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. The first observable difference is at $O(u^3)$: a $\sim 4.6\%$ larger black-hole shadow than GR, testable with next-generation EHT.
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; in DFD it is a photon sphere with no horizon. Theories agree through $O(u^2)$ and first differ at $O(u^3)$.
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 v3.3
Black-hole shadow size and EHT discriminator
From the Padé identity, DFD predicts a black-hole shadow approximately 4.6% larger than the GR prediction, arising from the first-difference $O(u^3)$ between the exponential lapse $e^{2u}$ and the truncated rational form. EHT 2019/2022 data on M87* and Sgr A* are consistent with both at current precision; next-generation EHT would discriminate.
Source: 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 $\alpha^{-1} = 137.036$
Closed-form derivation of $\alpha^{-1}$ from Chern–Simons quantisation on CP²×S³ topology:
$\alpha^{-1} = \dfrac{\pi^{3/2}}{24}\,\mathrm{Tr}(Y^2)\,k_{\max}\,\dfrac{k_{\max}+3}{k_{\max}+4}\,\left[1 + \dfrac{N_{\rm sp}}{g_F\,\mathrm{Tr}(Y^2)\,((k_{\max}+4)^2-1)}\right]$
With $\mathrm{Tr}(Y^2)=10$, $k_{\max}=60$, $N_{\rm sp}=7$, $g_F=8$, this gives $\alpha^{-1} = 137.035999854$ — sub-ppm vs CODATA 2022. Independently lattice-verified at $L = 6\text{–}16$ in the cited Monte Carlo analysis.
Source: Ab Initio Derivation of the Fine-Structure Constant · DOI 10.5281/zenodo.18045109
SU(2) Chern–Simons quantum shift & Bridge Lemma
Connects two independently derived quantities in the DFD microsector framework: the UV cutoff $k_{\max} = 62$ from lattice Chern–Simons simulations and 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:
$b = k_{\max} - h^\vee$
A consistency check between the $\alpha$-derivation and the fermion-mass programme.
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)
All nine charged fermion masses (e, μ, τ, u, d, s, c, b, t) derived from CP²×S³ topology using A₅ class geometry for amplitude factors and Spinᶜ bundle degrees for power-law exponents:
$m_f = A_f\,\alpha^{n_f}\,\dfrac{v}{\sqrt{2}}$
Reported as 1.42% mean error across three orders of magnitude in mass, with no per-fermion fitting parameter.
Source: Ab Initio Charged Fermion Mass Spectrum
CKM Wolfenstein parameters from CP² geometry
All four Wolfenstein parameters of the CKM quark-mixing matrix derived from CP² geometry, with reported 0.55% mean agreement against PDG values.
Source: Quark Mixing from CP² Geometry
Strong CP angle $\bar\theta = 0$ without axion
The strong CP problem is resolved structurally — $\bar\theta = 0$ derived from internal-space topology — with no axion required.
Source: Unified v3.3 Appendix L
Higgs VEV $v = M_P\,\alpha^8\,\sqrt{2\pi}$
The electroweak vacuum expectation value is derived as $v = M_P\,\alpha^8\,\sqrt{2\pi} \approx 246.09$ GeV, with 0.05% agreement with measurement. The tree-level Higgs mass is reported at $m_H \approx 123$ GeV.
Source: Unified v3.3
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; $\Sigma m_\nu = 61.4$ meV.
Source: DFD Neutrino Paper v7 — S2 Seesaw Closure · Unified v3.3
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
Derived as a spectral-action consequence of the master invariant:
$\dfrac{G\,\hbar\,H_0^2}{c^5} = \alpha^{57}$
via three lemmas (Kaluza–Klein reduction, uniform gauge normalisation with eigenvalue cancellation, finite-dimensionality hierarchy identification). Within the SH0ES range; the early-vs-late H₀ tension is resolved within the DFD framework.
Source: Unified v3.3 §XIX.D
Cosmological constant ratio $(3/8\pi)\,\alpha^{57}$
The cosmological-constant hierarchy:
$\dfrac{\rho_c}{\rho_{\rm Pl}} = \dfrac{3}{8\pi}\,\alpha^{57}$
Spans 122.7 orders of magnitude with no fine-tuning input — derived as a consequence of $\alpha^{57}$ mode counting in the spectral action.
Source: Unified v3.3 · 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. Resolves the long-noted $a_0 \sim cH_0$ proximity within current uncertainty in $H_0$.
Source: Two Numerical Relations · Unified v3.3 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 $G\hbar H^2/c^5 = \alpha^{57}$. Predicted ratio $a_\star(z=1)/a_\star(0) \approx 1.79$ in a ΛCDM background, $\approx 2.83$ in DFD's matter-only $\psi$-screen cosmology — falsifiable with JWST high-redshift rotation curves.
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. Cluster masses reported as 16/16 within ±10%.
Source: Unified v3.3 galactic section
$\psi$-screen cosmology and CMB peaks without dark matter
Reconstruction of the CMB acoustic peaks within the $\psi$-screen cosmology, without a separate cold-dark-matter component. Documented as consistent with the published low-$\ell$ deficit.
Source: 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.
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 double-transit asymmetry exponent prediction $\Gamma = 4$ vs the standard $\Gamma \approx 1$ is reported as observed at $\Gamma = 4.4 \pm 0.9$ in the cited analysis.
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.
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 v3.3
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