the
observed field as the argument, understate the excess the
comets exclude). Every reading that boosts the Sun’s own
field is excluded by this test. Under the environmentalconfinement axiom (§NL 6) the Galaxy’s mode at the Sun
is typically empty (ne = 0.46–0.51 < 1), so the Sun’s own
field is Newtonian and the Newtonian run of the integration above reproduces the spike; this is a postdiction,
since the axiom was stated to pass it. What is untouched:
everything without an external mode (the RAR, BTFR
and a0 results, which carry no external mode), the SolarSystem monopole at the planets (γ − 1 = e−τ , τ ≃ 7 × 102

413
at Saturn), Q2 , and the dwarf-spheroidal external-field
results, whose host fields lie below about 0.2 a0 observed
and below 0.03 a0 in the Newtonian baryonic field (Table XCIX), far below the axiom’s threshold. The axiom’s
own predictions are γ = 1 with no orientation term for
local pairs, and a step in the dynamics of bound subsystems at the edge of the frozen zone, R ≃ 13–16 kpc; the
outer-halo globular clusters beyond it show about a third
of the predicted boost (0.33 ± 0.08) (Table CXXXIII).

6.

The external-field sector: environmental
confinement

Theorem NL.13 (A host caps an embedded boost). Let
a subsystem of depth τs sit in a host of depth τh , scaleseparated in the sense of Rem. NL.2. (i) The additive
composition is the fugacity law of Thm BS.5: ν(τs + τh ) =
1/(1−z e−τs ) with z = e−τh , so as τs → 0 the subsystem’s
boost tends to ν(τh ) = 1 + nh , nh = 1/(eτh − 1). (ii)
Every composite depth that depends on the subsystem
only through τs , is continuous, and returns τh as τs →
0 has the same limit; the accumulator, the quadrature
and the enclosed-mass depths all qualify. (iii) Tying the
embedded quanta to the host’s, whether locked, bounded
or in exchange equilibrium, gives the same nh ; letting the
host keep one quantum gives nh e−τh , and two host quanta
per embedded quantum give nBE (2τh ).
Proof. (i) is a geometric series (symbolic residual 0). (ii) is
continuity at τs = 0. (iii) sums the geometric occupation
P (m) = (1 − e−τh ) e−mτh of the host mode.
At the Sun the Galaxy’s depth, taken on its Newtonian field, is τh = 1.09–1.15 (nh = 0.46–0.51), so every
composition of (ii) leaves the Sun’s own field boosted by
46–51% where that field is weak, and the nestings of (iii)
by 11–17%, where the Oort spike allows about 2% out
to 5 × 104 AU (Status NL.12). In joint equilibrium the
state with the host mode occupied and the Sun’s mode
ladder active outweighs the state with the host empty
and the ladder frozen by a factor of about 500 (the ladder
carries ln Z = 7.4 nats), so the statistics favour the boost
(computed). The freeze has to be stated.
Axiom NL.14 (Environmental confinement). A bound
subsystem keeps the µ-law boost of its own field only while
the Bose mode of its host is typically occupied, that is,
while the median occupation of the host mode is at least
one:
nh ≥ 1 ⇐⇒ τh ≤ ln 2 ⇐⇒ gN,h ≤ (ln 2)2 a0 ≃ 0.48 a0 ,
(NL7)
p
with τh = gN,h /a0 and gN,h the Newtonian field of the
baryons of every enclosing bound system at the subsystem.
Where the host mode is typically empty the subsystem’s
own field is exactly Newtonian. The host’s own field is
unchanged across the subsystem.

A subsystem’s
relative acceleration is νemb gN,s , with
p


νemb = ν
τs2 + τh2 for τh < ln 2 (the quadrature composition, §NL 10, item 2) and νemb = 1 for τh > ln 2;
its centre of mass moves in the host’s field. In the
frozen case the forward law is resolved by mode, ∇2 Φ =
∇ · [ν(τh )∇ΦN,h ] + ∇2 ΦN,s across the subsystem: the
host’s field keeps its boost, the subsystem’s own field is
Newtonian, and no term couples the two. The forward
force then exerts no net push on the subsystem, and the
reciprocal term of the action completion, which cancels
that push (Thm NL.9), is absent with it, together with
the test-body excess of about 0.51 that it gives at the Sun
(Status NL.11). In weak hosts the completed law stands
as printed. It multiplies the total Newtonian field by ν
of the combined depth, so across a subsystem the host’s
field carries a term in the subsystem’s depth, which the
mode-by-mode demand of the energy in App. AV does
not have. The satellite and wide-binary numbers of this
appendix use the completed law; the χ results of App. AV
use the mode-by-mode energy. The action form of the
mode-resolved law is not constructed.
The threshold is not a chosen number. For a Bose mode,
n < 1, τ > ln 2, P (n=0) > 12 and a median occupation of
zero are one statement, so the threshold is where the host’s
typical state becomes the vacuum. What the axiom adds
is that the embedded ladder follows the typical (median)
state of the host mode, and follows it sharply: the evenhanded median readout fixes where the switch sits, and
its sharpness is what is stated; the host’s own boost keeps
the mean, ν(τh ) = 1 + nh . The host mode is the one the
host’s baryons polarize (Rem. GR.9), so its depth is set
by the baryonic field; counting χ would freeze the cores of
rich clusters out to about 1 Mpc and slow their galaxies
by about 40% (estimate), which cluster Tully–Fisher work
does not show. The Rest-Mass Channel confines the µ-law
to bound low-acceleration systems (App. GR); the axiom
judges “low-acceleration” by the system’s environment,
with the Bose median as the criterion, and adds no field
and no constant. It realizes the possibility Vokrouhlický,
Nesvorný & Tremaine leave open, a response “screened
. . . in external gravitational fields comparable in strength
to the critical acceleration” [214].
a. The edge of the frozen zone. On the Milky Way’s
Newtonian baryonic field the threshold lies at R ≃ 13 kpc
for 6.5 × 1010 M⊙ of baryons and 16 kpc for 1011 M⊙
(computed). Inside it every bound subsystem is Newtonian: the Solar System, the wide binaries of the Gaia
catalogues near the Sun, open clusters and the inner globular clusters. Outside it they keep their boost; for the
outer-halo globular clusters this is contradicted by the
data (Table CXXXIII). A subsystem that crosses the edge
switches laws: the relative force of a 1 M⊙ pair at 104 AU
is 1.59 times the Newtonian force just outside and exactly
Newtonian inside (1.85 at 2 × 104 AU; forward law, the
reciprocal term adds to both; computed), where MOND’s
external-field effect declines smoothly. The data allow
the threshold depth anywhere between 0.16 (the classical dwarf spheroidals, all boosted) and 1.09 (the Sun),

414
R ≈ 8–66 kpc; the axiom puts it at ln 2. In an action the
step is a δ-function force (Rem. NL.5), so the zero-dipole
theorem (Cor. NL.18) holds away from the edge only; a
smoothing over ∆τ ≤ 0.15 changes none of the results of
Table CXXXIII (estimate). The comets do not need an
exact step: if their energy budget scales with the switch’s
m
weight, a Hill switch nm
h /(1 + nh ) passes them for m ≳ 4–
5, while a single coherent host mode, m = 1, gives 5.9
against an allowance of about 1 (computed); a derived
width is open.
The Oort spike and the planets are postdictions: the
axiom was stated to pass them. The local wide binaries
and the edge are its predictions. Draco is boosted and, like
every reading, under-predicts its dispersion; it does not
discriminate. The axiom has four costs: it is a postulate,
which the Bose statistics provably cannot supply; it is a
step, a δ-function force in an action, with a width that
is not derived; the action form of the law in the frozen
regime is not constructed, so Thm NL.4 is not extended
to that regime; and the external-field values that the
completed law gives at the Sun’s depth (Status NL.11)
are not realised at the Sun.

7.

The test record

a. Checks at the bound-matter
value a0 = 1.1966 ×
p
10−10 m s−2 . rM,⊙ = GM⊙ /a0 = 7039.671 AU (reproduced to 10−5R); τ⊙ (Saturn) = 734.8; the partitionof-unity identity w0 w gc dλ = ∇ΦN closes to 8 × 10−7
on the numerical grid; the ephemeris pipeline reproduces
Hees et al. [29] to 0.58% across all 56 rows of their Table 2
and 48/49 rows of Milgrom [215] Table 1, before it was
pointed at DFD.
b. The disc RAR, and its sensitivity to the one imported constant. On the depth of record, FB-OUT, the
disc RAR is untouched: its midplane depth equals the
local one exactly, and the completion moves the relation
by at most 0.002 dex (computed). The numbers below
belong to the accumulator reading, whose one imported
constant is the width w0 . For an exponential Milky Way
disc (Rd = 2.6 kpc, hz = 0.3 kpc), the displacement of
the radial acceleration relation between the scale-additive
τ and the local per-point τ is 0.016–0.018 dex r.m.s. over
1–30 kpc, against the observed intrinsic scatter floor of
0.057 dex [104]. Varying the one imported constant over
w0 ∈ [0.5, 2] gives total displacements of 0.016–0.073 dex
and shape-only displacements (r.m.s. about the mean,
which a RAR fit would absorb into normalisation) of
0.008–0.035 dex. The imported value w0 = 1 gives the
smallest total displacement of the family, and the only
member that exceeds the observed floor is w0 = 0.5.
8.

The distinctive signature: radial

Theorem NL.15 (Angular null). Clumps at equal
distance from the field point produce τscale-additive =

τvector-sum bit-identically (0.00 × 100 residual for N =
2, 4, 8, 16 spread over a 34◦ cone). Angular clumping
is invisible to the equation. The entire discriminator lives in log-distance from the tracer.
Proof. Immediate from Thm NL.1.2: equal distance
means equal λ, and the closed form contracts co-distant
contributions into their vector sum before any square root
is taken.
Theorem NL.16 (Master law). At fixed gbar , with
N contributors each supplying g/N separated by ∆ efolds, τnew /τ√
old runs monotonically from 1 (Thm NL.1.2,
∆ = 0) to N (Thm NL.1.3, ∆ → ∞), and in the
deep field ∆ log10 gobs → − 12 log10 N exactly. The ratio
is ≥ 1 everywhere: there is no crossover, and the
forward response is uniformly weaker-boost than
the local one, at every geometry. Inside a structure
generating field per e-fold the accumulator√reaches a fixed
point q ∗ = w0 w(q ∗ ) and τnew /τold → C √Λ, with Λ the
structure’s extent in e-folds, with C = q ∗ /2; for the
primary detector w = e−q this is the omega constant,
q ∗ = W (1) = 0.5671432904, giving C = 0.3765445825.
Corollary NL.17 (Lensing mass = dynamical mass).
DFD’s metric is one-function at first post-Newtonian order, g00 = −e−ψ , gij = e+ψ δij (Sec. IV). The PPN
parameter γ is read off the ratio of the two exponents; ψ
never enters as a source. Hence γ − 1 = 0 for arbitrary ψ
(symbolic residual 0), light deflection = 4GM/(c2 b) (residual 0), and the lensing kernel minus the dynamical kernel
vanishes to O(v 2 /c2 ), as in general relativity: the traceless
strain of the parent tensor enters the spatial metric at
second order (App. WS). No static field equation for
ψ — local, nonlocal, either branch — can touch
this; it is a property of the coupling, not of the
response. The reciprocal force of the action completion
(Thm NL.7) is not part of the metric: it acts on bodies
in weak hosts and not on light, so where it does not vanish (test bodies in weak hosts, the wide-binary term γrec ,
satellites of weak hosts) dynamics differs from lensing by
it; it vanishes for isolated bodies and spherical sources and
is below 0.002 dex at disc midplanes. RAQUAL cannot
reproduce it (its conformal coupling leaves lensing on the
Einstein-frame potential); TeVeS needs an extra vector
field to force it.
Corollary NL.18 (Zero dipole radiation, andRwhy only
this equation has it). The source term −(c2 /2) ψ(ρ − ρ̄)
is linear in ρ with one universal constant, so the scalar
charge is ρ, the dipole moment is D P
= M Xcm , and
dipole power ∝ M 2 |Ẍcm |2 vanishes iff a Fa = 0. The
scalar-tensor (αA − αB )2 term cannot appear, because
every body’s scalar charge equals its mass,
binding enP
ergy included (App. WS). Measured | F|/ max |F| on
three clumpy configurations: 3.19 × 10−10 , 1.08 × 10−9 ,
2.00 × 10−11 . By Rem. NL.5 a hard partition’s
step-independent
14% jump in W is a δ-function
...
force, so D would carry a δ ′ and its radiated dipole
power would be formally divergent at every merge

415
TABLE CXXXIII. Tests of the environmental-confinement axiom. Pulls are from the central values; the interval of Boufourou
(2026) is its 68% statistical interval.
Test

DFD with the axiom

Data

Result
−5

4

Sun Newtonian: 2 × 10 comets, 1 Gyr, CODE: centroid 3–4 × 10 ,
consistent
Galactic tide; median original
4.6% hyperbolic [213]
(postdiction)
1/a = 4.0 × 10−5 AU−1 , 77% in
[0, 0.75] × 10−4 , no hyperbolic originals
same
excluded
The printed readings 46–100% hyperbolic originals, or an
without the axiom
energy budget up to about 18 times the
allowance (quadrature: 17.6 against about
(accumulator,
1)
quadrature,
completion, window
variant, τs + 2τh )
τs + kτh , k ≥ 3
pass the comets (0–1.3% hyperbolic) but same
not adopted
change the host’s own depth, carry an
unfixed integer and cost the satellites
0.12–0.25 dex
Local wide binaries γ = 1.000, no orientation term of either Chae (2023) 1.43 ± 0.06; Chae et null camp; pulls
parity; a softer switch on the host’s state al. (2026) 1.60+0.17
+7.2, +4.3, +2.9,
−0.14 ; Yoon et al.
+0.12
that passes the comets allows at most
−0.3, +2.2, 0.0σ
(2025) 1.32−0.11 ; Banik et al.
γ − 1 ≃ 0.01–0.03 for 1–1.5 M⊙ pairs at (2024) α
+0.065
grav = −0.021−0.045
1–3 × 104 AU (computed)
(Newton 0, MOND 1); Boufourou
(2026) 1.045 [1.025, 1.068]; Saad
& Ting (2026) 1.00+0.24
−0.19
Planets
γ − 1 = 0 exactly
ephemerides
consistent
Q2
0 at every order; the Galactic tide is
DE440 (1.6 ± 1.8) × 10−27 ,
−0.9σ, −1.0σ
8.2 × 10−31 s−2
Cassini (3 ± 3) × 10−27 s−2
RAR, BTFR, a0 ,
unchanged: a host’s own field is never
SPARC
as printed
Milky Way disc RAR frozen, and an isolated galaxy has no host
mode
Milky Way satellites all boosted: dwarf spheroidals
satellite kinematics
as printed
nh = 5.6–28, LMC 4.2–5.3, Sagittarius
1.2–1.6
Draco
boosted (nh = 6.6–8.2); under-predicted McGaugh & Wolf (2010)
not a discriminator
by every reading
Cluster spirals
frozen only within about 25 kpc of a
Dale et al. (2001), Bösch et al. as printed
Coma-class centre; nh ≥ 2.0 beyond
(2013)
0.1 Mpc
The edge
a step at R ≃ 13–16 kpc
outer-halo globular clusters
outer boost
(R ≥ 16 kpc, 28 dispersion
contradicted: −5σ
profiles, Baumgardt catalogue
(error scaled for
v4): boost 0.33 ± 0.08 of the
χ2 /dof = 2.9), −9σ
predicted one; halo pairs not yet on 18 binary-robust
measured
clusters
Oort spike

crossing. The smooth scale-additive rule is the
only candidate partition for which the zero-dipole
theorem holds at all.

9.

The external-field effect: even anisotropy dies

Proposition NL.19 (EFE anisotropy). For the local
operator, µ reads the vector sum |gint + gext | and the
external-field effect is anisotropic, with closed form (symbolic residual 0)
Lext ≡

d ln µ
τ
= nBE (τ ),
d ln g gext
2

G∥ /G⊥ =

1
,
1 + Lext

where nBE = 1/(eτ − 1). For the scale-additive operator the external field is a separate scalar mode (a depth,
never a vector-sum argument), so the even (axisymmetric)
anisotropy vanishes, G∥ /G⊥ = 1 exactly for the axisaveraged ratio (Thm NL.10); the action completion allows
only an odd cos θ term proportional to the mass asymmetry. At the solar circle, where the Newtonian Galactic
field gN,e ≃ 1.19–1.33 a0 (observed 1.79–1.94 a0 ) gives
τext = 1.09–1.15, Lext = 0.27–0.28 and the local reading
predicts an anisotropy of about 21% against DFD’s exact
even-harmonic zero (computed); at the Sun Axiom NL.14
removes both the anisotropy and the odd term.
Falsifier NL.20 (Wide-binary sky-orientation modulation).

416
TABLE CXXXIV. Forward evaluations of the scale-additive equation. No fitted constants; every entry is a prediction, not
a fit. Bars are the published measurement uncertainties. The local-reduction Q2 entries are at the bound-matter value
a0 = 1.1966 × 10−10 m s−2 (Postulate ZS.11). The action completion of §NL 5 makes the Q2 row exact at first order: Q2 = 0
(Thm NL.8), and Axiom NL.14 makes it exact at every order. The disc-RAR entry is for the depth of record, FB-OUT, with its
completion; the accumulator reading gives 0.016–0.018 dex.
Test

Local reduction

Solar-system Q2

30.6–31.8e-27 0 (Sun Newtonian)
DE440
+16.1 to +16.8σ
−0.89σ
(1.6 ± 1.8)e-27
+9.2 to +9.6σ
−1.00σ
Cassini (3 ± 3)e-27

Bar

Milky Way disc RAR

untested

≤ 0.002 dex

0.057 dex

E, p, L, 320k steps

—

≤ 2.6 × 10−12
no secular drift

—

N -independence

exact

exact (theorem)

—

BTFR v 4 = GM a0

exact

exact

10−2

ν≥1

exact

exact

—

0 exactly

0 exactly

10−5

0

0 (and rescued)

—

PPN γ − 1
Dipole radiation

Let θ be the angle between a bound pair’s separation vector and the Galactic-centre direction. The completed
scale-additive equation predicts exactly zero even
(cos2 θ) dependence of the force boost γ on θ, at every separation and mass ratio (Thm NL.10; vobs /vN
inherits an even term ≲ 7 × 10−4 , zero for equal masses);
the only allowed modulation is odd, ∝ (m1 − m2 ) cos θ,
zero for equal-mass pairs. AQUAL/MOND predicts 6.88%
peak-to-peak at s = 104 AU and 46.65% at s ≈ 5×103 AU,
where gint = gext and the vector sum can cancel. Any
detection of an even θ-modulation above the measurement floor, or of any θ-modulation above that
floor in equal-mass pairs, falsifies the completed
scale-additive equation. Every pair of the Gaia catalogue lies inside the Milky Way’s frozen zone, where
Axiom NL.14 predicts no boost and no modulation of
either parity; there any detected modulation, even or
odd, or any boost falsifies the axiom. The test is a pure
geometry cut on the 1.1 × 106 pairs with > 99% bound
probability in the Gaia eDR3 catalogue of El-Badry, Rix
& Heintz [53] and requires no new observation. It is an
even-harmonic null for DFD and a signal for MOND, and
nulls are robust to the M/L, contamination and eccentricity systematics that dominate the amplitude tests.

10.

Scale-additive

Two further constructions

1. Directed enclosed mass with a smoothed
log-radial window: disc RAR 0.0133 dex, τ (R0 )
within +5.7% of the established value.
2. Square-root-outside-the-scale-integral
(“FBR∞
2
OUT”):
a
τ
=
|G
|
d
ln
r.
Band
complete0
r
0
R
ness Gr d ln r = g is
p a vector identity, so under non-reversal τ = gbar /a0 exactly: the disc

RAR displacement is zero as an identity (measured 1.03 × 10−4 dex disc, 1.83 × 10−4 dex Plummer — the quadrature error of an exact identity, falling 4× under grid doubling). Its exact
two-source composition (symbolic residual 0) is
a0 τ 2 = g1 (1−ϵ3 )+|g1 ϵ3 ê1 +g2 ê2 | with ϵ = dnear /dfar :
the crossover variable ϵ3 is derived, and for the Sun
in the Galaxy ϵ3 = 2.01 × 10−25 , so the functional
never forms the vector sum. For ϵ → 0 this depth
composes in quadrature, a0 τ 2 = g1 + g2 (as in
Thm NL.8), whereas thePaccumulator of item 3
composes additively, τ = i τi (Thm NL.1.3): two
printed readings of the scale-separated limit. In
weak host fields the data choose between them: the
rotation-curve shapes of Ursa Major members, the
outer gradients of 510 cluster spirals (Dale et al.
2001) and the stellar-mass Tully–Fisher relation at
z = 0.17 (Bösch et al. 2013) disfavour the additive composition at about 3–4σ and agree with the
quadrature form (computed on published values;
the host fields are partly estimates). Dwarf satellites test only the ceiling ν(τhost ) that both forms
share.
3. The accumulator of §NL 2.
All three evade the Cassini obstruction in the same
way: by never forming g⊙ + gext .
11.

No-go results that close the alternatives

Theorem NL.21 (Subdivision no-go for a literal per–
source
composition). Read the composition law τtot =
P
i τi literally,
p over a partition of a body into N pieces,
with τi =
gi /a0 per piece. Then τ is partitiondependent: for N equal co-located pieces τN /τ1 =

417
p
√
N 1/N = N exactly, and in the deep field gobs =
4
gbar /τ gives vflat
= GM a0 /N , i.e. vflat ∝ N −1/4 . A
11
galaxy of 10 stars therefore has its asymptotic rotation
speed wrong by a factor (1011 )1/4 = 562.
p
√ p
Proof. τN = N √g/(N a0 ) =
N g/a0 (symbolic).
More sharply, τ ∝ M is homogeneous of degree 12 in the
source while every linear functional of ρ is homogeneous of
degree 1; hence no kernel reproduces a literal per-source
τ , which is a functional of a partition of ρ rather than
of ρ. N -independence is restored exactly by Eq. (NL2),
because γ is linear in ρ (Thm NL.1.4).
Proposition NL.22 (MR-NG: no enclosed-mass law
can work). Let τ = F [M (x, r)] be built from the mass
enclosed within radius r of the field point, with the kernel
fixed by the point-mass normalisation. Put the field point
inside a sheet of surface density Σ and thickness h. For
h < r < R the ball of radius r contains at least πΣr2 , and
monotonicity gives
r
Z Rs
GπΣr2
πGΣ
R
τ ≥
d
ln
r
=
ln
≡ τamb .
2
a
r
a
h
0
0
h
This term is (i) strictly positive, (ii) independent of the
system’s total mass, (iii) logarithmically divergent as
h → 0. No window, smoothing or normalisation
removes it.
Status NL.23 (Size of the obstruction at the solar circle). With Σ = 0.1211 kg m−2 (58.0 M⊙ pc−2 ) and
ln(R0 /hz ) = ln(8.275/0.3) = 3.3172, at the bound-matter
value a0 = 1.1966 × 10−10 m s−2 (both depths scale as
−1/2
a0 , so the ratio does not depend
on it), τamb = 1.5281
p
against the established τ = gext /a0 = 1.0587: the ambient term alone is 1.4433× the entire correct answer. The full enclosed-mass law gives τ (R0 ) = 2.81 and
displaces the Milky Way RAR by 0.184 dex against the
0.057 dex observed floor. Replacing M (x, r) by r2 |genc |/G
makes the scale integral logarithmically divergent at large
r. The whole enclosed-mass class is closed.
Proposition NL.24 (FB-NG: no field-based law with the
root
p inside the scale integral can work). Let −dτ /d ln r =
C |Gr |/a0 , where Gr is any self-similar band of the
true Newtonian field and C is fixed by the point-mass
normalisation. If the shell-dipole density keeps one direction in r — proved for every axisymmetric density
decreasing in cylindrical radius, evaluated at a mid-plane
point, and
p for every decreasing spherical model — then
τ (x) ≥ gbar (x)/a0 for every admissible kernel, with
equality iff all the mass lies on a single sphere about x.
Proof. Two inputs, both forced: the exponent 12 (by the
√
point-mass normalisation)
pPand the concavity of ·. Jensen
P √
then gives i ai ≥
a , strictly for more than one
√
√ √i i2
nonzero band, since ( a+ b) −(a+b) = 2 ab > 0. Summing square roots across scales always exceeds the square
root of the sum. The bound is kernel-, mass-, geometry-

and a0 -independent, and it contains Prop. NL.22 as the
special case in which the bands discard direction.
Status NL.25. Milky Way disc displacement 0.1393 dex,
Plummer sphere 0.0577 dex, against the 0.057 dex floor;
0/2000 violations of the inequality against a 1.2 × 10−4
quadrature floor. The repair is one move: put the
root outside thep
scale integral (§NL 10, FB-OUT),
which restores τ = gbar /a0 as an identity.
Theorem NL.26 (ℏ no-go for a DFD-native black-hole
entropy). DFD’s per-mode entropy is exactly ℏ-free. With
the established scale-locking rule λ̄ = r and bath temperature kT = ℏc/rM ,
τ=

Emode
ℏc/r
rM
=
=
,
kT
ℏc/rM
r

∂τ
≡ 0,
∂ℏ

and the same holds for τ = rs /r, the leading-order exterior ψ (symbolic residual 0). But SBH /k = 4πGM 2 /(ℏc)
is exactly first order in 1/ℏ. Therefore a total entropy S = N s(τ ) is ℏ-free unless the mode count N
carries ℏ: any construction that returns A/(4ℓ2P )
must introduce the Planck length through a cell
measure, since no measure built from DFD’s own
scale-locked lengths {r, rM , rs , c, G, a0 } contains it.
Corollary NL.27 (Bekenstein–Hawking restated exactly). With N = M c2 /(ℏc/rs ) = 2GM 2 /(ℏc) =
A/(8πℓ2P ) quanta of reduced wavelength rs whose energies sum to M c2 , one has SBH /N = 2π exactly (symbolic). Bekenstein–Hawking is exactly 2πkB of entropy
per horizon-wavelength quantum. The same 2π is the
Rindler periodicity: DFD’s scale-locking λ̄ = ρ with Unruh kTloc = ℏc/(2πρ) gives τ = 2π exactly and universally
in the near-horizon zone of any horizon, hence a universal
boost ν = 1/(1 − e−2π ) = 1.0018709366. A finite Hilbert
space of dimension kmax + 1 = 61 cannot supply 2π nats
per mode: that requires dim H = e2π = 535.49, short by a
factor 2π/ ln 61 = 1.5284 in the exponent.
Theorem NL.28 (No causal linear response can make
a standing well). The CMB third-peak height requires
the zero-frequency (standing, DC) component of Ψ(k, z∗ ).
Let R be any causal, linear, time-translation-invariant
response acting on the source, with arbitrary amplitude
c (0) = R̂(0)fˆ(0): a linear
and arbitrary memory. Then Rf
filter cannot create power at a frequency the source does
not carry. An oscillating (acoustic) source has fˆ(0) = 0 at
the acoustic frequency, so no such R — of any amplitude
and any memory — generates DC power.
Proposition NL.29 (DFD’s only nonlinearity is rectification-free for sign-symmetric drives). The single escape
from Thm NL.28 is rectification. DFD’s deep-field nonlinearity is odd in the source and therefore rectification-free by
symmetry for sign-symmetric (half-wave-antisymmetric)
drives: its rectified DC coefficient is 0 to 5.05 × 10−18 ,
and about
p a non-zero operating point g0 the rectified DC is
0.83463 a0 /g1 g0 — odd in g0 , with finite gain. Finite

418
gain × zero DC = zero: the rectifier is an amplifier, never a generator. That conclusion holds only under the sign-symmetric hypothesis above: an in-phase even
harmonic √
breaks it and rectifies at first order, coefficient
−Γ( 54 )/(2 π Γ( 74 )) = −0.27821 (the gain above is exactly −3× this integral). DFD’s acoustic source has evenharmonic fraction ε = O(δ) with δ ≲ 10−4 (the corpus’s
own recombination baryon contrast is ∼10−5 , App. AE):
the rectified DC is ≤ 0.278 ε (597) ≤ 0.0166 of the baryon
potential against νreq = 6.364327 (Status NL.31) — short
by ≥ 380× (≥ 3800× at the corpus’s own 10−5 ): the
no-go survives on amplitude.
Theorem NL.30 (ψ cannot be the standing well). DFD’s
ψ quantum is exactly massless, mψ = 0, protected by the
shift symmetry ψ → ψ + ϵ of the printed action, whose
scalar sector contains no potential term in ψ (the action displayed in Theorem AE.1). On the Friedmann
background no scalar mode propagates (App. WS), so ψ
cannot sustain a pressureless, non-oscillating potential
well at recombination. This forbids ψ from being the
standing well the third peak requires.
Status NL.31 (The gap is a shape gap, not an amplitude gap). The requirement is νreq = ωm /ωb = 6.364327,
i.e. τreq = 0.1709376. DFD’s boost on the acoustic perturbations is ν(k) = 129, 71, 45, 41, 113, 172, 231, 311, 597
at k = 0.010 . . . 0.20 Mpc−1 : 6–94× more amplitude
than required, and nonetheless useless, for three independent reasons. (a) It multiplies AC where the requirement is DC (Thm NL.28). (b) Its k-shape is too
flat by 42.5× (ν varies by ×8.37 over k = 0.0159 → 0.20
where the requirement varies by ×355.7). (c) It makes the
fluid Jeans-unstable at k ≤ 0.025 Mpc−1 (νcrit = 12.85
against ν = 71), destroying the peaks rather than fixing them. The background channel is worse: the cap is
1/4
νmax = 1/(1 − e−1/(2α ) ) = 1.2206074697, a shortfall
of 5.2141×; and by Thm AE.1 (ψ couples to ρ − ρ̄) the
true background τ vanishes identically, so ν ≡ 1 and the
shortfall is the whole 6.3643×.
Status NL.32 (The third-peak height needs a cold clusterer). The CMB third-peak height still requires
a cold clusterer. DFD does not avoid the charge; it
derives the field that carries it — χ-matter, App. AV, with
Ωχ h2 computed from representation theory at the locked
kmax = 60, given the flux-period identification, the stated
count 158, the asserted premise P4′ (Lemma AV.27) and
two stated conventions (App. FK). Every no-CDM channel tested is closed, and the closure is a theorem about
shape, not a deficit of amplitude. This is consistent with,
and strengthens, Thm J.1 and Rem. J.3.
Remark NL.33 (Sign changes in the response are the
acoustic phase). The response’s sign changes are not a
new physical scale. There are 23 of them, spaced at π/rs
— that is the acoustic phase sin(krs ) = 0 and nothing
more — their count varies by 62–124% with the analysis
window, and their number collapses to zero as soon as
ωc h2 exceeds 10−2 .

12.

What is not derived

Named, with nothing booked on them; each item states
where it stands.
NL-A: The vacuum detector w(q) belongs to the accumulator reading only. The depth of record, FBOUT, has none: its band is the
enR ∞field of the
2
−4
closed mass dipole,
a
τ
=
3G
|D
(r)|
r
dr
x
0
0
R
with Dx (r) = |z−x|<r ρ(z)(z − x) d3 z, which reproduces its two-source composition and response
kernel exactly; the constant 3 is fixed by dimensions and completeness, and no free function enters
(computed).
NL-B: The width w0 belongs to the accumulator reading only: it is not fixed by the accumulator’s own
normalisations, w0 = 1 is imported from the sharpwindow derivation of §NL 2, and the sensitivity over
[0.5, 2] is in §NL 7. In FB-OUT the memory rate
is 3, fixed by dimensions and completeness, and no
width is chosen.
NL-C: The Rnonlocal variational term N
=
−(a20 /2) dx′ (∂Q/∂τ )(x′ ) δτ (x′ )/δΦN (x)
vanishes in the action completion (Thm NL.7),
and the nonlocality acts through the reciprocal force.
Its size follows from
QM (t, τ )−QQU (t) = −2ν ′ (t) t3 (τ −t)2 +O((τ −t)3 ):
the completed energy equals the local one to second
order in τFB − t. For a spherical density that
decreases outward the band never turns at any
point, so τFB = t everywhere, the reciprocal force
vanishes and the completed law is the local law.
For exponential discs of thickness 0.05–0.2 Rd the
reciprocal force on a midplane star and the change
of the forward field off the plane are each below 3%
and cancel to ≤ 0.002 dex between 0.5 and 8 Rd
(computed). Between a bound subsystem and its
host the environmental-confinement axiom applies
(§NL 6).
NL-D: For point masses the depth has a bounded Lipschitz kink at exact equidistance, where the order in
which two masses enter the ball swaps (codimension
one, no δ-function, energy still exactly conserved at
fixed a0 ); for any continuous density it is absent. It
is a property of point masses, not an open item.
NL-E: The lensing corollary (Cor. NL.17) rests on DFD’s
optical dictionary, the two core postulates n = eψ
and Φ = −c2 ψ/2, and on nothing else. The scaleadditive law is a static law of bound, slow, weak-field
systems, the only place it acts; its time-dependent
form is not claimed, and with Axiom WS.1 read, as
in Proposition WS.6, as the absence of a temporal
kinetic term for ψ, the corrections it would carry
there are of relative order (v/c)2 ≲ 10−6 .

419
NL-F: The informative failure, a no-go rather than an
open item. A fixed-width window in ln s cannot
chain, and fails the disc RAR at 0.157 dex (windowshape spread only 5.7%: structural, not an artefact).
A disc’s field is generated over ∆ ln s ≈ 1.33 e-folds,
so a unit window counts it as ≈ 2 modes, while the
Sun and the Galaxy are 18.955 e-folds apart and
must count as two. FB-OUT does what such a window cannot: a disc stays one mode, because τFB = t
at the midplane, and scale-separated systems split,
because they compose in quadrature.
NL-G: The environmental-confinement axiom (Axiom NL.14) is stated, not derived: the Bose statistics
cannot supply it (Thm NL.13). Its threshold is a
step; a smoothing width, and the action form of the
law in the frozen regime, are not derived.
13.

Effect on established results, and the items still
to be settled

a. The four quantities the completion shifts. Four
quantities are shifted by the completion; each is settled or
handed to a named observation. (i) The map from a0 to
the fitted RAR knee g † : on FB-OUT the completed law
at a disc midplane equals the local law to ≤ 0.002 dex
(computed), so g † = a0 up to the disc-geometry and fitting effects that any law of this form shares. (ii) The
Freeman transition surface density: it is the local law’s,
since the completed midplane force equals the local one
to about 0.001 dex in the inner disc (computed); its scale
is ΣM = a0 /2πG = 137 M⊙ pc−2 . (iii) Dwarf-spheroidal
dispersions: for an isolated dwarf the completed law is
the local law, since a spherical density that decreases
outward has τFB = t at every p
point; on the forward law a
2 + τ 2 ), which tends
satellite’s internal boost is ν( τsub
host
to the ceiling ν(τhost ) where the host dominates, and the
reciprocal term of the completion adds to it. On published
values four tidally safe Milky Way dwarfs sit 1.1σ above
the forward law’s own prediction (computed), and on the
dispersions and M31-centric distances published in 2020–
2026 the seven M31 dwarfs that are tidally safe at those
distances sit about 2–3σ above it, and the four selected
as tidally safe on earlier distances (And I, XIX, XXI and
XXIII), of which only XIX and XXIII are tidally safe at
the new ones, 5.7σ (computed); with the reciprocal term,
which the completed law of weak hosts carries, the excesses
fall to about 0.9σ, 2σ and 4.4σ (estimate). The ceiling
counts only the µ-law branch of the floor law: a satellite’s
own χ can hold its stars above it unless the host pushes
them, relative to that χ, harder than the satellite’s own
gravity (App. AV). Deeper dispersions of external-fielddominated M31 satellites decide it: a combined excess
of 5σ over the completed law, with population-synthesis
mass-to-light ratios, in satellites that pass an equilibrium
check and whose stars the host pushes harder than their
own gravity, falsifies the ceiling. (iv) The wide-binary amplitude: under the environmental-confinement axiom the

Sun’s host mode is typically empty, so local wide binaries
are Newtonian, γ = 1 exactly, with no orientation term;
the completed law’s amplitude (Status NL.11) applies to
pairs in host fields below (ln 2)2 a0 , beyond R ≃ 13–16 kpc
in the Milky Way. A confirmed boost of the Chae type
in local binaries falsifies the axiom.
b. The place this equation is most exposed, stated
plainly. On Crater II — where McGaugh [216] predicted
−1
σEFE = 2.1+0.9
and Caldwell et al. [217] mea−0.6 km s
−1
sure 2.7 ± 0.3 km s
from 62 member stars — the
same-µ comparison on the accumulator reading gives
σnew /σold = 0.888, i.e. ≈ 1.86 km s−1 , with the observed
host field as the argument; with the baryonic host field of
Table XCIX it has not been recomputed. MOND’s prediction overlaps the measurement within its own asymmetric
bar; 1.86 km s−1 sits further below it. On Crater II
the accumulator reading is in worse shape than
MOND, not better. On the depth of record Crater II
is boosted, well below the threshold of Axiom NL.14, and
sits under the ceiling ν(τhost ) like every satellite; it is
tidally disturbed, so it does not decide that ceiling, which
item (iii) above hands to deeper M31 data.

420
TABLE CXXXV. Effect of the scale-additive completion on results established elsewhere in this monograph.
Item

Result

Shift

PPN γ − 1 = 0
unchanged (theorem)
0 exactly
Zero dipole radiation unchanged, rescued
0 exactly
Lensing = dynamics unchanged where the reciprocal term vanishes 0 there
BTFR v 4 = GM a0 unchanged
→ 0 asymptotically
ν≥1
unchanged
none
Disc RAR
safe
≤ 0.002 dex vs 0.057

Appendix BS: Single-Mode Bose Statistics and the
Gravity Law

partition function is
Z1 (τ ) =

Appendix N obtains the interpolation function from a
saturation-union composition law (Assumption N.5),




µ(ψ1 + ψ2 ) = 1 − 1 − µ(ψ1 ) 1 − µ(ψ2 ) ,
(BS1)
and Lemma N.6 shows that this alone forces µ = 1 − e−cψ .
This appendix identifies what Eq. (BS1) is. Writing µ =
1/ν with ν ≡ gobs /gbar = Mdyn /Mbar , the composition
law says that the quantity 1 − µ is multiplicative over
independent contributions. That is the defining property
of a probability of independent events, and µ itself is then a
normalised inverse partition function. The galactic sector
of DFD is thereby read as single-mode Bose–Einstein
statistics at zero chemical potential. The reading adds
one stated input, the ground-state principle (Rem. BS.4),
and it supplies a temperature, a heat capacity and an
entropy, together with two zero-parameter observational
relations that contain no a⋆ and no a0 .
Throughout, τ > 0 is the dimensionless mode exponent,
k is Boltzmann’s constant, all entropies are quoted as the
dimensionless S/k, and
1
y ≡ gbar /a0 ,
x ≡ gobs /a0 ,
nBE (τ ) ≡ τ
.
e −1
(BS2)
Where a number is printed, a0 = 1.20 × 10−10 m s−2 is
an explicitly chosen empirical SPARC
√ benchmark. The
bound-matter prediction is a0 = 2 α cH0loc = 1.1966 ×
10−10 m s−2 (Theorem N.14, Postulate ZS.11). Tables in
this appendix retain the benchmark so their published
environment examples remain reproducible. The dimensionless theorems and inequalities are independent of this
benchmark. Nothing in this appendix is fitted.

1.

The Partition Function

Definition BS.1 (The gravitational mode). Associate
with each field point a single harmonic mode of the ψ
quantum with dimensionless Boltzmann exponent τ , in
thermal equilibrium at zero chemical potential. The mode
is a statistical reading of the response: under Axiom WS.1
the local ψ has no canonical momentum of its own, so the
mode is not a propagating quantum of ψ. Its canonical

∞
X

e−nτ =

n=0

1
.
1 − e−τ

(BS3)

Theorem BS.2 (The boost is the partition function).
With µ ≡ 1/ν and the identification ν = Z1 ,
ν = Z1 =

1
,
1 − e−τ

1
= 1 − e−τ = P(n = 0),
Z1
ν = 1 + nBE

µ=

(BS4)

Equivalently: µ is the probability that the mode is
empty, and ν is spontaneous plus stimulated response.
Proof. Eq. (BS3) is a geometric series with ratio e−τ < 1.
The Gibbs weight of the ground state is p0 = e0 /Z1 =
1/Z1 , so P(n = 0) = 1/Z1 = µ. Finally 1 + nBE =
−1
1 + (eτ − 1)P
= eτ /(eτ − 1) = (1 − e−τ )−1 = Z1 , and
−1
−nτ
⟨n⟩ = Z1
= nBE . All four residuals are
nne
identically zero.
Remark BS.3 (Relation to Appendix N). Theorem BS.2
is Lemma N.6 with the multiplicative function g = 1 − µ
identified as e−τ = P(n ≥ 1). The saturation-union
law (BS1) is exactly the statement that the occupation
probabilities P(n ≥ 1) of independent modes multiply.
Beyond that law the reading adds one stated input, the
ground-state principle of Rem. BS.4. The two routes share
Assumption N.5 in full and differ only in what is then
substituted for ψ: Appendix N uses the level response
Assumption N.3, this appendix uses the deep-field limit
(Theorem BS.9).
Remark BS.4 (The one stated input: the ground-state
principle). The identification ν = Z1 is one stated input
of the theory, the ground-state principle: the Newtonian
fraction of the response at a field point is the probability
that the point’s single ψ mode is empty, µ = P(n=0).
Fermi occupation would cap ν at 2 (Cor. BS.8), where
a deep SPARC point, at 2.70 × 10−12 m s−2 , needs 7.18.
Classical occupation, ⟨n⟩ = 1/τ , leaves a strong-field tail
that fails the approach-rate condition of App. C 1, and a
chemical potential caps the boost (Thm BS.5). A mode
of energy ℏc/r at one temperature per system has τ =
rM /r, the SOURCE branch; the TOTAL branch is not a

421
single-temperature mode. The branch itself is chosen by
data (§BS 7); one temperature per system is not derived.
The principle cannot come from a quadratic action: a
harmonic mode driven linearly by a static source responds
as χ = 2/ε whatever its occupation, because stimulated
emission andR absorption cancel. Its energy form is exact:
τ
QQU (τ ) = 4 0 v 3 [1 + nBE (v)] dv, one quantum per mode
below the local depth plus the Planck spectrum cut off
there, whose full value 4Γ(4)ζ(4) = 4π 4 /15 is the constant
of the Newtonian limit (computed).
Theorem BS.5 (Vanishing chemical potential, and why
rotation curves are flat). Let the mode carry fugacity
z = eµchem /kT ≤ 1, so that Z1 (τ, z) = (1 − ze−τ )−1 . Then
1
lim+ ν =
,
(BS5)
1−z
τ →0
which is finite for every z < 1 and divergent only at z = 1.
Hence
µchem = 0

(BS6)

is what the observed flatness of rotation curves requires,
to the precision of the largest measured boosts: any z < 1
caps the boost at 1/(1 − z), so gobs → gbar /(1 − z) at large
radius and every rotation curve eventually turns over into
a rescaled Keplerian decline.
Proof. Geometric series with ratio ze−τ < 1; take τ → 0+
(sympy limit, 1/(1 − z)). The physical statement follows
because gobs = ν gbar and gbar ∝ r−2 outside the baryons:
a bounded ν gives vc2 = rgobs ∝ r−1 .
Remark BS.6 (A host acts as a fugacity). A subsystem in
a host of depth τh composes with it as a mode of fugacity
z = e−τh , so the host caps the subsystem’s boost at 1/(1−
z) = ν(τh ) (App. NL, Thm NL.13); no composition built
from these statistics removes the cap, and the freeze of the
Sun’s field inside the Galaxy is the axiom of environmental
confinement (Axiom NL.14).
Theorem BS.7 (The µ-law response is never sub-Newtonian). For every τ > 0, ν = (1 − e−τ )−1 > 1, with
inf τ ν = 1 approached only as τ → ∞. The µ-law response
therefore cannot produce gobs < gbar at any acceleration.
Proof. 0 < 1−e−τ < 1 for τ ∈ (0, ∞), so ν −1 = nBE > 0.
Structurally: the mode’s ladder is bounded below by the
ground state, so detailed balance offers no de-excited level
for the response to fall into. There is no “n = −1”.
Corollary BS.8 (Fermi statistics is excluded by inspection). A fermionic quantum would give Z1F = 1 + e−τ ,
hence
1 < νF ≤ 2,

1
2 ≤ µF < 1.

(BS7)

A deep SPARC data point, at gbar = 2.70 × 10−12 m s−2 ,
requires ν = 7.18 (Table CXXXVI); more than a hundred points lie deeper, and the sample spans ν ≈ 1–60
(computed). The gravitational quantum is a boson;
Pauli exclusion would cap the boost at a factor 2,
short by 3.6× at that point alone.

2.

The Interpolation Function

Theorem BS.9 (The interpolation function, up to one
binary branch). Assume (i) Theorem BS.2; (ii) τ is a
function of a single local acceleration invariant made di2
mensionless by a0 ; (iii) the deep-field limit gobs
→ gbar a0 .
Then, among powers of gbar /a0 or of gobs /a0 , τ is fixed
up to the choice of which of the two enters, and exactly
two laws exist:
√
− gbar /a0
,
µ
=
1
−
e
√
SOURCE: τ = y =⇒
1
√
ν(y) =
1 − e− y
(BS8)
TOTAL: τ = x =⇒

µ(x) = 1 − e−x ,
x = gobs /a0

(BS9)

Both reproduce the Newtonian limit ν → 1 and the deepfield limit exactly; no third such power law is admissible, and statistics does not choose between these
two.
Proof. By Theorem BS.2, ν = (1 − e−τ )−1 , so
limτ →0 τ ν = 1 (i.e. ν ∼ 1/τ , µ ∼ τ ) and ν → 1 as
τ → ∞. SOURCE: with τ = τ (y), gp
obs = νgbar → gbar /τ ;
√
imposing gobs = gbar a0 gives τ = gbar /a0 . Allowing a
general power τ = A y p and solving the two resulting conditions returns the unique solution p = 12 , A = 1 — the
unit coefficient being the definition of a0 . TOTAL: with
τ = τ (x), gbar = µ gobs → τ gobs , and the same limit forces
τ = gobs /a0 . Direct evaluation confirms lim(ν 2 y) = 1 on
2
both branches, i.e. both satisfy gobs
→ gbar a0 exactly.
Remark BS.10 (What the two branches are). Eq. (BS8)
is the McGaugh–Lelli–Schombert radial-acceleration function [10], catalogued at Table LVIII as “RAR / Empirical”.
Eq. (BS9) is Bekenstein’s exponential form, catalogued
in the same table as “Exponential”. Both entries of
that catalog are outputs of one theorem, not independent phenomenology. Statistics does not select
between them. The selection is empirical (§BS 7), and
the deep-field observables — a0 , the BTFR and g † — are
structurally blind to the difference, which is the statistical
restatement of the deep-field blindness of the transition
region.
Remark BS.11 (Where this sits beside µ(s) = s/(1 + s)).
Theorem N.8 obtains µ(s) = s/(1 + s) by combining Eq. (BS1) with the level-response Assumption N.3,
keff (s) = k0 (1 + s), which is equivalent to τ = log(1 + s).
The composition step is common to both routes; the entire difference lies in what is assumed about τ , and §BS 3
reports what the level ratio implied by Assumption N.3
numerically is. The two asymptotic limits
are common to
√
all three functions (s/(1 + s), 1 − e− y , 1 − e−x ) and are
theorem-grade in every case; the three differ only in the
transition region, where they agree pairwise
to ≲ 0.09 dex
√
(≲ 0.02 dex for s/(1 + s) against 1 − e− y ) and where no

422
√

present galactic dataset separates s/(1 + s) from 1 − e− y
(§BS 7).

3.

The Magnitude of τ

Assumption N.3 reads τ as the logarithm of a Chern–
Simons level ratio, keff /k0 = eτ . The magnitudes are
recorded here without prejudice to that assumption’s
other uses.
Proposition BS.12 (The implied level ratio). With the
explicitly empirical benchmark a0 = 1.20 × 10−10 m s−2 ,
the level ratio keff /k0 = eτ demanded by Assumption N.3
is
environment

g/a0

eτ ,
τ = log(1 + s)

eτ√
,
τ = s

Saturn’s orbit
(a =
9.5826 AU)
Earth’s surface

5.3816 × 105

5.38 × 105

10318.60

8.1722 × 1010

8.17 × 1010

10124152.09

against DFD’s own Toeplitz truncation kmax = 60 (Theorem AB.1).
Even on the corpus’s own reading the ratio at Earth’s
surface exceeds kmax by nine orders of magnitude; on
the Bose reading it exceeds it by more than 10124150 .
As a topological level, τ is not a small integer in
any environment. As a Boltzmann exponent it is
unremarkable — e−733 is simply a very cold mode. The
identification τ = ε/kT is therefore not a reinterpretation
of convenience; it is the reading in which the numbers are
ordinary.

4.

Thermodynamics of the Mode

Theorem BS.13 (Entropy and heat capacity). For the
mode of Definition BS.1,
S
= ln Z1 + τ nBE = ln ν + τ nBE ,
(BS10)
k
C
dnBE
τ2
= −τ 2
=
= τ 2 ν(ν − 1),
k
dτ
4 sinh2 (τ /2)
(BS11)
with C/k → 1 as τ → 0 and C/k → 0 as τ → ∞.
P
Proof. Eq. (BS10) is the Gibbs entropy − n pn ln pn
−nτ
with pn = e
/Z1 ; the difference against the closed
form is identically zero. Eq. (BS11) is the Einstein

−1
heat capacity; the third equality uses 4 sinh2 (τ /2)
=
nBE (1 + nBE ) (residual 0) together with ν = 1 + nBE . The
limits are 1 and 0 by direct evaluation.
Corollary BS.14 (Newtonian gravity is a frozen quantum degree of freedom). C/k → 1 as τ → 0 is the classical equipartition (Dulong–Petit) value for one mode, and
C/k → 0 as τ → ∞ is quantum freeze-out. Hence

deep-MOND = the classical limit; Newtonian =
the quantum-frozen limit,
with the numerical values of Table CXXXVI. Solarsystem safety is therefore not a tuning and not a
screening mechanism. There is no occupied state to
boost: the mode entropy at Saturn’s orbit is 1.9 × 10−316
of one nat. No parameter was chosen to make this true,
and no fifth-force suppression is invoked. This holds at
every planet (τ ≃ 234 at Neptune’s orbit and 734 at Saturn’s). Beyond about 103 AU the Sun’s own mode is not
frozen by its own depth (τ ≃ 0.7 at 104 AU), and a host
acts as a fugacity that caps, but cannot remove, the boost
(Rem. BS.6); there the Sun’s field is Newtonian by the
environmental-confinement axiom (Axiom NL.14), a rule
about the Sun’s surroundings rather than a property of its
mode.
Theorem BS.15 (The temperature is the Hawking–Unruh geometric mean). Let TH (M ) = ℏc3 /(8πGM k) be
the Hawking temperature and TU (a0 ) = ℏa0 /(2πck) the
Unruh temperature at the DFD
pacceleration scale. Then,
with the MOND radius rM ≡ GM/a0 ,
TDFD = 4π

p

TH (M ) TU (a0 ) =

ℏc
k rM

(BS12)

with residual
identically zero, and the prefactor is derived:
√
4π = 8π · 2π, the geometric mean of Hawking’s 8π and
Unruh’s 2π. Consequently
ℏ
d ln TDFD
1
rM
tcoh ≡
,
= − , (BS13)
=
kTDFD
c
d ln M
2
exactly half the Hawking exponent.
2 2
2
2
Proof. TH TU =
p ℏ c a0 /(16π GM k ); its square root
times 4π is ℏc a0 /GM
√/k = ℏc/(krM ). The prefactor
√
identity is 8π · 2π = 16π 2 = 4π (residual 0). Evaluated at 60 digits: for the Sun, rM = 1.0531 × 1015 m,
TDFD = 2.1744 × 10−18 K, tcoh = 3.513 × 106 s; for a
6 × 1010 M⊙ disc, rM = 8.360 kpc, TDFD = 8.8769 ×
10−24 K, tcoh = 8.605 × 1011 s, at the bound-matter
a0 = 1.1966 × 10−10 m s−2 ; both equalities in Eqs. (BS12)
and (BS13) hold to relative 10−61 , i.e. to working precision.

Remark BS.16. tcoh = rM /c says the mode’s coherence
time is the light-crossing time of the MOND radius. It
contains no free parameter. In the present reading it
is an identity of the ensemble mean, not a dynamical
relaxation time of the response: all thermal quantities
in this appendix (temperature, entropy, heat capacity,
coherence time) are identities of the ensemble mean, no
fluctuation content is claimed, and no mode count per
resolution element is defined.
Theorem BS.17 (Equation of state, conditional on masslessness). If the ψ quantum is exactly massless, its dispersion is ε = ℏc|k|, its density of states in d spatial

423
p
TABLE CXXXVI. The gravitational mode across seven decades of τ , SOURCE branch (τ = gbar /a0 , empirical benchmark
a0 = 1.20 × 10−10 m s−2 ). Saturn’s and Earth’s rows are evaluated in log10 ; the omitted corrections are O(e−2τ ). Saturn’s row
uses a = 9.5826 AU; at a = 9.5549 AU one gets τ = 735.72 instead.
gbar (m s−2 )

environment

τ

ν

S/k

C/k

−12

a deep SPARC point 2.70 × 10
0.15000
7.1792
2.8981
0.99813
gbar = a0
1.20 × 10−10
1.00000
1.58198
1.04065
0.92067
Saturn’s orbit
6.4579 × 10−5
733.595
1 + 10−318.60
1.86 × 10−316
1.36 × 10−313
Earth’s surface
9.80665
2.85871 × 105 1 + 10−124152.09 2.34 × 10−124147 6.68 × 10−124142

dimensions is g(ε) ∝ εd−1 , and
d=3

P V = d1 U −−−→ P V = 13 U

(BS14)

exactly — the photon-gas equation of state — with the
Maxwell residual (∂S/∂V )T − (∂P/∂T )V ≡ 0.


R
Proof. F = kT g(ε) ln 1 − e−ε/kT dε; integrating by
parts with g ∝ εd−1 gives F = −U/d, and P =
−(∂F/∂V )T = −F/V since F ∝ V . Verified symbolically for d = 1, 2, 3, 4: P V − U/d = 0, F + U/d = 0 and
the Maxwell residual = 0 in every case.
Remark BS.18. The exponent is a falsifiable signature
of the dispersion relation: a massive quantum (ε ∝ p2 )
would give P V = 23 U . The antecedent mψ = 0 is not
proved here, and under Axiom WS.1 the local ψ has no
propagating quanta; compare the masslessness statement
(Theorem NL.30, App. NL), and Theorem BS.17 is graded
accordingly in Table CXXXVIII.

5.

Z rH
0

Example BS.21 (The Milky Way). With Mbar = 6 ×
1010 M⊙ one gets rM = 8.360 kpc; with the bound-matter
loc
−1
−1
clock
√ H0 = 72.09 km s Mpc (Postulate ZS.11; a0 =
2 α cH0 rescales rM accordingly), rH = 4158.6 Mpc, so
X = 13.1172 and
Stot /k = 99.88.



S(rM /r)
X2
d ln r =
+ X + K + O e−2X
k
2

(BS15)
with
2

Remark BS.20 (The domain matters). The lower limit is
r = 0, not r = rM . The frozen inner region contributes
0.70812 of the constant 0.72869 — 97.18%. Restricting
the integral to r ≥ rM replaces the constant by 0.0205774
and the closed form of Eq. (BS15) is then wrong.

The Total Entropy of a Galaxy

Theorem BS.19 (Closed form for the integrated mode
entropy). In the deep field τ (r) = rM /r. Counting one
mode per logarithmic radial shell out to the Hubble radius
rH = c/H0 , and writing X ≡ ln(rH /rM ),
Stot
=
k

whose product has no simple pole; multiplying by (1 + s),
1
1
M[S/k](s) = 2 + + K + O(s).
(BS18)
s
s
R
∞
With τH = e−X , Stot /k = τH (S/k) dτ /τ is the residue
at s = 0 of M[S/k](s) esX /s, i.e. the coefficient of s−1
in (s−3 + s−2 + Ks−1 )(1 + sX + s2 X 2 /2 + · · · ), which is
X 2 /2 + X + K. Against direct quadrature the neglected
remainder is 9.5×10−7 at X = 5, 8.4×10−14 at X = 13.12
and 4.0 × 10−24 at X = 25.

2

π
γ
−
− γ1 = 0.72869 39170 03930 6059 . . . ,
12
2
(BS16)
γ the Euler–Mascheroni constant and γ1 the first Stieltjes
constant.
K =

Proof. RThe Mellin
transform of the mode entropy is,

∞
using 0 τ s−1 − ln(1 − e−τ ) dτ = Γ(s)ζ(s + 1) and

R ∞ s−1 
τ
τ /(eτ − 1) dτ = Γ(s + 1)ζ(s + 1),
0
M[S/k](s) = Γ(s) ζ(1 + s) (1 + s),

(BS17)

confirmed against direct quadrature to relative 10−31 at
s = 0.5, 1.3, 2.7. Near s = 0, Γ(s) = s−1 − γ + (γ 2 /2 +
π 2 /12)s + O(s2 ) and ζ(1 + s) = s−1 + γ − γ1 s + O(s2 ),

(BS19)

A galaxy’s gravitational field carries of order one hundred
nats of entropy at one mode per e-fold of radius. At X ≃
13.12 the closed form’s neglected remainder is below 10−13 ,
so the value quoted is exact at the displayed precision.
6.

The Observables

The statistical reading is testable because τ appears in
two independent places: an amplitude ν (a mass budget)
and a shape u (a log-slope of the rotation curve). The
observable itself, its measurement on SPARC, its
systematics theorem and its falsifier are already
booked in Appendix AT 7; they are not restated here.
What this subsection supplies is the statistical provenance
of Eq. (AT16), together with two identities that follow
from the thermodynamics and have no counterpart there.
Theorem BS.22 (Statistical provenance of the (ν, u)
relation). The shape variable of Eq. (AT13), u ≡
−2 d ln ν/d ln gbar , evaluated on Definition BS.1 equals
the thermal energy of the mode in units of kT :
d ln ν
⟨E⟩
u = −
= τ nBE =
,
(BS20)
d ln τ
kT

424


and eliminating τ through τ = ln ν/(ν − 1) returns


ν
(BS21)
u = (ν − 1) ln
ν−1
— which is exactly Eq. (AT16) of Appendix AT 7. The
limits u → 0 as ν → 1+ and u → 1 as ν → ∞ are, in this
reading, quantum freeze-out and classical equipartition
respectively.
√
Proof. On the SOURCE branch τ = y, so d ln τ =
1
2 d ln y and −2 d ln ν/d ln y = −d ln ν/d ln τ ; with ν =
(1 − e−τ )−1 this is τ nBE , and ⟨E⟩/kT = τ ⟨n⟩ = τ nBE by
Theorem BS.2. Inversion of ν(τ ) and substitution give
Eq. (BS21); all residuals identically zero.
Remark BS.23 (What Theorem BS.22 closes and what
it does not). Eq. (AT16) uses µ = 1 − e−τ while Theorem N.8 gives µ = x/(1 + x). Theorem BS.9 supplies the
derivation of µ = 1 − e−τ from the same composition law
both routes already share. Which of the two is the law is
settled by the Solar System, not by statistics (Cor. BS.14;
App. C, strong-field approach rate; x/(1 + x) is the weakresponse form), and the two forms are not separated by
SPARC — SPARC provably cannot (Appendix AT 7).
Theorem BS.24 (The curvature relation). The mode
heat capacity is likewise τ -free in terms of ν:


C
ν
= ν(ν − 1) ln2
(BS22)
k
ν−1
and it is directly measurable, since
d2 ln gobs
C
= u + 4 w′ ,
w′ ≡
:
(BS23)
k
d(ln gbar )2
the mass discrepancy read against the curvature of the
radial acceleration relation. Both identities have residual
0.
Theorem BS.25 (An observable gravitational entropy).
Combining Eq. (BS10) with Eq. (BS20),


Mdyn
S
= ln
+u
(BS24)
k
Mbar
with u measured by the estimator u = 2 − 4(s − 12 )/β
of Appendix AT 7. At the SPARC sample-mean u =
0.7818 ± 0.0321, Eq. (BS21) returns ν = 2.650579 and
S/k = 1.7566 per mode. Every quantity on the right is
measured; nothing is fitted.
Corollary BS.26 (The κ2 /Ω2 form holds only for a
point mass). It is tempting to use the epicyclic identity
of Appendix AT 7, κ2 /Ω2 = 3 + β(1 − u/2), to write
S/k = ln(Mdyn /Mbar ) + (κ2 /Ω2 − 1). That substitution
is exact for a point mass and for nothing else: in general
 2


u
κ
−
1
−
u
=
(β
+
2)
1
−
,
(BS25)
Ω2
2
which vanishes only at β = −2 (residual 0). For a disc,
β ranges well away from −2 and the error is first order

in (β + 2). Eq. (BS24) with the measured u is the
general statement; the κ2 /Ω2 form holds only for
a point mass.
Remark BS.27 (Systematics). That distance and inclination errors move a galaxy exactly horizontally in the
(ln ν, u) plane, by (−1, 0) and (+2, 0) respectively, and
that the entire derivative tower is invariant, is Theorem AT.18 of Appendix AT 7; it holds symbolically to
fifth order (residuals 0) and is not restated. Its consequence for the present appendix is that the two dominant
SPARC systematics slide a galaxy along the horizontal,
whereas a failure of Eq. (BS21) is vertical.
7.

Confrontation, Including the Misses

Each law’s u(ν) below is derived from that law’s own
µ by the same

 operation (Eq. (AT13)); with L(ν) ≡
ln ν/(ν − 1) the closed forms are uSOURCE = (ν − 1)L,
uTOTAL = 2(ν − 1)L/[1 + (ν − 1)L], usimple = 2(ν −
1)/(2ν − 1) and ustandard = 2(ν 2 − 1)/(2ν 2 − 1), each
with residual 0 against direct differentiation of its µ. The
last two reproduce the n = 1 and n = 2 members of
Proposition AT.20 exactly, which is an independent check
of both derivations; the TOTAL row is new.
a. What the data say, and what they cannot. The
SPARC confrontation of Eq. (BS21) is Appendix AT 7
and is not repeated: χ2 /8 = 1.08, p = 0.37 on 2316 points
from 113 galaxies; n ≥ 2 excluded at 2∆ ln L ≥ +14.07;
and the limit, already recorded there, that these
data do not separate the SOURCE form from
µ = x/(1 + x) (+0.43) and provably cannot (ceiling
+3.99 for any a0 -free shape statistic). Three additions
belong here.
• Where the separating power lives. The gap
|uSOURCE − usimple | peaks at ν = 1.13955 with value
0.074850 and has decayed to 0.003685 by ν = 5. The
discriminating power sits in the Newton-adjacent
inner galaxy, the opposite of where deep-field tests
look — which is why pushing SPARC to ν → 1.1 is
worth more than adding deep-field points.
• The curvature channel fails as a discriminator.
DFD passes its own (ν, C) test of Eq. (BS22) at χ2 /8 =
1.36, p = 0.21, but scores 2∆ ln L = −0.68 against µ =
x/(1 + x) — worse than the slope channel — because
σC /σu ≥ 1.90 is an irreducible floor set by SPARC’s
3.63-dex dynamic range. Recorded as a negative result.
• Branch selection: SOURCE, on a statistic that
is not clean. The whole-sample β 2 -weighted mean u
leans TOTAL at ∆χ2 = −2.19; a post-hoc binning in
y with full bootstrap covariance gives ∆χ2 (TOTAL −
SOURCE) = +32.45 (0/60 mock misidentifications),
selecting SOURCE. Both numbers are stated because both are real, and on the selecting statistic
SOURCE’s own χ2 = 22.82 for four bins is not itself a
good fit. The branch of record is SOURCE, Eq. (BS8):

425
TABLE CXXXVII. The shape observable u as a function of the amplitude ν, for the two DFD branches and the two standard
MOND interpolation functions. No a0 , no mass-to-light ratio and no distance enters any column.
law

u(ν)

ν = 1.2

1.6

2.5

5.0

(ν − 1)L 0.358352 0.588498 0.766238 0.892574
2(ν − 1)L
DFD TOTAL
0.527627 0.740949 0.867650 0.943238
1 + (ν − 1)L
2(ν − 1)
MOND simple
0.285714 0.545455 0.750000 0.888889
2ν − 1
2(ν 2 − 1)
MOND standard
0.468085 0.757282 0.913043 0.979592
2ν 2 − 1
DFD SOURCE

the binned statistic selects it, and the corpus’s field
equations use it as the exact law, with µ = x/(1 + x) its
weak-response approximant (§VII); the SPARC data
do not separate it from that approximant (above).
b. The ν ≥ 1 falsifier is a corollary of Theorem BS.7.
Falsifier AT.22 — a measured Υ[3.6] > 0.58 M⊙ /L⊙ refutes DFD from SPARC alone — rests on “the µ-law response is never sub-Newtonian”. Theorem BS.7 is where
that premise is proved, from detailed balance and the
absence of a de-excited level, rather than read off the interpolation function. The falsifier is branch-independent
for exactly that reason.
c. Provenance of the empirical numbers. Every
closed form in this appendix was re-derived symbolically and every printed constant recomputed at 60 digits. The SPARC-derived numbers quoted above (χ2 /8,
2∆ ln L, ∆χ2 , u = 0.7818 ± 0.0321) are carried from
Appendix AT 7 and from the dedicated branch and curvature analyses; they were not re-run here. The y-binned
branch statistic was declared post-hoc in its source analysis, as was the Υ[3.6] inversion of Remark AT.21; the
χ2 /8 = 1.08 goodness-of-fit and the n ≥ 2 exclusions were
pre-registered.

8.

Grades

426

TABLE CXXXVIII. Tier and status of every result in this appendix, in the grading scheme of Appendix AG. Negative results
are listed with the positive ones.
result

where

tier

status

ν = 1 + nBE given ν = Z1
Thm BS.2 T
validated
ground-state principle µ = P(n=0)
Rem. BS.4 stated input adopted
µchem = 0 from flat curves
Thm BS.5 T
validated
a host caps an embedded boost at ν(τh )
Thm NL.13 T
motivates Axiom NL.14
ν > 1; no sub-Newtonian µ-law response
Thm BS.7 T
validated
Fermi excluded (1 < νF ≤ 2)
Cor. BS.8 T
validated
µ = 1 − e−τ , both branches
Thm BS.9 T
validated
which τ : SOURCE
§BS 7
E
adopted
S/k, C/k, Dulong–Petit reading
Thm BS.13 T
validated
solar-system safety at the planets is structural Cor. BS.14 T
validated
√
TDFD = 4π TH TU
Thm BS.15 T
open
P V = U/3 (given mψ = 0)
Thm BS.17 D
open
Stot /k = X 2 /2 + X + K
Thm BS.19 T
open
u = τ nBE = ⟨E⟩/kT
Thm BS.22 T
validated
C/k = ν(ν − 1) ln2 [ν/(ν − 1)]
Thm BS.24 T
validated
S/k = ln(Mdyn /Mbar ) + u
Thm BS.25 T
validated
κ2 /Ω2 form only at β = −2
Cor. BS.26 T
—
separation from µ = x/(1 + x)
§BS 7
—
not achieved, +0.43
curvature as a discriminator
§BS 7
—
failed, −0.68

427
ACKNOWLEDGMENTS

We thank the atomic clock groups at JILA and PTB for
valuable discussions regarding clock comparison methodologies and data interpretation. We also acknowledge the
SPARC collaboration for making their galaxy rotation
curve database publicly available.

[1] G. Alcock, Constitutive derivation of tensor gravitational
radiation from CP2 ×S3 spectral geometry in Density
Field Dynamics, Zenodo preprint (2026).
[2] Particle Data Group, S. Navas, et al., Electroweak model
and constraints on new physics (2024), review in Review
of Particle Physics, Phys. Rev. D 110, 030001 (2024).
[3] G. Alcock, Density Field Dynamics and the c-field: A
three-dimensional, time-emergent dynamics for gravity
and cosmology (2025), preprint, v1.
[4] G. Alcock, Density Field Dynamics: A complete unified
theory (2025), v1.0.
[5] C. M. Will, The confrontation between general relativity and experiment, Living Reviews in Relativity 17, 4
(2014).
[6] C. M. Will, Theory and experiment in gravitational
physics, Cambridge University Press (2018).
[7] G. Bertone and T. M. P. Tait, A new era in the search
for dark matter, Nature 562, 51 (2018).
[8] P. J. E. Peebles and B. Ratra, The cosmological constant
and dark energy, Reviews of Modern Physics 75, 559
(2003).
[9] V. C. Rubin, W. K. Ford, and N. Thonnard, Rotational
properties of 21 Sc galaxies with a large range of luminosities and radii, Astrophysical Journal 238, 471
(1980).
[10] S. S. McGaugh, F. Lelli, and J. M. Schombert, Radial
acceleration relation in rotationally supported galaxies,
Physical Review Letters 117, 201101 (2016).
[11] F. Zwicky, Die Rotverschiebung von extragalaktischen
Nebeln, Helvetica Physica Acta 6, 110 (1933).
[12] A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological
constant, Astronomical Journal 116, 1009 (1998).
[13] S. Perlmutter et al., Measurements of Ω and Λ from
42 high-redshift supernovae, Astrophysical Journal 517,
565 (1999).
[14] M. Milgrom, A modification of the Newtonian dynamics
as a possible alternative to the hidden mass hypothesis,
Astrophysical Journal 270, 365 (1983).
[15] B. Famaey and S. S. McGaugh, Modified Newtonian
Dynamics (MOND): Observational phenomenology and
relativistic extensions, Living Reviews in Relativity 15,
10 (2012).
[16] F. Lelli, S. S. McGaugh, J. M. Schombert, and M. S.
Pawlowski, One law to rule them all: The radial acceleration relation of galaxies, Astrophysical Journal 836,
152 (2017).
[17] B. P. Abbott et al., Gravitational waves and gammarays from a binary neutron star merger: GW170817 and
GRB 170817A, Astrophysical Journal Letters 848, L13
(2017).

[18] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis,
Modified gravity and cosmology, Physics Reports 513, 1
(2012).
[19] A. Joyce, B. Jain, J. Khoury, and M. Trodden, Beyond
the cosmological standard model, Physics Reports 568,
1 (2015).
[20] C. Brans and R. H. Dicke, Mach’s principle and a relativistic theory of gravitation, Physical Review 124, 925
(1961).
[21] J. D. Bekenstein, Relativistic gravitation theory for the
modified Newtonian dynamics paradigm, Physical Review D 70, 083509 (2004).
[22] T. P. Sotiriou and V. Faraoni, f (R) theories of gravity,
Reviews of Modern Physics 82, 451 (2010).
[23] A. Einstein, Über den Einfluß der Schwerkraft auf die
Ausbreitung des Lichtes, Annalen der Physik 340, 898
(1911).
[24] A. Einstein, Lichtgeschwindigkeit und Statik des Gravitationsfeldes, Annalen der Physik 343, 355 (1912).
[25] W. Gordon, Zur Lichtfortpflanzung nach der Relativitätstheorie, Annalen der Physik 377, 421 (1923).
[26] V. Perlick, Ray Optics, Fermat’s Principle, and Applications to General Relativity (Springer, 2000).
[27] J. Bekenstein and M. Milgrom, Does the missing mass
problem signal the breakdown of Newtonian gravity?,
Astrophysical Journal 286, 7 (1984).
[28] L. Blanchet and S. Marsat, Modified gravity approach
based on a preferred time foliation, Physical Review D
84, 044056 (2011), arXiv:1107.5264.
[29] A. Hees, W. M. Folkner, R. A. Jacobson, and R. S. Park,
Constraints on modified Newtonian dynamics theories
from radio tracking data of the Cassini spacecraft, Physical Review D 89, 102002 (2014), nine years of Cassini
range and Doppler tracking of Saturn give the SolarSystem quadrupole Q2 = (3 ± 3) × 10−27 s−2 on the
DE430 ephemeris., arXiv:1402.6950.
[30] R. S. Park, A. Hees, B. Famaey, H. Desmond, and A. Durakovic, Improved constraints on modified Newtonian
gravity from Cassini radio tracking data, Physical Review
D 10.1103/r7n8-kw38 (2026), Q2 = (1.6±1.8)×10−27 s−2
(1σ) from the DE440 dataset, a 40 per cent improvement
on Hees et al. (2014)., arXiv:2602.17884.
[31] G. Alcock, The physical origin of the refractive field
in Density Field Dynamics: Gravity as electromagnetic
vacuum loading, Zenodo preprint (2026).
[32] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer,
2001).
[33] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and
Quasilinear Elliptic Equations (Academic Press, 1968).
[34] E. Babichev, V. Mukhanov, and A. Vikman, k-Essence,
superluminal propagation, causality and emergent geom-

428
etry, JHEP 02, 101, arXiv:0708.0561 [hep-th].
[35] B. Bertotti, L. Iess, and P. Tortora, A test of general
relativity using radio links with the Cassini spacecraft,
Nature 425, 374 (2003).
[36] J. G. Williams, S. G. Turyshev, and D. H. Boggs,
Progress in lunar laser ranging tests of relativistic gravity,
Physical Review Letters 93, 261101 (2004).
[37] L. Shao and N. Wex, New tests of local Lorentz invariance of gravity with small-eccentricity binary pulsars,
Classical and Quantum Gravity 29, 215018 (2012).
[38] S. S. Shapiro, J. L. Davis, D. E. Lebach, and J. S. Gregory, Measurement of the solar gravitational deflection
of radio waves using geodetic very-long-baseline interferometry data, 1979–1999, Physical Review Letters 92,
121101 (2004).
[39] I. I. Shapiro, Fourth test of general relativity, Physical
Review Letters 13, 789 (1964).
[40] R. F. C. Vessot et al., Test of relativistic gravitation
with a space-borne hydrogen maser, Physical Review
Letters 45, 2081 (1980).
[41] J. Lense and H. Thirring, Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten
und Monde nach der Einsteinschen Gravitationstheorie,
Physikalische Zeitschrift 19, 156 (1918).
[42] I. Ciufolini and E. C. Pavlis, A confirmation of the general relativistic prediction of the Lense–Thirring effect,
Nature 431, 958 (2004).
[43] C. W. F. Everitt et al., Gravity Probe B: Final results
of a space experiment to test general relativity, Physical
Review Letters 106, 221101 (2011).
[44] N. Yunes and F. Pretorius, Fundamental theoretical bias
in gravitational wave astrophysics and the parametrized
post-Einsteinian framework, Physical Review D 80,
122003 (2009).
[45] R. Abbott et al., Tests of general relativity with GWTC3, Physical Review D 112, 084080 (2025).
[46] J. M. Weisberg, D. J. Nice, and J. H. Taylor, Timing measurements of the relativistic binary pulsar PSR
B1913+16, Astrophysical Journal 722, 1030 (2010).
[47] J. M. Weisberg and Y. Huang, Relativistic measurements from timing the binary pulsar PSR B1913+16,
Astrophysical Journal 829, 55 (2016).
[48] Event Horizon Telescope Collaboration, First M87 Event
Horizon Telescope results. I. The shadow of the supermassive black hole, Astrophysical Journal Letters 875,
L1 (2019).
[49] Event Horizon Telescope Collaboration, First Sagittarius
A* Event Horizon Telescope results. I. The shadow of
the supermassive black hole in the center of the Milky
Way, Astrophysical Journal Letters 930, L12 (2022).
[50] F. Lelli, S. S. McGaugh, and J. M. Schombert, SPARC:
Mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves, AJ 152, 157 (2016),
arXiv:1606.09251.
[51] K.-H. Chae, Breakdown of the Newton-Einstein standard
gravity at low acceleration in internal dynamics of wide
binary stars, The Astrophysical Journal 952, 128 (2023).
[52] I. Banik, C. Pittordis, W. Sutherland, et al., Strong
constraints on the gravitational law from Gaia DR3 wide
binaries, Monthly Notices of the Royal Astronomical
Society 527, 4573 (2024).
[53] K. El-Badry, H.-W. Rix, and T. M. Heintz, A million
binaries from Gaia eDR3: sample selection and valida-

tion of Gaia parallax uncertainties, Monthly Notices of
the Royal Astronomical Society 506, 2269 (2021), 1.3
million pairs with > 90 per cent bound probability; 1.1
million with > 99 per cent., arXiv:2101.05282.
[54] G. W. Angus, B. Famaey, and D. A. Buote, X-ray group
and cluster mass profiles in MOND: unexplained mass
on the group scale, Monthly Notices of the Royal Astronomical Society 387, 1470 (2008).
[55] M. T. Murphy, P. Molaro, A. C. O. Leite, G. Cupani,
S. Cristiani, V. D’Odorico, R. Génova Santos, C. J. A. P.
Martins, D. Milaković, N. J. Nunes, et al., Fundamental
physics with ESPRESSO: Precise limit on variations in
the fine-structure constant towards the bright quasar
HE 0515−4414, Astronomy & Astrophysics 658, A123
(2022), arXiv:2112.05819 [astro-ph.CO].
[56] J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum,
R. F. Carswell, and M. B. Bainbridge, Indications of a
spatial variation of the fine structure constant, Physical
Review Letters 107, 191101 (2011), arXiv:1008.3907
[astro-ph.CO].
[57] J. B. Whitmore and M. T. Murphy, Impact of instrumental systematic errors on fine-structure constant measurements with quasar absorption lines, Monthly Notices
of the Royal Astronomical Society 447, 446 (2015).
[58] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J.
Banday, R. B. Barreiro, N. Bartolo, et al., Planck 2018 results. VI. Cosmological parameters, Astronomy & Astrophysics 641, A6 (2020), arXiv:1807.06209 [astro-ph.CO].
[59] R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H. Yeh,
Big bang nucleosynthesis: Present status, Reviews of
Modern Physics 88, 015004 (2016), arXiv:1505.01076
[astro-ph.CO].
[60] G. Alcock, Ab initio derivation of the fine structure
constant from Density Field Dynamics, Zenodo preprint
(2026).
[61] M. E. Tobar, P. Wolf, S. Bize, G. Santarelli, and V. Flambaum, Testing local Lorentz and position invariance
and variation of fundamental constants by searching the
derivative of the comparison frequency between a cryogenic sapphire oscillator and hydrogen maser, Phys. Rev.
D 81, 022003 (2010).
[62] D. Hils and J. L. Hall, Improved Kennedy–Thorndike
experiment to test special relativity, Phys. Rev. Lett. 64,
1697 (1990).
[63] K. Beeks et al., Fine-structure constant sensitivity of the
Th-229 nuclear clock transition, Nature Communications
16, 9147 (2025).
[64] R. Lange, N. Huntemann, J. M. Rahm, C. Sanner,
H. Shao, S. Weyers, and E. Peik, Improved limits for violations of local position invariance from atomic clock comparisons, Physical Review Letters 126, 011102 (2021).
[65] S. Blatt, A. D. Ludlow, G. K. Campbell, et al., New
limits on coupling of fundamental constants to gravity
using 87 Sr optical lattice clocks, Physical Review Letters
100, 140801 (2008).
[66] G. Alcock, Dataset and analysis package for “solar-locked
differential in ion–neutral optical frequency ratios”, Zenodo dataset, MIT License (2025).
[67] G. Alcock, Solar-locked differential in ion–neutral optical
frequency ratios: Empirical evidence for a reproducible
heliocentric phase modulation, Zenodo preprint (2025).
[68] ROCIT Collaboration, Final publishable report: Robust
optical clocks for international timescales (18SIB05),

429
EMPIR Final Report (2024), project 18SIB05 ROCIT.
[69] T. Ooi, J. F. Doyle, C. Zhang, J. S. Higgins, J. Ye,
K. Beeks, T. Sikorsky, T. Schumm, et al., Frequency
reproducibility of solid-state thorium-229 nuclear clocks,
Nature 650, 72 (2026).
[70] V. V. Flambaum and A. F. Tedesco, Dependence of
nuclear magnetic moments on quark masses and limits
on temporal variation of fundamental constants from
atomic clock experiments, Physical Review C 73, 055501
(2006).
[71] J. S. Higgins et al., Temperature sensitivity of a thorium229 solid-state nuclear clock, Physical Review Letters
134, 113801 (2025).
[72] G. Alcock, Composition-dependent bounds on scalarfield coupling to nuclear decay rates, Zenodo preprint
(2026).
[73] K. Beloy, M. I. Bodine, T. Bothwell, S. M. Brewer,
S. L. Bromley, J.-S. Chen, J.-D. Deschênes, J. Helmcke,
D. B. Hume, C. J. Kennedy, et al., Frequency ratio
measurements at 18-digit accuracy using an optical clock
network, Nature 591, 564 (2021).
[74] D. Lee, K. Kim, Z. Z. Hu, B. Lewis, W. Warfield,
K. Zhou, A. L. Collopy, J. A. Sherman, A. S. Kowligy,
P. B. Patel, J. D. Roslund, A. Cingöz, M. M. Boyd, and
J. Ye, Phase-continuous comparison of three all-optical
time scales over 20 days, arXiv preprint arXiv:2608.00910
(2026).
[75] G. D’Amico, G. Rosi, S. Zhan, L. Cacciapuoti, M. Fattori,
and G. M. Tino, Canceling the gravity gradient phase
shift in atom interferometry, Physical Review Letters
119, 253201 (2017).
[76] C. Overstreet, P. Asenbaum, T. Kovachy, R. Notermans,
J. M. Hogan, and M. A. Kasevich, Effective inertial
frame in an atom interferometric test of the equivalence
principle, Physical Review Letters 120, 183604 (2018).
[77] A. Roura, Circumventing Heisenberg’s uncertainty principle in atom interferometry tests of the equivalence
principle, Physical Review Letters 118, 160401 (2017).
[78] M. D. Ernst, Permutation methods: A basis for exact
inference, Statistical Science 19, 676 (2004).
[79] G. Alcock, Unexplained bright–dim intensity asymmetries in SOHO/UVCS Lyman-α data, Submitted (revised) to Solar Physics; under review (2026), theoryneutral empirical companion: 163/321 days significant
at 5% FDR; temporal-split permutation methodology.
[80] Y. Benjamini and Y. Hochberg, Controlling the false
discovery rate: A practical and powerful approach to
multiple testing, Journal of the Royal Statistical Society:
Series B 57, 289 (1995).
[81] J. Cohen, Statistical Power Analysis for the Behavioral
Sciences, 2nd ed. (Lawrence Erlbaum Associates, 1988).
[82] ALPHA Collaboration, E. K. Anderson, et al., Observation of the effect of gravity on the motion of antimatter,
Nature 621, 716 (2023).
[83] R. Akbari et al., Be+ -assisted, simultaneous confinement of more than 15000 antihydrogen atoms, Nature
Communications 16, 10106 (2025), order-of-magnitude
improvement in antihydrogen trapping efficiency.
[84] I. M. H. Etherington, On the definition of distance in
general relativity, Philosophical Magazine 7, 15, 761
(1933).
[85] G. F. R. Ellis, On the definition of distance in general
relativity: I. M. H. Etherington (Philosophical Magazine
ser. 7, vol. 15, 761 (1933)): Editorial note, General

Relativity and Gravitation 39, 1047 (2007).
[86] R. F. L. Holanda, J. A. S. Lima, and M. B. Ribeiro,
Testing the distance-duality relation with galaxy clusters
and type Ia supernovae, Astrophysical Journal Letters
722, L233 (2010).
[87] N. Liang, Z. Li, P. Wu, S. Cao, K. Liao, and Z.-H.
Zhu, A consistent test of the distance-duality relation
with galaxy clusters and Type Ia supernovae, Monthly
Notices of the Royal Astronomical Society 436, 1017
(2013), arXiv:1104.2497.
[88] D. Scolnic et al., The Pantheon+ analysis: The full
dataset and light-curve release, Astrophys. J. 938, 113
(2022), arXiv:2112.03863.
[89] D. Brout et al., The Pantheon+ analysis: Cosmological constraints, Astrophys. J. 938, 110 (2022),
arXiv:2202.04077.
[90] DESI Collaboration, A. G. Adame, et al., DESI 2024
VI: Cosmological constraints from the measurements
of baryon acoustic oscillations, JCAP 2025 (02), 021,
arXiv:2404.03002.
[91] M. S. Madhavacheril et al., The Atacama Cosmology
Telescope: DR6 gravitational lensing map and cosmological constraints, Astrophys. J. 962, 113 (2024),
arXiv:2304.05203.
[92] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J.
Banday, R. B. Barreiro, N. Bartolo, et al., Planck 2015
results. XXI. The integrated Sachs-Wolfe effect, Astronomy & Astrophysics 594, A21 (2016), arXiv:1502.01595
[astro-ph.CO].
[93] B. Stölzner, A. Cuoco, J. Lesgourgues, and M. Bilicki,
Updated tomographic analysis of the integrated SachsWolfe effect and implications for dark energy, Physical
Review D 97, 063506 (2018), arXiv:1710.03238 [astroph.CO].
[94] I. Tutusaus et al., Measurement of the Weyl potential evolution from the first three years of Dark Energy Survey
data, Nature Communications 15, 9295 (2024), modelindependent Weyl potential 2–3σ shallower at low z.
[95] G. Gu, X. Wang, Y. Wang, G.-B. Zhao, L. Pogosian,
et al., Dynamical dark energy in light of the DESI
DR2 baryonic acoustic oscillations measurements, Nature Astronomy (2025), dataset-dependent preference
for w(z) ̸= −1, arXiv:2504.06118.
[96] C. Pittordis and W. Sutherland, Wide binaries from
Gaia DR3: testing GR vs. MOND with realistic triple
modelling, Open Journal of Astrophysics (2025), see
also Hernandez (2025) MNRAS 537:2925 for opposing
interpretation.
[97] L. Wenzl et al., The Atacama Cosmology Telescope DR6:
gravitational lensing × BOSS EG test, Physical Review
D 111, 043535 (2025).
[98] A. H. Wright, B. Stölzner, M. Asgari, et al., KiDS-legacy:
Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey, Astronomy and Astrophysics
703, A158 (2025), arXiv:2503.19441.
[99] B. Stölzner, A. H. Wright, M. Asgari, et al., KiDS-legacy:
Consistency of cosmic shear measurements and joint cosmological constraints with external probes, Astronomy
and Astrophysics 702, A169 (2025), arXiv:2503.19442.
[100] V. Calafut, P. A. Gallardo, E. M. Vavagiakis, et al.,
The Atacama Cosmology Telescope: Detection of the
pairwise kinematic Sunyaev–Zel’dovich effect with SDSS
DR15 galaxies, Physical Review D 104, 043502 (2021),

430
arXiv:2101.08374.
[101] P. A. Gallardo et al., Test of the gravitational force
law on cosmological scales using the kinematic Sunyaev–
Zel’dovich effect, Physical Review Letters (2026), published 15 April 2026; DOI 10.1103/rk8v-rcm3.
[102] M. F. Atiyah and I. M. Singer, The index of elliptic
operators: III, Annals of Mathematics 87, 546 (1968).
[103] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral
asymmetry and Riemannian geometry. I, Mathematical
Proceedings of the Cambridge Philosophical Society 77,
43 (1975).
[104] P. Li, F. Lelli, S. McGaugh, and J. Schombert, Fitting
the radial acceleration relation to individual SPARC
galaxies, Astronomy & Astrophysics 615, A3 (2018),
residual r.m.s. scatter about the fitted RAR is 0.057 dex.
[105] A. G. Riess, D. Scolnic, G. S. Anand, L. Breuval,
S. Casertano, L. M. Macri, S. Li, W. Yuan, C. D. Huang,
S. Jha, et al., JWST validates HST distance measurements: Selection of supernova subsample explains differences in JWST estimates of local H0 , The Astrophysical
Journal 977, 120 (2024), combined JWST three-method
result: H0 = 72.6 ± 2.0 km/s/Mpc, arXiv:2408.11770.
[106] A. G. Riess, S. Li, G. S. Anand, W. Yuan, L. Breuval,
et al., The perfect host: JWST cepheid observations in
a background-free SN Ia host confirm no bias in Hubbleconstant measurements (2025), arXiv:2509.01667.
[107] H0DN Collaboration, S. Casertano, G. Anand, R. I.
Anderson, R. Beaton, A. Bhardwaj, et al., The local
distance network: a community consensus report on
the measurement of the Hubble constant at 1% precision, Astronomy & Astrophysics 708, A166 (2026),
arXiv:2510.23823.
[108] W. L. Freedman, B. F. Madore, I. S. Jang, T. J. Hoyt,
A. J. Lee, and K. A. Owens, Status report on the ChicagoCarnegie Hubble Program (CCHP): Measurement of the
Hubble constant using the Hubble and James Webb
Space Telescopes, Astrophys. J. 985, 203 (2025), TRGB
(HST+JWST): H0 = 70.39 ± 1.9 km/s/Mpc; claims
consistency with ΛCDM, arXiv:2408.06153.
[109] B. I. Ciocan et al., MUSE-DARK III: The evolution
of the radial acceleration relation at intermediate redshifts, Astronomy & Astrophysics 709, L16 (2026),
arXiv:2604.22613.
[110] G. T. Alcock, Is the acceleration scale of the radial acceleration relation tracking the cosmic expansion rate?,
Monthly Notices of the Royal Astronomical Society 551,
10.1093/mnras/stag1485 (2026), published 2026 September 7; MNRAS vol. 551, issue 4 (2026 October); article
ID stag1485. Accepted 2026 August 3, in original form
2026 July 10; article type Letter. Open access, CC-BY
4.0. Fits a0 (z) = A(1 + z)γ to the MUSE–DARK binned
RAR scale: γ = 0.78 ± 0.15 (statistical), 0.83 ± 0.13 with
the SPARC z ≃ 0 anchor; frozen a0 excluded at 5.3σ
√
(6.6σ anchored), ρm scaling at 4.8σ on the published
bin errors read as 1σ; a0 = A E(z) the informationcriterion-preferred one-parameter form.
[111] A. Nestor Shachar, S. H. Price, N. M. Förster Schreiber,
R. Genzel, T. T. Shimizu, L. J. Tacconi, H. Übler,
et al., RC100: Rotation curves of 100 massive starforming galaxies at z = 0.6–2.5 reveal little dark matter on galactic scales, Astrophys. J. 944, 78 (2023),
arXiv:2209.12199.

[112] G. Alcock, Epoch evolution of the MOND crossover scale
in Density Field Dynamics:
An epoch-consistency argu√
ment for a⋆ (z) = 2 α cH(z), Companion note to the
DFD Unified Review (this paper), Zenodo (2026), conditional extension of Appendix N under the Section XIX.D
epoch-consistency rule.
[113] A. H. Chamseddine and A. Connes, Universal formula for
noncommutative geometry actions: unification of gravity
and the standard model (1996), A. H. Chamseddine
and A. Connes, “Universal formula for noncommutative
geometry actions: unification of gravity and the standard
model,” Phys. Rev. Lett. 77, 4868 (1996).
[114] A. Chamseddine and A. Connes, (1997), A. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys. 186, 731 (1997).
[115] A. D. Sakharov, Vacuum quantum fluctuations in
curved space and the theory of gravitation (1967),
A. D. Sakharov, “Vacuum quantum fluctuations in
curved space and the theory of gravitation,” Dokl. Akad.
Nauk Ser. Fiz. 177, 70–71 (1967); Sov. Phys. Dokl. 12,
1040–1041 (1968); reprinted in Gen. Rel. Grav. 32, 365–
367 (2000).
[116] G. Alcock, Tree-level no-drive theorem for the minimal
optical-metric EM sector in Density Field Dynamics:
λbare = 1 from the pure gauge-invariant action, Companion note to the DFD Unified Review (this paper),
Zenodo (2026), minimal-sector baseline for Appendix R.
[117] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H. Freeman, 1973).
[118] V. A. Dzuba and V. V. Flambaum, Sensitivity of optical
transitions to variation of the fine-structure constant,
Physical Review A 97, 040501(R) (2018).
[119] R. Epstein and I. I. Shapiro, Post-post-Newtonian deflection of light by the Sun, Physical Review D 22, 2947
(1980).
[120] G. W. Richter and R. A. Matzner, Second-order contributions to gravitational deflection of light in the
parametrized post-Newtonian formalism, Physical Review D 26, 1219 (1982).
[121] F. Hofmann and J. Müller, Relativistic tests with lunar
laser ranging, Classical and Quantum Gravity 35, 035015
(2018).
[122] J. F. Bell and T. Damour, A new test of conservation laws
and Lorentz invariance in relativistic gravity, Classical
and Quantum Gravity 13, 3121 (1996).
[123] I. H. Stairs, Pulsars in binary systems: Probing binary
stellar evolution and general relativity, Living Reviews
in Relativity 6, 5 (2003).
[124] D. F. Bartlett and D. Van Buren, Equivalence of active
and passive gravitational mass using the moon, Physical
Review Letters 57, 21 (1986).
[125] E. Peik and C. Tamm, Nuclear laser spectroscopy of the
3.5 eV transition in Th-229, Europhysics Letters 61, 181
(2003).
[126] LIGO Scientific Collaboration and Virgo Collaboration,
GW170817: Observation of gravitational waves from a
binary neutron star inspiral, Physical Review Letters
119, 161101 (2017).
[127] E. Witten, Search for a realistic Kaluza–Klein theory
(1981), E. Witten, “Search for a realistic Kaluza–Klein
theory,” Nucl. Phys. B 186, 412–428 (1981).
[128] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9
(Springer, 1972).

431
[129] W. Fulton and J. Harris, Representation Theory: A
First Course, Graduate Texts in Mathematics, Vol. 129
(Springer, 1991).
[130] H. B. Lawson and M.-L. Michelsohn, Spin Geometry
(Princeton University Press, 1989).
[131] T. Friedrich, Dirac Operators in Riemannian Geometry,
Graduate Studies in Mathematics, Vol. 25 (American
Mathematical Society, 2000).
[132] E. Witten, Quantum field theory and the Jones polynomial, Communications in Mathematical Physics 121,
351 (1989).
[133] C. Y. Tsang et al., Tracking the baryon number with nuclear collisions, Science 10.1126/science.ads5962 (2026),
STAR Collaboration.
[134] D. Kharzeev, Can gluons trace baryon number?, Physics
Letters B 378, 238 (1996), arXiv:hep-ph/9602314.
[135] G. C. Rossi and G. Veneziano, A possible description
of baryon dynamics in dual and gauge theories, Nuclear
Physics B 123, 507 (1977).
[136] F. Nicastro, J. Kaastra, Y. Krongold, S. Borgani,
E. Branchini, R. Cen, M. Dadina, C. W. Danforth,
M. Elvis, F. Fiore, et al., Observations of the missing
baryons in the warm-hot intergalactic medium, Nature
558, 406 (2018).
[137] J.-P. Macquart et al., A census of baryons in the universe
from localized fast radio bursts, Nature 581, 391 (2020).
[138] D. Eckert, S. Ettori, E. Pointecouteau, R. F. J. van der
Burg, and S. I. Loubser, The gravitational field of XCOP galaxy clusters, Astronomy & Astrophysics 662,
A123 (2022), arXiv:2205.01110 [astro-ph.CO].
[139] C. Burke, M. Hilton, and C. Collins, Coevolution of
brightest cluster galaxies and intracluster light using
CLASH, Monthly Notices of the Royal Astronomical
Society 449, 2353 (2015).
[140] M. Montes and I. Trujillo, Intracluster light at the Frontier – II. The Frontier Fields Clusters, Monthly Notices
of the Royal Astronomical Society 474, 917 (2018).
[141] S. Walker et al., The physics of galaxy cluster outskirts,
Space Science Reviews 215, 7 (2019).
[142] M. Cappellari, Full spectrum fitting with photometry
in pPXF: stellar population versus dynamical masses,
non-parametric star formation history and metallicity
for 3200 LEGA-C galaxies at redshift z ≈ 0.8, Monthly
Notices of the Royal Astronomical Society 526, 3273
(2023), arXiv:2208.14974 [astro-ph.GA].
[143] G. W. Pratt, J. H. Croston, M. Arnaud, and
H. Böhringer, Galaxy cluster X-ray luminosities and
temperatures from the REXCESS sample, Astronomy
& Astrophysics 498, 361 (2009), arXiv:0809.3784 [astroph].
[144] A. B. Mantz, S. W. Allen, R. G. Morris, R. W. Schmidt,
A. von der Linden, and O. Urban, Cosmology and astrophysics from relaxed galaxy clusters – II. cosmological
constraints, Monthly Notices of the Royal Astronomical
Society 440, 2077 (2014), see also Mantz et al. 2016,
MNRAS 463, 3582 (arXiv:1606.03407), arXiv:1402.6212
[astro-ph.CO].
[145] G. Alcock, Ab initio derivation of the charged fermion
mass spectrum from Density Field Dynamics, Zenodo
preprint (2026).
[146] X. Dai and D. S. Freed, η-invariants and determinant
lines, Journal of Mathematical Physics 35, 5155 (1994),
arXiv:hep-th/9405012.

[147] D. S. Freed and G. W. Moore, Twisted equivariant matter, Annales Henri Poincaré 14, 1927 (2013),
arXiv:1208.5055.
[148] A. H. Chamseddine and A. Connes, Noncommutative
geometry as a framework for unification of all fundamental interactions including gravity. Part I (2010),
A. H. Chamseddine and A. Connes, Noncommutative geometry as a framework for unification of all fundamental
interactions including gravity. Part I, Fortsch. Phys. 58,
553 (2010).
[149] J. Ellis and M. K. Gaillard, Strong and weak CP violation, Nuclear Physics B 150, 141 (1979).
[150] I. B. Khriplovich and A. I. Vainshtein, Infinite renormalization of θ-term and Jarlskog invariant for CP violation,
Nuclear Physics B 414, 27 (1994), hep-ph/9308334.
[151] I. Garcı́a-Etxebarria and M. Montero, Dai–Freed anomalies in particle physics, Journal of High Energy Physics
2019, 003 (2019), arXiv:1808.00009.
[152] S. M. Koushiappas, A cosmological uncertainty relation and late-universe acceleration, Physical Review D
(2026), arXiv:2604.27771.
[153] V. A. Kostelecký and N. Russell, Data tables for Lorentz
and CPT violation, Reviews of Modern Physics 83, 11
(2011).
[154] N. Ashby, T. E. Parker, and B. R. Patla, A null test of
general relativity based on a long-term comparison of
atomic transition frequencies, Nature Physics 14, 822
(2018).
[155] M. Filzinger, S. Hausser, M. Steinel, et al., Improved
limits on the coupling of ultralight bosonic dark matter to
photons from optical atomic clock comparisons, Physical
Review Letters 130, 253001 (2023).
[156] T. M. Fortier, N. Ashby, J. C. Bergquist, et al., Precision
atomic spectroscopy for improved limits on variation of
the fine structure constant and local position invariance,
Physical Review Letters 98, 070801 (2007).
[157] T. Rosenband, D. B. Hume, P. O. Schmidt, et al., Frequency ratio of Al+ and Hg+ single-ion optical clocks;
metrology at the 17th decimal place, Science 319, 1808
(2008).
[158] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, 2011).
[159] L. C. Evans, Partial Differential Equations, 2nd ed.
(American Mathematical Society, 2010).
[160] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach
spaces, American Journal of Mathematics 93, 265 (1971).
[161] G. Alcock, Pairing-symmetry selection rules for the
Cooper-pair mass anomaly from A5 microsector representation theory, Zenodo preprint (2026).
[162] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni,
T. Schwetz, and A. Zhou, The fate of hints: updated
global analysis of three-flavor neutrino oscillations, JHEP
2024 (12), 216, arXiv:2410.05380 [hep-ph].
[163] A. Papapetrou, Eine Theorie des Gravitationsfeldes
mit einer Feldfunktion, Zeitschrift für Physik 139, 518
(1954).
[164] H. Yilmaz, New approach to general relativity, Physical
Review 111, 1417 (1958).
[165] P. Boonserm, T. Ngampitipan, A. Simpson, and
M. Visser, Exponential metric represents a traversable
wormhole, Physical Review D 98, 084048 (2018).
[166] K.-H. Chae, F. Lelli, H. Desmond, S. S. McGaugh, P. Li,
and J. M. Schombert, Testing the strong equivalence

432
principle: Detection of the external field effect in rotationally supported galaxies, The Astrophysical Journal
904, 51 (2020).
[167] K.-H. Chae, H. Desmond, F. Lelli, S. S. McGaugh, and
J. M. Schombert, Testing the strong equivalence principle.
II. relating the external field effect in galaxy rotation
curves to the large-scale structure of the universe, The
Astrophysical Journal 921, 104 (2021).
[168] A. Paranjape and R. K. Sheth, The phenomenology
of the external field effect in cold dark matter models,
Monthly Notices of the Royal Astronomical Society 517,
130 (2022).
[169] H. Desmond, The underlying radial acceleration relation,
Monthly Notices of the Royal Astronomical Society 526,
3342 (2023).
[170] L. J. Tacconi, R. Genzel, and A. Sternberg, The evolution
of the star-forming interstellar medium across cosmic
time, Annu. Rev. Astron. Astrophys. 58, 157 (2020),
arXiv:2003.06245.
[171] M. Milgrom, High-redshift rotation curves and MOND
(2017), arXiv:1703.06110.
[172] R. Genzel, N. M. Förster Schreiber, H. Übler, P. Lang,
T. Naab, R. Bender, L. J. Tacconi, et al., Strongly
baryon-dominated disk galaxies at the peak of galaxy
formation ten billion years ago, Nature 543, 397 (2017),
arXiv:1703.04310.
[173] H. Übler, N. M. Förster Schreiber, R. Genzel, E. Wisnioski, S. Wuyts, et al., The evolution of the Tully-Fisher
relation between z ∼ 2.3 and z ∼ 0.9 with KMOS3D ,
Astrophys. J. 842, 121 (2017), arXiv:1703.04321.
[174] R. Genzel, S. H. Price, H. Übler, N. M. Förster Schreiber,
T. T. Shimizu, L. J. Tacconi, et al., Rotation curves
in z ∼ 1–2 star-forming disks: Evidence for cored
dark matter distributions, Astrophys. J. 902, 98 (2020),
arXiv:2006.03046.
[175] H. J. Mo and S. D. M. White, An analytic model for
the spatial clustering of dark matter haloes, Monthly
Notices of the Royal Astronomical Society 282, 347
(1996), arXiv:astro-ph/9512127.
[176] R. K. Sheth and G. Tormen, Large-scale bias and
the peak background split, Monthly Notices of the
Royal Astronomical Society 308, 119 (1999), arXiv:astroph/9901122.
[177] M. Blomqvist, H. du Mas des Bourboux, N. G. Busca,
V. de Sainte Agathe, J. Rich, C. Balland, J. E. Bautista,
K. Dawson, A. Font-Ribera, J. Guy, J.-M. Le Goff,
N. Palanque-Delabrouille, M. M. Pieri, I. Pérez-Ràfols,
G. Rossi, D. P. Schneider, A. Slosar, and C. Yèche,
Baryon acoustic oscillations from the cross-correlation
of Lyα absorption and quasars in eBOSS DR14, Astronomy & Astrophysics 629, A86 (2019), arXiv:1904.03430
[astro-ph.CO].
[178] N. Hand, G. E. Addison, E. Aubourg, N. Battaglia,
E. S. Battistelli, D. Crichton, S. Das, J. Dunkley, et al.,
Evidence of galaxy cluster motions with the kinematic
Sunyaev–Zel’dovich effect, Physical Review Letters 109,
041101 (2012), arXiv:1203.4219 [astro-ph.CO].
[179] H. Desmond, A. Hees, and B. Famaey, On the tension
between the radial acceleration relation and Solar system quadrupole in modified gravity MOND, Monthly
Notices of the Royal Astronomical Society 530, 1781
(2024), AQUAL/QUMOND: 8.7σ tension between the
SPARC RAR and the Cassini quadrupole under fiducial

assumptions, relieved to 1.9σ only by removing bulge
galaxies., arXiv:2401.04796.
[180] X. Wu, Model-independent test of the cosmic distance
duality relation with recent observational data, arXiv
e-prints (2026), arXiv:2603.27616.
[181] S. Necco and R. Sommer, The Nf = 0 heavy quark
potential from short to intermediate distances, Nucl.
Phys. B 622, 328 (2002), hep-lat/0108008.
[182] B. Lucini and M. Teper, SU (N ) gauge theories in four
dimensions: exploring the approach to N = ∞, JHEP
2001 (06), 050, hep-lat/0103027.
[183] C. J. Morningstar and M. J. Peardon, The glueball
spectrum from an anisotropic lattice study, Phys. Rev.
D 60, 034509 (1999), hep-lat/9901004.
[184] R. G. Edwards, U. M. Heller, and T. R. Klassen, Accurate scale determinations for the Wilson gauge action,
Nucl. Phys. B 517, 377 (1998), hep-lat/9711003.
[185] M. Teper, SU (N ) gauge theories in 3+1 dimensions,
Phys. Rev. D 59, 014512 (1998), hep-lat/9804008.
[186] G. S. Bali and K. Schilling, Running coupling and the
Lambda parameter from SU (3) lattice simulations, Phys.
Rev. D 47, 661 (1993), see also G. S. Bali, Phys. Rept.
343 (2001) 1.
[187] T. Lippert et al., SU (3) dynamical fermions: SESAM
and TχL results for the Wilson action, Nucl. Phys. B
(Proc. Suppl.) 60A, 311 (1998), SESAM and TχL Collaborations, hep-lat/9707004.
[188] S. Güsken et al., The static potential in full QCD, Nucl.
Phys. B (Proc. Suppl.) 63, 385 (1998), hep-lat/9710075.
[189] A. Jaffe and E. Witten, Quantum Yang–Mills theory,
Clay Mathematics Institute Millennium Problem statement (2000).
[190] S. K. Donaldson and P. B. Kronheimer, The Geometry
of Four-Manifolds (Oxford University Press, 1990).
[191] P. Shah, T. M. Davis, D. Bacon, et al., The Dark Energy Survey: detection of weak lensing magnification
of supernovae and constraints on dark matter haloes,
Monthly Notices of the Royal Astronomical Society 532,
932 (2024), arXiv:2406.05047.
[192] A. Berlin, Z. Bogorad, P. W. Graham, and H. Ramani,
Cavendish tests of millicharged particles, Physical Review Letters (2026), arXiv:2510.25825.
[193] A. Berlin, Z. Bogorad, P. W. Graham, and H. Ramani,
Electric accumulation of millicharged particles, Physical
Review D (2026), arXiv:2510.25834.
[194] DESI Collaboration, M. Abdul-Karim, et al., DESI DR2
results II: Measurements of baryon acoustic oscillations
and cosmological constraints, Phys. Rev. D 112, 083515
(2025), arXiv:2503.14738.
[195] F.-Y. Cyr-Racine, F. Ge, and L. Knox, Symmetry of
cosmological observables, a mirror world dark sector,
and the Hubble constant, Phys. Rev. Lett. 128, 201301
(2022), arXiv:2107.13000.
[196] D. W. Pesce, J. A. Braatz, M. J. Reid, A. G. Riess,
D. Scolnic, J. J. Condon, F. Gao, C. Henkel, C. M. V.
Impellizzeri, C. Y. Kuo, and K. Y. Lo, The Megamaser Cosmology Project. XIII. Combined Hubble constant constraints, Astrophys. J. Lett. 891, L1 (2020),
arXiv:2001.09213.
[197] J. B. Jensen, J. P. Blakeslee, M. Cantiello, M. Cowles,
G. S. Anand, et al., The TRGB-SBF project. III. refining
the HST surface brightness fluctuation distance scale
calibration with JWST, The Astrophysical Journal 987,
87 (2025), arXiv:2502.15935.

433
[198] C. Vogl et al., No rungs attached: A distance-ladder free
determination of the Hubble constant through type II
supernova spectral modelling, Astron. Astrophys. 702,
A41 (2025), arXiv:2411.04968.
[199] K. Hotokezaka, E. Nakar, O. Gottlieb, S. Nissanke,
K. Masuda, G. Hallinan, K. P. Mooley, and A. T. Deller,
A Hubble constant measurement from superluminal motion of the jet in GW170817, Nature Astronomy 3, 940
(2019), arXiv:1806.10596.
[200] J. Carrick, S. J. Turnbull, G. Lavaux, and M. J. Hudson,
Cosmological parameters from the comparison of peculiar
velocities with predictions from the 2M++ density field,
Monthly Notices of the Royal Astronomical Society 450,
317 (2015), arXiv:1504.04627.
[201] TDCOSMO Collaboration, TDCOSMO 2025: Cosmological constraints from strong lensing time delays, Astronomy & Astrophysics 704, A63 (2025), primary
+3.1
H0 = 71.6+3.9
−3.3 ; +SLACS+SL2S combined 74.3−3.7 ; su+4.1
persedes the TDCOSMO-IV+SLACS 67.4−3.2 value,
arXiv:2506.03023.
[202] A. G. Riess, W. Yuan, L. M. Macri, et al., A comprehensive measurement of the local value of the Hubble constant with 1 km s−1 Mpc−1 uncertainty from the Hubble
Space Telescope and the SH0ES team, The Astrophysical Journal Letters 934, L7 (2022), H0 = 73.04 ± 1.04
km/s/Mpc, arXiv:2112.04510.
[203] J. P. Blakeslee, J. B. Jensen, C.-P. Ma, P. A. Milne, and
J. E. Greene, The Hubble constant from infrared surface
brightness fluctuation distances, The Astrophysical Journal 911, 65 (2021), H0 = 73.3 ± 2.5 km/s/Mpc, 40–99
Mpc, arXiv:2101.02221.
[204] R. Stiskalek, H. Desmond, and I. Banik, Testing the local
supervoid solution to the Hubble tension with direct
distance tracers, Mon. Not. R. Astron. Soc. 543, 1556
(2025).
[205] A. Großardt, J. Bateman, H. Ulbricht, and A. Bassi, Optomechanical test of the Schrödinger–Newton equation,
Physical Review D 93, 096003 (2016), arXiv:1510.01696.
[206] H. Yang, H. Miao, D.-S. Lee, B. Helou, and Y. Chen,
Macroscopic quantum mechanics in a classical spacetime,
Physical Review Letters 110, 170401 (2013).

[207] D. N. Page and C. D. Geilker, Indirect evidence for
quantum gravity, Physical Review Letters 47, 979 (1981).
[208] S. M. Saad and Y.-S. Ting, No gravitational anomaly
in wide binaries from forward modeling of 3D orbits, The Open Journal of Astrophysics 9 (2026),
arXiv:2603.11015.
[209] K.-H. Chae, B.-C. Lee, X. Hernandez, V. G. Orlov,
D. Lim, D. A. Turnshek, and Y.-W. Lee, Detection
of gravitational anomaly at low acceleration from a
highest-quality sample of 36 wide binaries with accurate
3D velocities, arXiv e-prints (2026), arXiv:2601.21728,
arXiv:2601.21728.
[210] Y. Yoon, Y. Tian, and K.-H. Chae, Probing the nature
of gravity in the low-acceleration limit: wide binaries of
extreme separations with perspective effects, The Astrophysical Journal 992, 102 (2025), arXiv:2507.13177.
[211] X. Hernandez, V. Verteletskyi, L. Nasser, and A. AguayoOrtiz, Statistical analysis of the gravitational anomaly
in Gaia wide binaries, Monthly Notices of the Royal Astronomical Society 528, 4720 (2024), arXiv:2309.10995.
[212] H. Boufourou, Estimator forensics for the wide-binary
gravity test: the eccentricity–triple coupling manufactures a pseudo-signal, and a pre-registered protocol for
Gaia DR4, arXiv e-prints (2026), arXiv:2608.24556,
arXiv:2608.24556.
[213] M. Królikowska and P. A. Dybczyński, The catalogue of
cometary orbits and their dynamical evolution, Astronomy & Astrophysics 640, A97 (2020).
[214] D. Vokrouhlický, D. Nesvorný, and S. Tremaine, Testing
MOND on small bodies in the remote solar system, The
Astrophysical Journal 968, 47 (2024), arXiv:2403.09555.
[215] M. Milgrom, MOND effects in the inner Solar system,
Monthly Notices of the Royal Astronomical Society 399,
474 (2009), arXiv:0906.4817.
[216] S. S. McGaugh, MOND prediction for the velocity dispersion of the “feeble giant” Crater II, The Astrophysical
Journal Letters 832, L8 (2016).
[217] N. Caldwell, M. G. Walker, M. Mateo, E. W. Olszewski,
S. Koposov, V. Belokurov, G. Torrealba, A. GeringerSameth, and C. I. Johnson, Crater 2: an extremely cold
dark matter halo, The Astrophysical Journal 839, 20
(2017).