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\begin{document}

\title{Density Field Dynamics: A Complete Unified Theory}

\author{Gary Alcock}
\email{gary@gtacompanies.com}
\affiliation{Independent Researcher, Los Angeles, CA, USA}

\date{v4.0 --- July 4, 2026\\[2pt]
\makebox[0.41\columnwidth]{\scriptsize
Foundations: 19 Aug 2025~\cite{Alcock2025cfield};
Unified v1.0: 25 Dec 2025~\cite{Alcock2025unified}}}

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\begin{abstract}
\small
\noindent\textit{This v4.0 release supersedes v3.3 (Zenodo, prior public version) and
consolidates the full DFD theorem, falsifier, and outstanding-problem closure program
into a single canonical document.}

\medskip
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
$\psi(\mathbf{x},t)$ establishing an optical refractive index $n = e^\psi$.
Light propagates according to the eikonal of the optical metric
$d\tilde{s}^2 = -c^2 dt^2/n^2 + d\mathbf{x}^2$, while matter responds to the
effective potential $\Phi = -c^2\psi/2$. This framework has an \textbf{optical scalar sector}
$\psi$ that governs clock rates, refraction, and quasi-static dynamics, together with a
\textbf{transverse-traceless radiative sector} $h^{\rm TT}_{ij}$ for gravitational waves,
derived as the spin-2 irreducible component of the same zero-mode parent tensor on
$\mathbb{C}P^2 \times S^3$ whose trace yields $\psi$~\cite{AlcockConstitutive2026}.
It reproduces all classic tests of general relativity in the weak-field limit
($\gamma = \beta = 1$, 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 $\mu(x) = x/(1+x)$
and scale $a_* = 2\sqrt{\alpha}\,cH_0$, \textbf{both derived from $S^3$ topology}
(Appendix~\ref{app:mu-derivation}; the $H_0$-free form
$a_0 = 2\alpha^{29}\sqrt{c^7/G\hbar}$ is an action-level theorem,
Appendix~\ref{app:alpha57-closure}). A dedicated model-independent SPARC shape analysis further finds $n_{\rm opt}=1.15\pm0.12$ (95\% CI $[1.00,1.50]$) in the family $\mu_n(x)=x/(1+x^n)^{1/n}$, with DFD's $n=1$ inside the confidence region and Standard MOND's $n=2$ strongly disfavored.

\textbf{This paper presents DFD as a unified framework:}
%
(1)~\textbf{Fine-structure constant:} $\alpha^{-1} = 137.036$ from the microsector spectral action on $\mathbb{C}P^2 \times S^3$
with Toeplitz truncation at $k_{\max} = 60$. The derivation is \textbf{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 matching experiment at the $5.6\times10^{-9}$ fractional level ($0.0056$~ppm residual).
\textbf{Verified by lattice Monte Carlo} (L6--L16; 9/10 at L16 with $p < 0.01$, mean $+1.1\%$);
%
(1b)~\textbf{Weinberg angle:} $\sin^2\theta_W = 3/13 = 0.2308$ from gauge partition $(3,2,1)$ and canonical trace normalization
(Appendix~\ref{app:complete-params}). The 5/3 GUT normalization factor is \textbf{derived}, not assumed.
Agreement with $\sin^2\theta_W^{\rm exp} = 0.23122$: \textbf{0.2\%} (tree-level vs $\overline{\rm MS}$);
%
(1c)~\textbf{Strong coupling:} $\alpha_s(M_Z) = 0.1187$ from $\Lambda_{\rm QCD} = M_P \alpha^{19/2} = 61.20$ MeV
and the unique proper-time$\to\overline{\rm MS}$ matching factor $\sqrt{4\pi}$ (Appendix~\ref{app:complete-params}).
Agreement with PDG 2024 ($0.1180 \pm 0.0009$): \textbf{$0.8\sigma$};
%
(2)~\textbf{Higgs hierarchy:} $v = M_P \times \alpha^8 \times \sqrt{2\pi} = 246.09$ GeV
(observed: 246.22 GeV, \textbf{0.05\% error})---the 17 orders of magnitude are topological, not fine-tuned.
The Higgs quartic $\lambda_H = 1/8$ from dimension counting (Appendix~\ref{app:complete-params}) gives tree-level $m_H = 123$ GeV;
%
(3)~\textbf{Nine charged fermion masses:} $m_f = A_f \alpha^{n_f} v/\sqrt{2}$ with sector-dependent exponents
achieves \textbf{1.42\% mean error} at leading order. Prefactors computed via explicit Yukawa operator: CP$^2$ kernels
$K_d = J_3$, $K_u = I_4$ fixed by symmetry (Lemma~\ref{lem:kernel}), QCD factors from $b_0 = 7$,
generation operator $G = {\rm diag}(2/3,1,1)$ derived from primed microsector trace (Theorem~\ref{thm:G-derived}, Appendix~\ref{app:microsector});

(4)~\textbf{CKM pattern (v4.0):} Wolfenstein parameters fit the integer pattern
$(\lambda, A, \bar{\rho}, \bar{\eta}) = (31, 108, \tfrac{43}{2}, 49) \times \alpha$, with the magnitude integers arising from $\mathbb{C}P^2$ line bundle cohomology and the CP-even apex $\bar\rho = \tfrac{43}{2}\alpha = 21.5\alpha$ selected by the Euler-projection postulate of the strengthened DFD--SD branch (Appendices~\ref{app:complete-params} and~\ref{app:ckm-update}). \emph{Per-channel agreement against PDG~2024:} $\lambda$ at $0.5\%$ ($1.8\sigma$), $\bar\eta$ at $1.5\%$ ($0.7\sigma$), $A$ at $4.6\%$ ($2.4\sigma$, sensitive to the open inclusive/exclusive $|V_{cb}|$ tension), $\bar\rho$ at $1.3\%$ ($0.2\sigma$). The earlier ``$0.55\%$ mean agreement'' summary statistic was computed against PDG 2018--2020 reference values and is now explicitly retracted as a headline; the magnitude integers are preserved as suggestive of cohomological origin, while the apex is now selected at theorem grade \emph{inside the strengthened DFD--SD branch} by the Euler-projection postulate (Appendix~\ref{app:ckm-update}), which --- consistent with Anti-Theorem~AT-7$'$ --- is a new structural postulate rather than a consequence of the pre-strengthened axioms, and which supersedes the earlier post-hoc half-integer Cayley correction. The branch predicts $\gamma = 66.3093^\circ$ and $J = 2.9731\times10^{-5}$. PMNS from tribimaximal base + charged lepton corrections;
%
(5)~\textbf{Strong CP (theorem):} $\bar\theta = 0$ to all loop orders, with $\arg\det(M_u M_d)$ kept real by the det-orthogonal structure; all-orders, the CP anomaly vanishes because the mapping torus has even dimension (8), forcing $\eta = 0$ by spectral symmetry (Appendices~\ref{app:strongcp_selection_rule},~\ref{app:ckm-update}). \textbf{No axion required.} The minimal real-kernel flavor sector gives $J = 0$ (Theorem~\ref{thm:ckm-obstruction}); the observed nonzero CKM phase ($J \neq 0$) is supplied by the conjugation-odd, determinant-orthogonal Berry offset of the \emph{strengthened DFD--SD branch} (Postulate~Y.10, Appendix~\ref{app:ckm-update}), which preserves $\bar\theta = 0$;

(6)~\textbf{G--H$_0$ invariant (spectral-action-derived):} The dimensionless constraint $G\hbar H_0^2/c^5 = \alpha^{57}$ is now derived
via Gaussian mode integration on the finite-dimensional microsector (Appendix~\ref{app:O_alpha57_clean}): the exponent 57 is topologically forced by primed-determinant
scaling on the finite microsector state space; the per-mode suppression factor $\alpha$ follows from uniform gauge normalization with exact eigenvalue cancellation (Lemma~O.4); the identification with the physical hierarchy uses the finite dimensionality of the microsector to eliminate all UV ambiguities (Lemma~O.5).
This predicts $H_0 = 72.09$ km/s/Mpc, matching JWST distance-ladder measurements (SH0ES JWST combined: $72.6 \pm 2.0$ km/s/Mpc, $0.3\sigma$ agreement)
but disagreeing with Planck CMB-inferred $H_0 = 67.4 \pm 0.5$ km/s/Mpc at $9.4\sigma$ (Planck statistical uncertainty)---the ``Hubble tension'' is interpreted as a $\psi$-screen optical bias;
%
(7)~\textbf{UVCS test:} Ly-$\alpha$/O~VI asymmetry ratio $R = \Gamma \times (\sigma_{\rm OVI}/\sigma_{\rm Ly\alpha})^2$ with
$\Gamma_{\rm obs} = 4.4 \pm 0.9$ matching DFD's double-transit prediction $\Gamma = 4$ ($0.4\sigma$); standard physics predicts $\Gamma = 1$;
%
(8)~\textbf{CMB from baryon loading $+$ derived $\chi$:} Peak ratio $R = 2.34$ from baryon loading,
peak location $\ell_1 = 220$ from $\psi$-lensing with $\Delta\psi = 0.30$, third-peak height from the derived $\chi$-matter dark matter --- a cold, collisionless component of mass $5.09$~eV whose abundance $\Omega_\chi h^2 = 0.1182$ is computed \emph{forward} from the finite $SU(2)_{60}$ Chern--Simons vacuum ($-1.5\sigma$ from Planck), with no initial-condition angle and no fitted density (App.~\ref{app:chi-matter}, Thm.~\ref{thm:chi-postinflation});
%
(9)~\textbf{Quantitative $\psi$-screen reconstruction:} $\Delta\psi(z=1) = 0.27 \pm 0.02$ from $H_0$-independent
distance ratios---the ``accelerating expansion'' is reinterpreted as an optical effect requiring no dark energy;

(10)~\textbf{Clock sector and Majorana scale (Appendix~\ref{app:P_kalpha_MR}):} the electromagnetic-sector proposal $k_\alpha = \alpha^2/(2\pi)$ and the scale $M_R = M_P\alpha^3$ are derived from the Appendix~O protocol. In the present version the clock sector is interpreted in a channel-resolved way: same-ion E3/E2 measurements strongly constrain any pure $\alpha$-sector coupling law, while cross-species and nuclear-clock channels remain the primary DFD discriminators. The cavity--atom sector is likewise treated with geometric cancellation at tree level and a residual screened signal rather than the earlier order-unity slope picture; 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 $\mathcal{O}(1\,\mathrm{kHz})$;
%
(10b)~\textbf{Neutrino mass spectrum (Appendix~\ref{app:neutrino}):} Closed at zero continuous parameters; one binary branch choice (Branch B over the equally canonical Branch A) is selected by the oscillation data, with everything downstream $\alpha$-locked (Revised Statement in Appendix~\ref{app:neutrino}).
Branch B exponents $k = \alpha^{-3/11}$, $r = \alpha^{-7/20}$ from microsector integers; absolute scale
$m_3 = (14/13)\pi M_P\alpha^{14}$ from finite-$d$ priming. Predictions: $\Delta m^2_{21} = 7.48 \times 10^{-5}$ eV$^2$,
$\Delta m^2_{31} = 2.51 \times 10^{-3}$ eV$^2$ (NuFIT 6.0: $\chi^2 = 0.025$, $p = 0.99$);
$\sum m_\nu = 61.46$ meV; combined hierarchy exponent $k^2r^2 = \alpha^{-137/110}$ (numerator is $\alpha^{-1}$);
%
(11)~\textbf{Dust branch from microsector (Appendix~\ref{app:temporal-completion}):} The temporal kinetic function $K(\Delta)$ is
derived from the same $S^3$ saturation-union composition law that fixed $\mu(x)$. Key results: (i) temporal deviation
invariance is forced by the composition law; (ii) the unique temporal segment scalar is $\Delta = (c/a_0)|\dot\psi - \dot\psi_0|$;
(iii) with $K'(\Delta) = \mu(\Delta)$, the dust branch emerges with $w \to 0$, $c_s^2 \to 0$. A no-go lemma proves the naive
quadratic identification gives $w \to 1/2$ (not dust). Full $P(k)$ matching is a program item, not a theorem.
%
(12)~\textbf{Screen-closure theorems (Sec.~\ref{sec:psitomo-closure-theorems}):} Two $\psi$-screen estimators (SNe, CMB)
reconstruct $\Delta\psi_{\rm screen}$ independently; a third estimator (duality) serves as a metric-consistency check
($\Delta\psi_{\rm dual}=0$ by Etherington reciprocity). Together they imply overdetermined closure identities:
(i) SN reconstructs $\Delta\psi_{\rm screen} - \mathcal{M}$
(single global constant); (ii) anisotropy maps must match on overlapping sky ($\ell \ge 1$). A $\chi^2_{\mathcal{M}}$ test across
redshift bins provides a quantitative falsifier. No dynamical assumption about $\mu(x)$ or growth required.

\textbf{Additional sectors included in the present master review:}
%
(13)~\textbf{Antimatter gravity (Sec.~\ref{sec:antimatter}):} Species-dependent sensitivities $\sigma_A$ from non-metric
$\psi$-sector couplings predict matter--antimatter differential acceleration $\Delta a_{H\bar{H}}/a \approx 2|\sigma_{\bar{H}} - \sigma_H|$.
At the metric level, DFD reproduces GR's universal free fall; C-odd couplings ($n_B$, $n_L$) could produce percent-level signals
testable by ALPHA-g. Antihydrogen probes parameter-space directions inaccessible to ordinary-matter EP tests;
%
(14)~\textbf{EM--$\psi$ coupling (Appendix~\ref{app:em-psi-coupling}):} Parameter $\lambda$ controls electromagnetic back-reaction
on $\psi$. Existing cavity stability provides an \emph{accidental} bound $|\lambda - 1| \lesssim 3 \times 10^{-5}$.
An intentional $2\omega$ modulation search could reach $|\lambda - 1| \sim 10^{-14}$ with existing apparatus;
%
(15)~\textbf{IBVP well-posedness (Sec.~\ref{sec:ibvp}):} Theorem-grade existence, uniqueness, and continuous dependence
for the initial-boundary value problem on bounded domains. Energy estimates with Gronwall bound ensure stability.
Finite speed of propagation guarantees causality;
%
(16)~\textbf{Late-time observations (Sec.~\ref{sec:obs-status}):} DES Y3 Weyl potential 2--3$\sigma$ shallower at low $z$
(supportive); DESI DR2 $w(z) \neq -1$ hints (consistent with $\psi$-screen); wide binaries active/contested;
$E_G$ and KiDS-Legacy show mild tension.

\textbf{New structural results:}
%
(17)~\textbf{GR as the Pad\'e approximant of DFD (Appendix~\ref{app:pade-identity}):}
The lapse-squared scalar $\mathcal{L}(u) \equiv c^2/|g_{tt}|$ of isotropic-coordinate Schwarzschild is mathematically identical to the squared $[1,1]$ Pad\'e approximant of DFD's exponential. Writing $P_{m,m}(u)$ for the diagonal $[m,m]$ Pad\'e approximant of $e^u$ (so $P_{m,m}(u)\approx e^u$): $\mathcal{L}_{\rm GR}(u) = [P_{1,1}(u)]^2 = [(1+u/2)/(1-u/2)]^2$, while DFD's matter-coupling physical metric gives $\mathcal{L}_{\rm DFD}(u) = \exp(2u)$. The identity extends to a Pad\'e hierarchy in which $[P_{m,m}(u)]^2 = \exp(2u) + O(u^{2m+1})$: \textbf{GR is the $m=1$ slot; DFD is the $m \to \infty$ entire-function limit.} Agreement through $O(u^2)$ by construction reproduces the post-Newtonian parameter $\beta = 1$ in both theories (the spatial-curvature parameter $\gamma = 1$ for DFD follows separately from the physical metric, Sec.~\ref{sec:ppn}). First divergence at $O(u^3)$: the Pad\'e pole at $u=2$ (Schwarzschild horizon in isotropic coordinates) is a structural artifact of the rational approximation; DFD's explicit exterior solution $\psi = 2GM/(c^2 r)$ is finite at every $r > 0$, with the radius $r = 2GM/c^2$ appearing as the DFD photon sphere (not a horizon), producing the $4.6\%$ larger shadow (Sec.~\ref{sec:strong-dfd-regime}) consistent with EHT data on M87$^\star$ and Sgr A$^\star$ at present precision. A firewall theorem distinguishes DFD from Yilmaz-type exponential metrics, which arise within GR as wormhole solutions with exotic matter (Theorems~\ref{thm:no-wormhole},~\ref{thm:no-exotic});
%
(18)~\textbf{Uniqueness of the internal manifold (Appendix~\ref{app:uniqueness}):}
The manifold $X = \mathbb{C}P^2 \times S^3$ used throughout DFD is not an ansatz. It is the unique compact Riemannian manifold satisfying nine axioms: compactness, dimension 7, Spin$^c$ structure, K\"ahler-factor product structure, $\chi_{\rm top} = 3$ (three generations), gauge partition $(3,2,1)$, $\pi_3 = \mathbb{Z}$ (proton stability), even mapping-torus dimension (Strong-CP), and finite Toeplitz truncation at $k_{\max} = 60$ (Theorem~\ref{thm:uniqueness}). The forced decomposition $7 = 4 + 3$ gives $M_c = \mathbb{C}P^2$ uniquely (Fano surface with $\chi=3$) and $M_g = S^3$ uniquely (simply-connected 3-dim Lie group with $\pi_3 = \mathbb{Z}$). The integers $\{3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137\}$ are therefore cohomological invariants of a uniquely forced manifold, not free parameters.

(19)~\textbf{Quantum sector (Sec.~\ref{sec:quantum}, Appendix~\ref{app:qm-status}):} the single-particle Schr\"odinger 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 $\mathbb{C}P^2$ K\"ahler structure; the imports are explicitly inventoried --- the value of $\hbar$, the variable-$N$ field-theoretic map, and the $\mathrm{Cl}(4,0)$ enlargement supplying the Dirac $\beta$ --- with the Born rule conditional and single-outcome selection not derived (a residue shared by all interpretations). The derived Schr\"odinger--Newton nonlinearity is an experimental discriminator (levitated-optomechanics coherence);
%
\textbf{Distance duality and screen clarification:}
%
(E1)~\textbf{Distance duality corrected (Sec.~\ref{sec:cosmology}):}
Etherington's reciprocity theorem holds exactly in DFD's optical metric.
The erroneous $e^{\Delta\psi}$ factor in the distance duality relation from an earlier internal draft
is deleted: $D_L = (1+z)^2 D_A$ exactly. Notation is disambiguated: $\Delta\psi_{\rm screen}$
(distance bias, Estimators A and C) vs.\ $\Delta\psi_{\rm dual} = 0$ (DDR consistency, Estimator B).
The $\psi$-screen program is retained, but the reciprocity statement is now explicit and version-independent.

The gauge emergence framework on $\mathbb{C}P^2 \times S^3$ yields the Standard Model gauge group,
with $N_{\text{gen}} = 3$ entering as a discrete input consistent with the index structure (App.~F),
and proton stability from $S^3$ winding (conditional on axiom V7).
The extended derivations --- theorem-grade closures and promotions, the
new-territory frontier, the CKM/$\tau$ and series IV--VI derivations, and the
mathematical-foundation and quantum-gravity closures --- are documented in the
companion volume \emph{Density Field Dynamics v4.0: Extended Derivations and
Frontier Predictions}. The present paper states
the final theory: axioms, theorems with proofs, predictions, and falsifiers.

\textbf{DFD introduces no continuous fit parameters.} The discrete topological sector carries no
continuous freedom: hypercharge integrality admits $q_1 \in \{3,6\}$ with $q_1 = 3$ selected,
the determinant-line lift gives $\mathcal{O}(9)$ at that selection (with $N_{\rm gen} = 3$ a discrete
input), and the five chiral multiplet types fix the padding (status remarks: App.~F).
Within the bundle decomposition $E = \mathcal{O}(a) \oplus \mathcal{O}^{\oplus n}$, minimal-padding uniquely selects
$(a,n) = (9,5)$ with $k_{\max} = 60$. A trials-factor audit (App.~\ref{app:CE:combined-evidence}) enumerates the entire admissible discrete space ($8.0\times10^{9}$ addresses) and proves that landscape scanning cannot account for the $\alpha$ match. One scale measurement ($H_0$ or $G$) then determines all
dimensionful quantities via $G\hbar H_0^2/c^5 = \alpha^{57}$.
This paper presents the mathematical formulation and demonstrates that
DFD constitutes a unified framework for gravity and particle physics, falsifiable with
current experimental technology.

\medskip
This release additionally sharpens two sectors. The CKM mixing magnitudes are
obtained at zero continuous free parameters from the $\mathbb{C}P^2$ line-bundle
cohomology pattern $(\lambda, A, \bar\rho, \bar\eta) = (31, 108, \tfrac{43}{2}, 49)\,\alpha$
over $k_{\max}=60=|A_5|$ (Appendix~\ref{app:ckm-update}),
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
$\gamma=66.3093^\circ$ and $J=2.9731\times10^{-5}$ (undetermined by the
pre-strengthened axioms per AT-7$'$; Appendix~\ref{app:ckm-update}). Separately, the
light-propagation law $c_1 = c/n$ and the matter-acceleration law
$\mathbf{a} = (c^2/2)\nabla\psi$ are unified as the null and timelike sectors of a
single optical-metric wave equation, with the exact identity
$\mathbf{a} = -\tfrac14\nabla(c_{\rm light}^2)$ (Appendix~\ref{app:master-equation}).

\end{abstract}
\maketitle

%-----------------------------------------------------------------------------
% Compact Scope and status block (post-abstract, pre-TOC).
%-----------------------------------------------------------------------------
\noindent\textbf{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 canonical audit ledger of this release (after Tier-1
closures T173--T182, reconciliation, Tier-2 closures T184--T189
(T190 synthesis), Tier-3 closures T191--T197 (T200 synthesis), 
Tier-4 closures T201--T210 (T211 synthesis), and Tier-5+6 closures
T212--T227 (T230 synthesis); \emph{development-record syntheses T183 / T190 / T200 / T211 / T230, not
reprinted in this deposit})
is: \textbf{91 theorem-grade closures, 0 internal inconsistencies}, with the genuinely open
items named rather than hidden --- the $A_s$ $32\pi$ prefactor and the interacting
many-body QM map (see \S\ref{sec:openproblems}; the $Q_\chi=1$ $\chi$-clustering channel
is now closed by the adopted Rest-Mass Channel axiom, App.~GR.3a). The $\chi$ relic
abundance, formerly an open overshoot, is now theorem-grade (the Finite $SU(2)_{60}$ CS
Vacuum Relic, $\Omega_\chi h^2=0.118$ at $-1.5\sigma$; App.~AV, Step~5b), its only
non-DFD input the standard cosmological relic-redshift
under the 100-item complete-programmatic-closure roadmap (composition recorded in the T211/T230 development syntheses)
($91 + 2~\mathrm{declined~anti{\text{-}}theorems}
+ 7~\mathrm{out{\text{-}}of{\text{-}}scope} = 100$),
\textbf{$\sim$190 cumulative theorems},
\textbf{16 anti-theorems},
\textbf{$\sim$80 Lean stubs (planned)} with 5 discharged lemmas
(statement-only stubs use \texttt{sorry} bodies),
\textbf{$\sim$175+ active falsifiers} indexed by T190V / T200V / T211V / T230V enumerations,
27 Python modules, $\sim$9 external residuals, and
\textbf{0 internal Tier-1--6 mathematical gaps} (complete programmatic closure).
Claims should be evaluated by the tier labels and falsifier IDs in the master ledger, not as a
single undifferentiated claim.

\bigskip

\noindent\textbf{Companion volume.} The extended derivations --- theorem-grade
closures and promotions, the new-territory frontier, the CKM/$\tau$ and series
IV--VI derivations, and the mathematical-foundation and quantum-gravity closures ---
are documented separately in \emph{Density Field Dynamics v4.0: Extended
Derivations and Frontier Predictions}. The
present paper contains the final theory only.

\medskip

\onecolumngrid
\noindent\textbf{Revision ledger (principal corrections in the current release; editorial pass of 9 June 2026).}
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Earlier statement & Current (final) status\\
\midrule
\parbox[t]{0.40\linewidth}{CKM apex $\bar\rho=19\alpha$} & \parbox[t]{0.52\linewidth}{Superseded by the strengthened DFD--SD branch, $\bar\rho=\tfrac{43}{2}\alpha$ (Euler-projection postulate; App.~\ref{app:ckm-update}).}\\[5pt]
\parbox[t]{0.40\linewidth}{Minimal real DFD gives CKM CP} & \parbox[t]{0.52\linewidth}{Minimal real kernel gives $J=0$ (theorem); nonzero $J$ requires the DFD--SD det-orthogonal offset.}\\[5pt]
\parbox[t]{0.40\linewidth}{Direct $k_a a^2/c^2$ MOND source} & \parbox[t]{0.52\linewidth}{Replaced by the nonlinear kinetic crossover; rotation curves from the AQUAL field equation (App.~\ref{app:alpha57-closure}).}\\[5pt]
\parbox[t]{0.40\linewidth}{$a_0$ calibrated from SPARC/RAR} & \parbox[t]{0.52\linewidth}{$a_0=2\alpha^{29}\sqrt{c^7/G\hbar}$ derived from constants; SPARC validates, not calibrates.}\\[5pt]
\parbox[t]{0.40\linewidth}{Black-hole base temperatures} & \parbox[t]{0.52\linewidth}{Retired in the minimal optical-exponential branch (no finite-radius horizon); nonminimal-branch only.}\\[5pt]
\parbox[t]{0.40\linewidth}{2PN lensing ``matches GR exactly''} & \parbox[t]{0.52\linewidth}{Branch-labeled: 1PN is branch-safe theorem; the 2PN coefficient is branch-dependent.}\\[5pt]
\parbox[t]{0.40\linewidth}{``$0.55\%$ CKM mean agreement''} & \parbox[t]{0.52\linewidth}{Retracted (computed against PDG 2018--2020); per-channel agreement reported instead.}\\[5pt]
\parbox[t]{0.40\linewidth}{Fermion-mass mean error quoted as ``$0.047\%$''} & \parbox[t]{0.52\linewidth}{Corrected (June 2026) to the table-derived $1.42\%$ at leading order; the refined $0.61\%$/$0.082\%$ values are archive-claimed (priming/RGE theorems). Forensic note: the printed $0.047\%$ traces to a further archive-stage statistic (an archived refinement-stage table, T95) never integrated into this manuscript and propagated onto the leading-order table by a global edit; it reproduces as the nine-charged-fermion mean of that archive table with the top-quark residual in pole convention ($0.0473\%$).}\\[5pt]
\parbox[t]{0.40\linewidth}{Matter-wave $T^3$ protocol (App.~W) screened-coupling form} & \parbox[t]{0.52\linewidth}{Harmonized (June 2026) to the derived kinematic phase of the matter-wave section; per-facility estimates and the falsification threshold restated from the derived scaling.}\\[5pt]
\parbox[t]{0.40\linewidth}{Quantum framework unlisted among inputs} & \parbox[t]{0.52\linewidth}{Added (June 2026) as an explicit background-input row in the DoF ledger and as an open program item; no derivation of QM is claimed.}\\[5pt]
\parbox[t]{0.40\linewidth}{``Five independent routes'' to $\alpha^{-1}$} & \parbox[t]{0.52\linewidth}{Reconciled (June 2026): the companion volume's independence statement now defers to the categorical engine count --- three structurally distinct engines, five presentations.}\\[5pt]
\parbox[t]{0.40\linewidth}{Glueball normalization chain (companion volume)} & \parbox[t]{0.52\linewidth}{Correction note added (June 2026): the printed coefficient forms do not close numerically and the flavor-scheme input is mislabeled; $m_{0^{++}}=1.69\pm0.10$~GeV retained as target, scaling $m\propto\Lambda_{\rm QCD}$ retained.}\\[5pt]
\parbox[t]{0.40\linewidth}{Pairwise-kSZ consistency stated unconditionally} & \parbox[t]{0.52\linewidth}{Growth-regime note added (June 2026): the verdict rests on the Hubble-EFE envelope; the zero-external-field regime instead predicts suppressed amplitude (App.~\ref{app:galaxy-bias}); adjudication is a named open item.}\\
\bottomrule
\end{tabular}
\end{center}
\twocolumngrid

\bigskip

\tableofcontents

%=============================================================================
% PART I: FOUNDATIONS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 1: INTRODUCTION
%-----------------------------------------------------------------------------
\input{section_introduction}

%-----------------------------------------------------------------------------
% SECTION 2: MATHEMATICAL FORMALISM
%-----------------------------------------------------------------------------
\input{section_formalism}

%-----------------------------------------------------------------------------
% SECTION 3: MATHEMATICAL WELL-POSEDNESS
%-----------------------------------------------------------------------------
\input{section_wellposedness}

%=============================================================================
% PART II: CONTACT WITH KNOWN PHYSICS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 4: PPN ANALYSIS
%-----------------------------------------------------------------------------
\input{section_ppn}

%-----------------------------------------------------------------------------
% SECTION 5: GRAVITATIONAL WAVES
%-----------------------------------------------------------------------------
\input{section_gravitational_waves}

%=============================================================================
% PART III: STRONG FIELDS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 6: STATIC STRONG FIELDS
%-----------------------------------------------------------------------------
\input{section_strong_fields}

%=============================================================================
% PART IV: GALACTIC DYNAMICS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 7: DEEP-FIELD REGIME
%-----------------------------------------------------------------------------
\input{section_galactic}

%=============================================================================
% PART V: THE α-RELATIONS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 8: PARAMETER-FREE NUMERICAL RELATIONS
%-----------------------------------------------------------------------------
\input{section_alpha_relations}

%-----------------------------------------------------------------------------
% SECTION 8B: GAUGE COUPLING VARIATION AND HIGH-ENERGY IMPLICATIONS
%-----------------------------------------------------------------------------
\input{section_gauge_coupling}

%-----------------------------------------------------------------------------
% SECTION 8C: CONVENTION-LOCKED α FROM THE MICROSECTOR
%-----------------------------------------------------------------------------
\input{section_alpha_locked}

%=============================================================================
% PART VI: LABORATORY TESTS AND PREDICTIONS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 9: ATOMIC CLOCK PHENOMENOLOGY
%-----------------------------------------------------------------------------
\input{section_clocks}

%-----------------------------------------------------------------------------
% SECTION 10: CAVITY-ATOM LPI TEST
%-----------------------------------------------------------------------------
\input{section_cavity_atom}

%-----------------------------------------------------------------------------
% SECTION 11: MATTER-WAVE INTERFEROMETRY
%-----------------------------------------------------------------------------
\input{section_matter_wave}

%-----------------------------------------------------------------------------
% SECTION 11A: SOLAR CORONAL TESTS
%-----------------------------------------------------------------------------
\input{section_solar_corona}

%-----------------------------------------------------------------------------
% SECTION 11B: ANTIMATTER GRAVITY TESTS
%-----------------------------------------------------------------------------
\input{section_antimatter}

%=============================================================================
% PART VII: FRONTIERS AND OPEN PROBLEMS
%=============================================================================

%-----------------------------------------------------------------------------
% SECTION 12: COSMOLOGICAL IMPLICATIONS
%-----------------------------------------------------------------------------
\input{section_cosmology}

%-----------------------------------------------------------------------------
% SECTION 13: QUANTUM AND GAUGE SECTOR (CONDITIONAL)
%-----------------------------------------------------------------------------
\input{section_quantum}

%-----------------------------------------------------------------------------
% SECTION 14: OPEN PROBLEMS AND LIMITATIONS
%-----------------------------------------------------------------------------
\input{section_openproblems}

%-----------------------------------------------------------------------------
% SECTION 15: DERIVATION OF THE GRAVITATIONAL CONSTANT
%-----------------------------------------------------------------------------
\input{section_G_derivation}

%-----------------------------------------------------------------------------
% SECTION 16: CONCLUSIONS AND OUTLOOK
%-----------------------------------------------------------------------------
\input{section_conclusions}

%=============================================================================
% APPENDICES
%=============================================================================
\appendix

%-----------------------------------------------------------------------------
% APPENDIX A: NOTATION AND CONVENTIONS
%-----------------------------------------------------------------------------
\input{appendix_A}

%-----------------------------------------------------------------------------
% APPENDIX B: DETAILED DERIVATIONS
%-----------------------------------------------------------------------------
\input{appendix_B}

%-----------------------------------------------------------------------------
% APPENDIX C: $\mu$-FUNCTION CATALOG
%-----------------------------------------------------------------------------
\input{appendix_C}

%-----------------------------------------------------------------------------
% APPENDIX D: EXPERIMENTAL PROTOCOLS
%-----------------------------------------------------------------------------
\input{appendix_D}

%-----------------------------------------------------------------------------
% APPENDIX E: DATA TABLES
%-----------------------------------------------------------------------------
\input{appendix_E}

%-----------------------------------------------------------------------------
% APPENDIX F: RIGOROUS FOUNDATIONS FOR GAUGE EMERGENCE
%-----------------------------------------------------------------------------
\input{appendix_F}

%-----------------------------------------------------------------------------
% APPENDIX G: DERIVATION OF α-RELATIONS FROM GAUGE EMERGENCE
%-----------------------------------------------------------------------------
\input{appendix_G}

%-----------------------------------------------------------------------------
% APPENDIX H: HIGGS AND YUKAWA SECTOR FROM GAUGE EMERGENCE
%-----------------------------------------------------------------------------
\input{appendix_H}

%-----------------------------------------------------------------------------
% APPENDIX I: FULL CLUSTER SAMPLE ANALYSIS
%-----------------------------------------------------------------------------
\input{appendix_I}

%-----------------------------------------------------------------------------
% APPENDIX J: DERIVATION OF THE ψ-CMB SOLUTION
%-----------------------------------------------------------------------------
\input{appendix_J}

%-----------------------------------------------------------------------------
% APPENDIX K: MICROSECTOR PHYSICS
%-----------------------------------------------------------------------------
\input{appendix_K}

%-----------------------------------------------------------------------------
% APPENDIX L: STRONG CP SELECTION RULE
%-----------------------------------------------------------------------------
\input{appendix_L_strongcp}

%-----------------------------------------------------------------------------
% APPENDIX M: DOUBLE-TRANSIT ENHANCEMENT ANALYSIS
%-----------------------------------------------------------------------------
\input{appendix_M_double_transit}

% Appendix N: μ(x) derivation
\newpage
\input{appendix_N_mu_derivation}

%-----------------------------------------------------------------------------
% APPENDIX O: α^{57} MODE-COUNT EXPONENT AND G-H0-α DICTIONARY
%-----------------------------------------------------------------------------
\newpage
\input{appendix_O_alpha57}

%-----------------------------------------------------------------------------
% APPENDIX P: CLOCK COUPLING AND MAJORANA SCALE
%-----------------------------------------------------------------------------
\newpage
\input{appendix_P_kalpha_MR}

%-----------------------------------------------------------------------------
% APPENDIX Q: TEMPORAL COMPLETION AND MATTER POWER SPECTRUM
%-----------------------------------------------------------------------------
\newpage
\input{appendix_Q_temporal_completion}

%-----------------------------------------------------------------------------
% APPENDIX R: EM-ψ BACK-REACTION COUPLING
%-----------------------------------------------------------------------------
\newpage
\input{appendix_R_em_psi_coupling}

%-----------------------------------------------------------------------------
% APPENDIX S: STANDARD MODEL EXTENSION DICTIONARY
%-----------------------------------------------------------------------------
\newpage
\input{appendix_S_sme_dictionary}

%-----------------------------------------------------------------------------
% APPENDIX T: FAMILY AND CLOCK-TYPE PARAMETRIZATION
%-----------------------------------------------------------------------------
\newpage
\input{appendix_T_family_clock}

%-----------------------------------------------------------------------------
% APPENDIX U: MATHEMATICAL WELL-POSEDNESS
%-----------------------------------------------------------------------------
\newpage
\input{appendix_U_wellposedness}

%-----------------------------------------------------------------------------
% APPENDIX V: EXTENDED PHENOMENOLOGY AND NUMERICAL METHODS
%-----------------------------------------------------------------------------
\newpage
\input{appendix_V_extended_phenom}

%-----------------------------------------------------------------------------
% APPENDIX W: EXPERIMENTAL PROTOCOLS AND SENSITIVITY ANALYSES
%-----------------------------------------------------------------------------
\newpage
\input{appendix_W_protocols}

%-----------------------------------------------------------------------------
% APPENDIX X: NEUTRINO MASS SPECTRUM FROM DFD MICROSECTOR
%-----------------------------------------------------------------------------
\newpage
\input{appendix_X_neutrino}

%-----------------------------------------------------------------------------
% APPENDIX Y: FINITE YUKAWA OPERATOR, A_f PREFACTORS, AND A_5 SPECIES PROJECTORS
%-----------------------------------------------------------------------------
\newpage
\input{appendix_Y_finite_yukawa}

%-----------------------------------------------------------------------------
% APPENDIX Z: COMPLETE PARAMETER DERIVATION
%-----------------------------------------------------------------------------
\input{appendix_Z_complete_params}

%-----------------------------------------------------------------------------
% APPENDIX AA: PADE IDENTITY % GR as the [1,1]^2 Pade approximant of DFD's exponential lapse-squared
%-----------------------------------------------------------------------------
\newpage
% Override section numbering: after Z, use AA, AB (double letters) manually
\renewcommand{\thesection}{AA}
\setcounter{section}{0}
\input{appendix_AA_pade}

%-----------------------------------------------------------------------------
% APPENDIX AB: UNIQUENESS OF THE INTERNAL MANIFOLD % Deriving CP^2 x S^3 from vacuum axioms
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AB}
\setcounter{section}{0}
\input{appendix_AB_uniqueness}

%-----------------------------------------------------------------------------
% APPENDIX AC: LINEAR GROWTH CLOSURE % EFE-screened first-order growth theorem
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AC}
\setcounter{section}{0}
\input{appendix_AC_linear_growth}

%-----------------------------------------------------------------------------
% APPENDIX AD: REGULAR-MODULE FORCING % Minimal faithful real-bimodule theorem
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AD}
\setcounter{section}{0}
\input{appendix_AD_bimodule}

%-----------------------------------------------------------------------------
% APPENDIX AE: OPTICAL-METRIC SIGMA_8 CLOSURE % Theorem-grade closure of the spherical top-hat sigma_8 statistic from the
% FRW-derived no-EFE deep-MOND optical lensing inversion.
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AE}
\setcounter{section}{0}
\input{appendix_AE_sigma8_optical}

\newpage
\renewcommand{\thesection}{GR}
\setcounter{section}{0}
\input{appendix_GR_growth_regime}

\newpage
\renewcommand{\thesection}{AF}
\setcounter{section}{0}
\input{appendix_AF_galaxy_bias}

\newpage
\renewcommand{\thesection}{AG}
\setcounter{section}{0}
\input{appendix_AG_confrontation_matrix}

%-----------------------------------------------------------------------------
% [Appendices AH--AM (wave-closure theorems, new-territory frontier, cross-validation
%  audits, and the development trail) are RELOCATED to the companion volume
%  the Extended Derivations companion volume
%  (DFD_Closure_Archive.tex). The main paper carries only the final theory.]
%-----------------------------------------------------------------------------

%-----------------------------------------------------------------------------
% APPENDIX AN: THE OPTICAL-METRIC MASTER EQUATION
% Unifies light propagation (c_1 = c/n) and matter acceleration
% (a = (c^2/2) grad psi) as the null and timelike sectors of one wave equation.
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AN}
\setcounter{section}{0}
\input{appendix_AN_master_equation}

%-----------------------------------------------------------------------------
% APPENDIX AO: THE CP SECTOR (FINAL)
% Consolidated CP-sector theorem package: real-kernel no-go (J=0);
% det-orthogonal strong-CP protection (theta-bar = 0, Dai-Freed anomaly form,
% no axion); Y.10 det-orthogonal geometric CP offset; cohomological magnitude
% skeleton (lambda=31a, A=108a, etabar=49a); Euler-projection postulate fixing
% the apex rhobar = (43/2)a; predictions gamma = 66.3093 deg, J = 2.9731e-5.
% Supersedes the old symplectic-obstruction / gamma = 4pi/11 treatment and the
% delta_CP ~ 68 deg estimate of Appendix K.
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AO}
\setcounter{section}{0}
\input{appendix_AO_ckm_update}

%-----------------------------------------------------------------------------
% APPENDIX AP: CORPUS-CLOSURE THEOREMS (alpha^57 hierarchy, MOND scale, A+ rigor)
% Full theorem package for the gravitational/MOND sector: alpha^57 clock
% hierarchy (60-3=57 mode count + primed Gaussian determinant), action-derived
% prefactor a0 = 2 sqrt(alpha) c H0, A12 invariant a0^2 G hbar/c^7 = 4 alpha^58,
% k_a = 3/(8 alpha) gauge-emergence, S^3 stationarity Xi* = 3/2, AQUAL kinetic
% galaxy law, and the A+ rigor discipline (branch separation, BH-temperature
% retirement, A23 repair, 2PN branch labelling, CKM J=0 minimal/DFD-SD).
%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AP}
\setcounter{section}{0}
\input{appendix_AP_alpha57_closure}

%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AQ}
\setcounter{section}{0}
\input{appendix_AQ_qcd_lattice}

%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AR}
\setcounter{section}{0}
\input{appendix_AR_ym_gap}

%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AS}
\setcounter{section}{0}
\input{appendix_AS_lattice_provenance}

%-----------------------------------------------------------------------------
\newpage
\renewcommand{\thesection}{AT}
\setcounter{section}{0}
\input{appendix_AT_new_results}

\renewcommand{\thesection}{AU}
\setcounter{section}{0}
\input{appendix_AU_dark_energy}

\newpage
\renewcommand{\thesection}{AV}
\setcounter{section}{0}
\input{appendix_chi_matter}

\newpage
\renewcommand{\thesection}{AW}
\setcounter{section}{0}
\input{appendix_AW_ew_precision}

\newpage
\renewcommand{\thesection}{AX}
\setcounter{section}{0}
\input{appendix_AX_rotating_exterior}

\newpage
\renewcommand{\thesection}{AY}
\setcounter{section}{0}
\input{appendix_AY_crosssector}

%=============================================================================
% TOP-20 OPEN-PROBLEMS RESOLUTION APPENDICES (added 2026-06-27)
%=============================================================================
\renewcommand{\thesection}{SL}
\setcounter{section}{0}
\input{appendix_SL_snlensing_discriminator}

\renewcommand{\thesection}{TP}
\setcounter{section}{0}
\input{appendix_TP_third_peak_resolution}

\renewcommand{\thesection}{QM}
\setcounter{section}{0}
\input{appendix_QM_derivation_status}

\newpage
\renewcommand{\thesection}{PK}
\setcounter{section}{0}
\input{appendix_PK_pipeline}

\newpage
\renewcommand{\thesection}{YM}
\setcounter{section}{0}
\input{appendix_YM_massgap_status}

\newpage
\renewcommand{\thesection}{FM}
\setcounter{section}{0}
\input{appendix_FM_fermion_masses}

\newpage
\renewcommand{\thesection}{LS}
\setcounter{section}{0}
\input{appendix_LS_los_screen}

\newpage
\renewcommand{\thesection}{CL}
\setcounter{section}{0}
\input{appendix_CL_cmb_likelihood}

\newpage
\renewcommand{\thesection}{FK}
\setcounter{section}{0}
\input{appendix_FK_fitted_knobs}

\renewcommand{\thesection}{SM}
\setcounter{section}{0}
\input{appendix_SM_scalaron_mass}

\newpage
\renewcommand{\thesection}{CE}
\setcounter{section}{0}
\input{appendix_CE_combined_evidence}

\newpage
\renewcommand{\thesection}{RM}
\setcounter{section}{0}
\input{appendix_RM_predictions_roadmap}

%=============================================================================
% ACKNOWLEDGMENTS
%=============================================================================

\begin{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.
\end{acknowledgments}

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