This page is a navigational index for researchers and AI assistants looking for a specific result, formula, or derivation in the Density Field Dynamics corpus. Each entry names the topic, gives the formula or result as it appears in the source paper, and links to the paper where the full derivation lives.
Schwarzschild light deflection — second-order coefficient $15\pi/4$
Expanding the optical-metric path integral $T = c^{-1}\int e^\psi\,ds$ to $O(\psi^2)$ for a point mass yields:
$\alpha = 4\epsilon + \dfrac{15\pi}{4}\,\epsilon^2 + O(\epsilon^3)$, with $\epsilon = GM/(c^2 b)$
matching the standard 2PN Schwarzschild coefficient (Epstein & Shapiro 1980; Will, Theory and Experiment in Gravitational Physics). In DFD this is not enforced as a constraint — it falls out of the Padé identity.
Why DFD reproduces it automatically. The lapse-squared scalar in GR's isotropic-coordinate Schwarzschild form satisfies the exact identity
$L_{\rm GR}(u) = \left[\dfrac{1+u/2}{1-u/2}\right]^2 = \left[P_{1,1}(e^u)\right]^2$, $L_{\rm DFD}(u) = e^{2u}$, $u = GM/(\rho c^2)$
GR is the $m=1$ slot of a Padé hierarchy whose entire-function limit is $e^{2u}$. The two agree through $O(u^2)$ by construction, so DFD reproduces $4\epsilon + (15\pi/4)\epsilon^2$ structurally — no per-result tuning. In v5.0 the agreement goes further: with the scalar-longitudinal piece of the field kept, the exterior of a static star is exactly Schwarzschild in DFD's coordinates, so the earlier $O(u^3)$ shadow difference is retired.
Comparison with other exponential-lapse formulations. Several prior frameworks also produce exponential lapses (Papapetrou 1954, Yilmaz 1958, Dicke 1957, Puthoff polarisable vacuum 1999/2002, Broekaert 2008), so the Padé identity itself holds for any of them. What is structurally distinctive about DFD: it is not embedded in GR (the Yilmaz exponential, for example, is itself a GR solution interpretable as a wormhole with exotic matter), and DFD's flat-$\mathbb{R}^3$ elliptic field equation admits no throat and requires no exotic matter.
Source: General Relativity as the Padé Approximant of DFD · Unified Derivations v1.2
Padé identity: GR as the $[P_{1,1}(e^u)]^2$ truncation of DFD
An exact mathematical identity in the lapse-squared scalar:
$L_{\rm GR}(u) = \left[\dfrac{1+u/2}{1-u/2}\right]^2 = \left[P_{1,1}(e^u)\right]^2$, $L_{\rm DFD}(u) = e^{2u}$, with $u = GM/(\rho c^2)$
GR is the $m=1$ slot of a Padé hierarchy; DFD is the entire-function limit. The Schwarzschild horizon at $r = 2GM/c^2$ is the Padé pole of the $m=1$ truncation. The pure exponential would put no horizon there, but v5.0 keeps the scalar-longitudinal piece of the parent strain, and the Einstein–Hilbert term then makes the exterior of a static star the Schwarzschild metric in DFD's coordinates. The pole is therefore a horizon of DFD's black holes as well.
Source: General Relativity as the Padé Approximant of Density Field Dynamics
Parametrised Post-Newtonian parameters $\gamma$, $\beta$
Full PPN expansion in the weak-field, slow-motion limit yields all ten PPN parameters at 1PN order:
$\gamma = \beta = 1$, $\xi = \alpha_1 = \alpha_2 = \alpha_3 = \zeta_1 = \zeta_2 = \zeta_3 = \zeta_4 = 0$
DFD is observationally indistinguishable from GR for current solar-system tests at 1PN.
Source: Parametrized Post-Newtonian Analysis of Density Field Dynamics
Nordtvedt parameter $\eta$ and lunar laser ranging constraint
Lunar laser ranging constrains the Nordtvedt parameter and $\dot G/G$. DFD's structure satisfies $|\dot G/G| < 10^{-13}\,{\rm yr}^{-1}$ consistent with a fixed vacuum stiffness. Discussed alongside the equivalence-principle implications.
Source: Variant Extensions · Unified v5.0
Black holes: Schwarzschild and Kerr horizons
In the rest-mass sector, v5.0 shows that DFD's gravity is general relativity's written in absolute time. Static stars obey the TOV equation, black holes have Schwarzschild's and Kerr's horizons (Kerr beyond first order in the spin is argued), and two-body dynamics is GR's through second post-Newtonian order. The earlier prediction of a shadow about 4.6% larger than GR's is retired. The same gravity completion makes Newton's law an exact zero-parameter identity of the quantized $\psi$ sector, with a Kohn-null laboratory discriminator.
Source: Unified v5.0 · General Relativity as the Padé Approximant of DFD
Tensor gravitational-wave speed $c_T = c$ exactly
$\psi$ and the transverse-traceless tensor $h^{\rm TT}_{ij}$ are derived as irreducible components of a single zero-mode parent tensor on CP²×S³. Lichnerowicz rigidity (CP² gap $= 8/R_1^2$, S³ gap $= 12/R_2^2$, $b_1 = 0$) rules out unwanted scalar/vector graviton modes, and the result $c_T = c$ holds exactly — satisfying the GW170817 constraint $|c_T - c|/c < 10^{-15}$.
Source: Constitutive Derivation of Tensor Gravitational Radiation
Well-posedness of the $\psi$ equation
Rigorous PDE analysis of the DFD field equation establishes existence, uniqueness, and regularity of weak solutions in appropriate Sobolev spaces, with energy conservation and asymptotic boundary conditions derived.
Source: Well-Posedness of the ψ Equation
Electromagnetic stiffness of the vacuum $K_0$
The vacuum is treated as a physical storage medium with vacuum force scale
$K_0 = \dfrac{c^4}{8\pi G} \approx 4.82 \times 10^{42}\,{\rm N}$, equivalently $K_0 = \dfrac{\hbar H_0^2}{8\pi\,\alpha^{57}\,c}$
The dimensionless field $\psi$ measures fractional loading; the optical refractive index is $n = e^\psi$. The DFD field equation is interpreted as a stress–strain constitutive equilibrium of the vacuum.
Source: Physical Origin of the Refractive Field
Cauchy functional-equation uniqueness of $n = e^\psi$
The exponential form $n = e^\psi$ is proved unique under multiplicative composition of successive vacuum loadings, via Cauchy's exponential functional equation.
Source: Physical Origin of the Refractive Field
Fine-structure constant $\alpha^{-1} = 137.036$
A closed form for $\alpha^{-1}$ built from Chern–Simons data on CP²×S³:
$\alpha^{-1} = \dfrac{\pi^{3/2}}{24}\,\mathrm{Tr}(Y^2)\,k_{\max}\,\dfrac{k_{\max}+3}{k_{\max}+4}\,\left[1 + \dfrac{N_{\rm sp}}{g_F\,\mathrm{Tr}(Y^2)\,((k_{\max}+4)^2-1)}\right]$
With $\mathrm{Tr}(Y^2)=10$, $k_{\max}=60$, $N_{\rm sp}=7$, $g_F=8$, this gives $\alpha^{-1} = 137.03599985$, 4.9 ppb from CODATA 2022. The step that identifies the hypercharge-trace coefficient with $1/e^2$ is stated as not derived, so v5.0 calls the result a closed form, not a derivation.
Source: Ab Initio Derivation of the Fine-Structure Constant · DOI 10.5281/zenodo.18045109
SU(2) Chern–Simons quantum shift & Bridge Lemma
The original paper connects a UV cutoff $k_{\max} = 62$ to the topological coefficient $b = 60$ from the heat kernel on CP². The connection is the quantum shift $k \to k + h^\vee$ in SU(2) Chern–Simons theory, where $h^\vee = 2$ is the dual Coxeter number:
$b = k_{\max} - h^\vee$
In v5.0, $k_{\max} = 60$ follows from a determinant-line count on CP², conditional on stated inputs including the electroweak identification, with no simulation input. It links the $\alpha$ closed form to the fermion mass tower.
Source: The Bridge Lemma
Uniqueness of CP²×S³ as the internal manifold
Six-axiom uniqueness theorem proving CP²×S³ is the unique internal manifold for DFD's spectral completion. The product structure $K = K_C \times K_G$ is forced by the logical incompatibility of chirality ($w_2 \ne 0$) with Lie-group parallelisability ($w_2 = 0$) on a single connected manifold; Matsushima–Lichnerowicz rules out CP²#CP²; Cartan classification fixes $S^3 \cong SU(2)$.
Source: Uniqueness of the Internal Manifold · DOI 10.5281/zenodo.19415584
Charged fermion mass spectrum (9 masses)
The nine charged fermion masses (e, μ, τ, u, d, s, c, b, t) from CP²×S³ topology, with A₅ class geometry for amplitude factors and Spinᶜ bundle degrees for the power-law exponents, which are derived:
$m_f = A_f\,\alpha^{n_f}\,\dfrac{v}{\sqrt{2}}$
v5.0 reports a 1.42% leading-order mean error across three orders of magnitude in mass. Seven of the prefactors are data-selected and labelled as such.
Source: Ab Initio Charged Fermion Mass Spectrum
CKM Wolfenstein parameters from CP² geometry
The CKM Wolfenstein magnitude integers derived from CP² line-bundle cohomology, with the apex fixed by the Euler-projection postulate; v5.0 reports per-channel agreement (λ at +1.8σ, |V_cb| at −1.75σ) and predicts γ = 66.31°. The earlier “0.55% mean agreement” summary statistic is retracted.
Source: Quark Mixing from CP² Geometry
Strong CP angle $\bar\theta = 0$ without axion
In v5.0, $\bar\theta = 0$ holds at tree level, with $\theta_{\rm YM} = 0$ fixed by the spectral action, and loop corrections are bounded ($\lesssim 10^{-16}$). No axion is required.
Source: Unified v5.0 Appendix L
Higgs VEV $v = M_P\,\alpha^8\,\sqrt{2\pi}$
The electroweak vacuum expectation value is written as $v = M_P\,\alpha^8\,\sqrt{2\pi} \approx 246.09$ GeV, 0.05% from measurement. The exponent is derived and the $\sqrt{2\pi}$ prefactor is fitted. The conjectured Higgs quartic $1/8$ gives $m_H = 124.6$ GeV, 0.40% low.
Source: Unified v5.0
Neutrino mass-squared splittings
Predicted $\Delta m^2$ splittings reported as $\chi^2 = 0.025$, $p = 0.99$ against NuFIT 6.0 in the cited fit; $\Sigma m_\nu = 61.5$ meV, $m_\beta = 9.15$ meV.
Source: DFD Neutrino Paper v7 — S2 Seesaw Closure · Unified v5.0
Cooper-pair mass anomaly $\delta = \sqrt{3}\,\alpha^2$
Predicted Cooper-pair mass-anomaly coefficient:
$\delta = \sqrt{3}\,\alpha^2 = 92.23\;{\rm ppm}$
derived from A₅ microsector representation theory with pairing-symmetry selection rules. Tate et al. (1989) measured $92 \pm 21$ ppm in niobium. The framework predicts universality for s-wave superconductors and vanishing for d-wave.
Source: Cooper-Pair Selection Rule
Hubble constant $H_0 = 72.09$ km/s/Mpc
Read from the master invariant of the spectral action:
$\dfrac{G\,\hbar\,H_0^2}{c^5} = C\,\alpha^{57}$
The exponent 57 is derived via three lemmas (Kaluza–Klein reduction, uniform gauge normalisation with eigenvalue cancellation, finite-dimensionality hierarchy identification); the coefficient $C$ rests on one measured input. On bound-matter clocks ($C = 1$) the G derivation gives $H_0 = 72.09$ km/s/Mpc with no parameter, and seven Cepheid-free local measurements give $71.87 \pm 0.96$. v5.0 reads both measured rates as right through one stated two-clock postulate: the early clock reads $H_0/\lambda$, and a stated same-era rule predicts $H_0^{\rm early} = 67.79$ km/s/Mpc. Deriving the size of $\lambda$ is open.
Source: Unified v5.0 Appendix ZS
Cosmological constant ratio $(3/8\pi)\,\alpha^{57}$
The cosmological-constant hierarchy:
$\dfrac{\rho_c}{\rho_{\rm Pl}} = \dfrac{3}{8\pi}\,C\,\alpha^{57}$
Spans 122.7 orders of magnitude. The exponent follows from $\alpha^{57}$ mode counting in the spectral action and the coefficient $C$ is measured. This vacuum term is DFD's only dark energy.
Source: Unified v5.0 Appendix AU · Alpha Rosetta Stone
MOND crossover scale $a_\star = 2\sqrt{\alpha}\,cH_0$
The galactic transition acceleration is tied to the cosmic Hubble rate by the structural identity:
$a_\star = 2\sqrt{\alpha}\,c H_0$
derived from S³ topology by variational stationarity of a coarse-grained IR functional, giving $1.1966 \times 10^{-10}$ m/s² on bound-matter clocks. It accounts for the long-noted $a_0 \sim cH_0$ proximity.
Source: Two Numerical Relations · Unified v5.0 Appendix N
Epoch evolution $a_\star(z) = 2\sqrt{\alpha}\,cH(z)$
Promotes the $z=0$ result to all epochs via the same epoch-consistency rule used for $G\hbar H^2/c^5 = \alpha^{57}$. v5.0 predicts $a_\star(z)/a_\star(0) = H(z)/H_0$ exactly, 1.78 at $z = 1$ on the measured clock; the earlier matter-only value 2.83 is not a DFD prediction. The relation is published in MNRAS (doi:10.1093/mnras/stag1485) and stated as contested: its exclusion of a frozen $a_\star$ rests on the published bin errors, and massive discs at $z \approx 2$, with molecular gas counted, do not show the rise. JWST-era rotation curves will test it.
Source: Epoch Evolution of the MOND Crossover Scale
SPARC galactic shape exponent $n_{\rm opt} = 1.15 \pm 0.12$
Model-independent shape analysis of SPARC rotation curves yields $n_{\rm opt} = 1.15 \pm 0.12$ (95% CI [1.00, 1.50]) — disfavouring MOND's $n=2$ in the same analysis. For clusters, v5.0 finds that the μ-law alone falls short (relaxed 1.57 ± 0.08, merging 1.99 ± 0.16) and that the shortfall is the derived $\chi$ at 0.84 ± 0.12 of the cosmic mix (an estimate); the earlier no-$\chi$ cluster calibration is not supported.
Source: Unified v5.0 galactic section
The CMB from the derived $\chi$ on the measured clock
The original $\psi$-screen paper reconstructed the large-scale CMB features with a global $\psi$ screen. v5.0 retires that global map: the optical metric leaves $\theta_*$ exactly invariant, and DES's detection of ordinary supernova lensing retired the optical-distance dressing. The acoustic peaks are carried by the derived $\chi$ component ($m_\chi = 32.005$ eV, $\Omega_\chi h^2 = 0.1193$, −0.6σ, conditional on the flux-period identification) rather than a fitted CDM density. On the measured clock the peak ratio is 2.20 (Planck 2.21), the third-peak height is $H_3/H_2 = 0.977$, $H_3/H_1 = 0.444$ (within 0.6σ), and $100\theta_* = 1.04110$, $\ell_1 \approx 220.4$.
Source: Unified v5.0 · ψ-Screen Cosmology · Large-Scale Power Suppression
Mach's principle: Bondi–Samuel taxonomy mapping
Against the Bondi–Samuel taxonomy: DFD satisfies Mach 3, Mach 10, and effectively Mach 8; partially satisfies Mach 1, 2, 6; fails Mach 4, 5, 7, 9. The $\psi$ rest frame is the operationally preferred frame, dynamically determined by cosmic matter; the flat $\mathbb{R}^3$ kinematic substrate remains absolute.
Source: Mach's Principle in DFD
ROCIT solar-locked frequency-ratio modulation
Analysis of public ROCIT 2022 data identifies a coherent perihelion-phase modulation in the Yb⁺/Sr ion–neutral ratio:
$A = (-1.045 \pm 0.078) \times 10^{-17}$, $Z = 13.47\sigma$, $p_{\rm emp} \approx 2 \times 10^{-4}$
with phase-consistent residuals in Yb/Sr (neutral–neutral) at $Z = 3.7\sigma$ and Yb³⁺/Sr (cavity–atom) at $Z = 13.5\sigma$. SYRTE control ratios (Rb/Cs, Yb/Rb, Yb/Cs) consistent with zero modulation, as predicted for facility/architecture-specific signature. The v5.0 book treats ROCIT-type results as suggestive rather than definitive: they motivate replication, and in the core theory co-located clocks stay in step.
Source: ROCIT Ion–Neutral Differential · ROCIT Cavity–Atom Modulation
SOHO/UVCS bright–dim Lyman-α asymmetry
Permutation-test analysis of 334 daily SOHO/UVCS Lyman-α sequences (2007–2009): 163/321 (51%) day–radial bins exhibit statistically significant bright–dim intensity asymmetry, FDR-controlled. Released as an anomaly inviting independent investigation. The earlier double-transit exponent $\Gamma = 4$ is retired in v5.0; DFD predicts the standard coronal ratio.
Source: Unexplained Bright–Dim Asymmetries · DFD: Gravity is Light §12.3
Local Position Invariance (LPI) test design
Multiple papers develop the cavity–atom LPI test design, including a sharp slope prediction for the sector-resolved test, formal proofs of geometric cancellation in standard configurations, and protocol details for co-located cavity–atom frequency-ratio comparisons. In v5.0 the cavity–atom geometric cancellation is derived, and four independent data streams support it, including JILA's 205-day silicon-cavity/Sr record (arXiv:2608.00910). The sharpest extended-channel number is a perihelion-locked annual modulation of the Th-229 nuclear clock, about 350 Hz half-amplitude (minimal DFD: zero).
Source: Sharp Slope Prediction for Sector-Resolved LPI · Sector-Resolved LPI Test · Geometric Cancellation Proof · Optical-Metric LPI Test
Tree-level no-drive theorem $\lambda_{\rm bare} = 1$
In the minimal optical-metric EM sector, the linear-in-$\psi$ EM source produced by the gauge-invariant action is proportional to the energy density $(E^2/c^2 + B^2)/(2\mu_0)$ rather than the stress invariant $(E^2/c^2 - B^2)/(2\mu_0)$. For ideal standing-wave cavity modes with energy equipartition, this carries no $2\omega$ component after volume integration — yielding $\lambda_{\rm bare} = 1$ at tree level. The accidental laboratory bound $|\lambda - 1| < 3 \times 10^{-5}$ is consistent with the prediction.
Source: Tree-Level No-Drive Theorem · EM→ψ Back-Reaction Bounds
Horndeski-class translation of DFD
For readers familiar with scalar-tensor literature, DFD admits an embedding in the Horndeski class. The unified treatment provides the explicit Horndeski functions and notes that the embedding is a translation layer for comparison purposes; the fundamental DFD description is the flat-arena formulation.
Source: Unified v4.0
Poynting-like vector in scalar-refractive electrodynamics
Multiple papers develop Poynting-like energy-flux expressions in the scalar-refractive electrodynamics formulation, including the dual-sector treatment.
Source: Scalar Refractive Gravity & Dual-Sector EM · Unified Derivations v1.2
Scalar oscillation eigenfrequency $\omega_\psi$
The scalar field $\psi$ admits eigenfrequency $\omega_\psi$ for cavity-confined modes, used in the EM→ψ back-reaction analysis and in the c-field foundational paper.
Source: EM→ψ Back-Reaction Bounds · DFD and the c-Field