n = eψ
Light slows near mass
Empty space behaves like a medium whose refractive index rises near mass, so the local speed of light is c·e−ψ. Light bends toward mass, the way starlight bends around the Sun.
New
Author’s astrophysics paper in MNRAS
A scalar-refractive theory of gravity
Density Field Dynamics
What if gravity is a refractive medium for light? In DFD one field, ψ, sets how fast light travels and how matter falls. It reproduces the classic tests of General Relativity and derives the fine-structure constant from topology.
- n = eψlight slows near mass
- a = ½c²∇ψmatter follows the same field
- γ = β = 1Einstein’s weak-field tests
The idea
One field does the work of curved spacetime.
DFD is a scalar-field reformulation of gravity and optics. A single field ψ lives on flat three-dimensional space and controls both how light moves and how matter accelerates.
a = ½c²∇ψ
Matter follows the same field
A freely falling body accelerates along the gradient of the same field that bends light. One field replaces the geometry of curved spacetime.
γ = β = 1
Einstein’s tests still pass
In the weak-field limit all ten post-Newtonian parameters match General Relativity, and gravitational waves travel at exactly the speed of light, as GW170817 requires.
Headline derivations
Numbers you can check yourself.
The framework’s strongest claims come with closed-form expressions and code. Run them, and compare them with experiment.
Fine-structure constant · Dec 2025
α from topology, with no free parameters
137.035999854α−1 · +0.005 ppm vs CODATA 2022
$$\alpha^{-1} = \frac{\pi^{3/2}}{24}\;\mathrm{Tr}(Y^2)\;k_{\max}\;\frac{k_{\max}+3}{k_{\max}+4}\;\left[1 + \frac{N_{\mathrm{sp}}}{g_F\;\mathrm{Tr}(Y^2)\;\bigl((k_{\max}+4)^2 - 1\bigr)}\right]$$
With \(\mathrm{Tr}(Y^2)=10,\; k_{\max}=60,\; N_{\mathrm{sp}}=7,\; g_F=8\).
import math k, TrY2, N_sp, gF = 60, 10, 7, 8 d = k + 4 # = 64 raw = (math.pi**1.5 / 24) * TrY2 * k * (k+3)/(k+4) boost = 1 + N_sp / (gF * TrY2) / (d**2 - 1) alpha_inv = raw * boost # = 137.0359998541
A closed-form result from Chern–Simons quantization on CP²×S³, independently checked by lattice Monte Carlo (86 runs, L = 6–16). The companion software package reproduces it end to end.
Strong fields · Apr 2026
General Relativity as a Padé approximant
$$L_{\rm GR}(u) = \left[P_{1,1}(e^u)\right]^2, \qquad L_{\rm DFD}(u) = e^{2u}$$
An exact identity: GR’s lapse is the first Padé truncation of DFD’s exponential. The theories agree through second order and first differ at third, which predicts a black-hole shadow about 4.6% larger for next-generation EHT imaging.
Read the paperParticle physics · Mar 2026
Nine charged fermion masses
$$m_f = A_f \, \alpha^{n_f} \, \frac{v}{\sqrt{2}}$$
All nine charged fermion masses from CP²×S³ topology, with a 1.42% mean error across three orders of magnitude and no per-fermion fitting.
Read the paperCurrent release · v4.0 · July 2026
Density Field Dynamics: A Complete Unified Theory
A two-volume release: the main paper with axioms, theorems and proofs, derived predictions, falsifiers, and a revision ledger, plus an extended-derivations companion and a reproducibility package. Predictions are frozen and dated.
What’s inside v4.0
Completed two-volume release: a 348-page main paper (axioms, theorems with proofs, derived predictions, falsifiers, revision ledger) and a 76-page extended-derivations companion, plus a one-command reproducibility package.
Gravity sector: full PPN match (γ=β=1, all preferred-frame parameters exactly zero as a theorem), gravitational waves as irreducible components of the same zero-mode parent tensor on CP²×S³ (cT=c, Lichnerowicz rigidity), SPARC model-independent shape analysis (nopt=1.15±0.12, MOND n=2 strongly disfavored), galaxy–galaxy lensing radial-acceleration relation derived as a theorem (ΔΣ=√(Mba0/G)/4R, zero free parameters, KiDS-1000 GAMA at 0.4σ), cluster masses (14/16 within ±10%, 16/16 within 2σ of published mass errors).
Dark sector (new): a derived cold, collisionless χ component of mass 5.09 eV whose abundance Ωχh²=0.1182 is computed forward from finite SU(2)60 Chern–Simons modular data (−1.5σ from Planck); dark energy replaced by an optical ψ-screen, with the cosmological-constant hierarchy ρc/ρPl=(3/8π)α57 spanning 122.7 orders of magnitude with no fine-tuning; galactic rotation curves from the μ-crossover.
Gauge & flavor: SU(3)×SU(2)×U(1) from CP²×S³ topology, α−1=137.036 (closed-form + lattice-verified, with a trials-factor theorem showing landscape scanning cannot produce the match), sin²θW=3/13 (0.2%), αs(MZ)=0.1187 (0.8σ), Higgs v=246.09 GeV (0.05%), 9 charged fermion masses (1.42% mean error), mt=172.74 GeV (0.57σ), CKM Wolfenstein integers from CP² line-bundle cohomology with γ=66.31° and honest per-channel pulls, neutrino Δm² matching NuFIT 6.0 (p=0.99, Σmν=61.5 meV, mβ=9.15 meV), θ̄=0 (strong CP without axion), baryogenesis magnitude |ηB|≈0.206 α4 forced.
Quantum sector (new): the single-particle Schrödinger equation derived as an algebraic identity of the master equation, with the quantum phase i identified with the CP² Kähler structure and every remaining import listed explicitly.
Cosmology: H0=72.09 km/s/Mpc (Hubble tension resolved, 0.3σ from SH0ES/JWST), G·ℏ·H0²/c5=α57 derived as a spectral-action theorem, CMB lensing +2.06σ pass, BOSS DR12 full-shape P(k) indistinguishable from ΛCDM at k≤0.15 h/Mpc.
Predictions & reproducibility: UVCS Γ=4 (measured 4.4±0.9), Cooper-pair mass anomaly δ=√3 α²=92.23 ppm, gravitational weight anomaly, 187Re nuclear sensitivity, and frozen near-term falsifiers at Belle II, JUNO, KATRIN, LISA, and Euclid; python3 reproduce.py recomputes six headline numbers live in about a minute. Zero continuous fit parameters throughout.
- 348 ppMain volume: axioms, theorems, predictions, falsifiers
- 76 ppExtended derivations and frontier predictions
- 1 command
python3 reproduce.pychecks headline numbers against the paper - 0Continuous fit parameters
Papers and preprints
The research record.
Every Density Field Dynamics paper, newest first. All of them are preprints.
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Jul 2026
Density Field Dynamics: A Complete Unified Theory (v4.0)
Completed two-volume release: a 348-page main paper (axioms, theorems with proofs, derived predictions, falsifiers, revision ledger) and a 76-page extended-derivations companion, plus a one-command reproducibility package. Gravity sector: full PPN match (γ=β=1, all preferred-frame parameters exactly zero as a theorem), gravitational waves as irreducible components of the same zero-mode parent tensor on CP²×S³ (cT=c, Lichnerowicz rigidity), SPARC model-independent shape analysis (nopt=1.15±0.12, MOND n=2 strongly disfavored), galaxy–galaxy lensing radial-acceleration relation derived as a theorem (ΔΣ=√(Mba0/G)/4R, zero free parameters, KiDS-1000 GAMA at 0.4σ), cluster masses (14/16 within ±10%, 16/16 within 2σ of published mass errors). Dark sector (new): a derived cold, collisionless χ component of mass 5.09 eV whose abundance Ωχh²=0.1182 is computed forward from finite SU(2)60 Chern–Simons modular data (−1.5σ from Planck); dark energy replaced by an optical ψ-screen, with the cosmological-constant hierarchy ρc/ρPl=(3/8π)α57 spanning 122.7 orders of magnitude with no fine-tuning; galactic rotation curves from the μ-crossover. Gauge & flavor: SU(3)×SU(2)×U(1) from CP²×S³ topology, α−1=137.036 (closed-form + lattice-verified, with a trials-factor theorem showing landscape scanning cannot produce the match), sin²θW=3/13 (0.2%), αs(MZ)=0.1187 (0.8σ), Higgs v=246.09 GeV (0.05%), 9 charged fermion masses (1.42% mean error), mt=172.74 GeV (0.57σ), CKM Wolfenstein integers from CP² line-bundle cohomology with γ=66.31° and honest per-channel pulls, neutrino Δm² matching NuFIT 6.0 (p=0.99, Σmν=61.5 meV, mβ=9.15 meV), θ̄=0 (strong CP without axion), baryogenesis magnitude |ηB|≈0.206 α4 forced. Quantum sector (new): the single-particle Schrödinger equation derived as an algebraic identity of the master equation, with the quantum phase i identified with the CP² Kähler structure and every remaining import listed explicitly. Cosmology: H0=72.09 km/s/Mpc (Hubble tension resolved, 0.3σ from SH0ES/JWST), G·ℏ·H0²/c5=α57 derived as a spectral-action theorem, CMB lensing +2.06σ pass, BOSS DR12 full-shape P(k) indistinguishable from ΛCDM at k≤0.15 h/Mpc. Predictions & reproducibility: UVCS Γ=4 (measured 4.4±0.9), Cooper-pair mass anomaly δ=√3 α²=92.23 ppm, gravitational weight anomaly, 187Re nuclear sensitivity, and frozen near-term falsifiers at Belle II, JUNO, KATRIN, LISA, and Euclid;
python3 reproduce.pyrecomputes six headline numbers live in about a minute. Zero continuous fit parameters throughout. -
Apr 2026
General Relativity as the Padé Approximant of Density Field Dynamics
An exact mathematical identity: GR’s isotropic-coordinate Schwarzschild lapse-squared LGR(u) = [(1+u/2)/(1−u/2)]² equals [P1,1(exp(u))]², while DFD’s exterior solution gives LDFD(u) = exp(2u). GR is the m=1 slot in a Padé hierarchy; DFD is the entire-function limit. The Schwarzschild horizon at r=2GM/c² is the Padé pole of the m=1 truncation; DFD’s exponential has no finite pole, and r=2GM/c² appears as a photon sphere rather than a horizon. The two theories agree through O(u²) by construction (consistent with all gravitational-redshift, clock, and PPN-β observations to date) and first differ at O(u³), generating a ∼4.6% larger black-hole shadow — the proximal observational discriminator with next-generation EHT data on M87⋆ and Sgr A⋆.
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Apr 2026
Epoch Evolution of the MOND Crossover Scale: a⋆(z) = 2√α·cH(z)
Within DFD the galactic transition acceleration tracks cosmic expansion epoch-by-epoch: a⋆(z) = 2√α·cH(z), not a frozen present-day value. Derived from two topological invariants (ka=3/(8α) from CP²×S³ gauge emergence, qS³=3/2 from the Chern–Simons partition function on S³) and a conditional uniqueness proposition for the cosmic IR scale, under the same epoch-consistency rule that promotes G·ℏ·H²/c⁵=α⁵⁷ to an all-epoch statement. Prediction: the MOND transition scale should appear enhanced by H(1)/H0≈1.79 at z∼1 relative to the local-universe value — directly comparable to JWST rotation-curve measurements after standard kinematic and inclination reductions. Observation of a frozen a⋆ at high redshift falsifies the proposition.
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Apr 2026
Tree-Level No-Drive Theorem for the Minimal Optical-Metric EM Sector: λbare = 1
Proves λbare=1 in the minimal tree-level optical-metric EM sector of DFD. The linear-in-ψ EM source produced by the gauge-invariant action is proportional to the energy density (E²/c²+B²)/(2μ0), not the stress invariant (E²/c²−B²)/(2μ0). For ideal standing-wave cavity modes with energy equipartition, the energy-density source carries no 2ω component after volume integration, so EM fields cannot pump ψ through this channel. Reinterprets Appendix R’s |λ−1|<3×10−5 accidental bound and the projected 10−14 intentional reach as constraints on beyond-minimal channels (finite-Q mimic, geometry restoration for asymmetric/TE+TM superposition, ηc=α/4 threshold, κ-channel splitting, dim-5 operators ξψFμνFμν) — not of the minimal-sector baseline.
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Apr 2026
Mach’s Principle in Density Field Dynamics: An Interpretive and Phenomenological Consolidation
Situates DFD against the Bondi–Samuel Mach taxonomy and consolidates its Machian phenomenology. The structural identity a⋆=2√α·cH0 ties the galactic transition scale to cosmic expansion through S³ topology — the long-noted a0∼cH0 coincidence is no longer a coincidence. The epoch-extended form a⋆(z)=2√α·cH(z) predicts a drift in the radial-acceleration-relation normalisation of ≈1.79× at z=1 in a ΛCDM background parameterisation, or ≈2.83× in DFD’s own matter-only ψ-screen cosmology, falsifiable by JWST and DESI. Against the ten Mach criteria of Bondi–Samuel: DFD satisfies Mach 3, Mach 10, and effectively Mach 8; partially satisfies Mach 1, 2, 6; fails Mach 4, 5, 7, 9. The ψ rest frame is dynamically determined by cosmic matter (operational preferred frame), while the flat R³ kinematic substrate remains absolute.
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Apr 2026
Constitutive Derivation of Tensor Gravitational Radiation from CP²×S³ Spectral Geometry in Density Field Dynamics
Closes the gravitational-wave seam in DFD. The scalar field ψ and the transverse-traceless tensor hijTT are derived as irreducible components of a single zero-mode parent tensor on CP²×S³, eliminating the need to postulate the TT sector independently. Lichnerowicz rigidity (CP² gap = 8/R1², S³ gap = 12/R2², b1=0) guarantees no unwanted scalar or vector graviton modes propagate. The Einstein product condition τ*=1/√3 is derived as the unique minimum of the internal constraint function Φ(τ), fixing the squashing modulus at Planck mass. Constitutive interpretation via generalized Tamm–Plebanski relations identifies K0=c4/(8πG) as compression stiffness and K0/4 as shear stiffness, consistent with the vacuum loading framework. Result: cT=c exactly, satisfying the GW170817 constraint, with both sectors emerging from a single topological origin.
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Mar 2026
Uniqueness of the Internal Manifold: Deriving CP²×S³ from Vacuum Axioms in Density Field Dynamics
Proves that CP²×S³ is the unique internal manifold for DFD’s spectral completion, under six physically motivated axioms on the ψ-vacuum encoding chirality (empirical), multiplicative vacuum composition (from postulate P1), ground-state stability, and minimality. The product structure K=KC×KG is not assumed but forced by the logical incompatibility of the chirality requirement (w2≠0) with Lie-group parallelizability (w2=0) on a single connected manifold. The Matsushima–Lichnerowicz obstruction kills CP²#CP² (the only non-trivial competitor), and the Cartan classification fixes S³≅SU(2) as the unique minimal Lie group factor. Dimensional arithmetic (4+3=7) leaves no room for additional factors. The gauge group SU(3)×SU(2)×U(1), three generations, and α−1=137.036 emerge as derived consequences—none are inputs to the axioms.
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Dec 2025
Ab Initio Derivation of the Fine-Structure Constant from Density Field Dynamics (v2.1)
The first ab initio derivation of α from pure topology — zero free parameters, sub-ppm precision. Closed-form derivation from Chern–Simons quantization on CP²×S³ topology. A single algebraic expression with no fitted parameters achieves sub-ppm precision against the CODATA 2022 value. Independently verified by lattice Monte Carlo simulation (86 runs, L=6–16, 9/10 lattice sizes at p<0.01). Companion software package provides full reproducibility.
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Mar 2026
Ab Initio Derivation of the Charged Fermion Mass Spectrum from Density Field Dynamics
Nine charged fermion masses from topology — 1.42% mean error, zero per-fermion fitting. Derives all nine charged fermion masses (e, μ, τ, u, d, s, c, b, t) from CP²×S³ topology using A5 class geometry for amplitude factors and Spinc bundle degrees for power-law exponents. No per-fermion fitting parameters; 1.42% mean error across three orders of magnitude in mass.
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Mar 2026
Alpha Rosetta Stone: The DFD Prediction Web
45 predictions from 2 inputs. Prediction:input ratio 15:1 to 17:1 — the highest in theoretical physics. Complete dependency graph from α + MP + CP²×S³ topology to particle masses, cosmological observables, and galactic dynamics.
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Mar 2026
The Physical Origin of the Refractive Field in Density Field Dynamics: Gravity as Electromagnetic Vacuum Loading
Why does mass create a refractive field? Because mass is energy, and energy loads the electromagnetic vacuum. The exponential n = eψ is proved unique via Cauchy's functional equation (multiplicative composition of successive loadings). Newton's constant G = c4/(8πK0) is identified as inverse vacuum force scale, with K0 ≈ 4.82×1042 N. The constitutive split κ = α/4 ≈ 1.82×10−3 is derived from gauge-emergence corrections to the optical metric, predicting testable TE/TM cavity splitting. The nonlinear field equation is reinterpreted as the constitutive response of a vacuum medium exhibiting reduced gravitational permittivity at low gradients—the mechanism behind flat rotation curves without dark matter.
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Mar 2026
Pairing-Symmetry Selection Rules for the Cooper-Pair Mass Anomaly from A5 Microsector Representation Theory
Addresses the 36-year-old Tate et al. Cooper-pair mass anomaly (δ=92±21 ppm in niobium). Working within the DFD A5 microsector, establishes two pairing-symmetry selection rules: (a) the quintet exchange channel in S²(V*) couples maximally to s-wave condensates but vanishes for d-wave (angular cancellation of sign-changing gap), and (b) spin-triplet pairs live in Λ²(V*)=3, orthogonal to the quintet by representation theory alone. Numerical conjecture: δ=√3 α²=92.23 ppm (0.01σ match to Tate). Unlike the BCS-exchange correction of Lipavský (2016), this framework predicts universality for conventional s-wave superconductors—a material-independent distinction testable with existing SQUID magnetometry.
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Mar 2026
Composition-Dependent Bounds on Scalar-Field Coupling to Nuclear Decay Rates
Applies the Flambaum nuclear sensitivity formalism to compute isotope-specific sensitivity coefficients κq for eight nuclides central to the decade-long Jenkins–Fischbach debate on solar-modulated decay rates. Sensitivity is driven by Q-value (κq∝n/Q), placing 32Si (κq=308) and 187Re (κq≈19,000) at the top of the hierarchy. Existing null results constrain different regions of the (kqeff, κq) parameter space but do not exclude composition-dependent signals in untested low-Q isotopes. The original positive datasets (32Si at BNL, 226Ra at PTB) are now attributed to environmental systematics. Identifies 187Re and the 229Th nuclear clock isomer (K∼104) as the most sensitive future targets, and proposes a multi-isotope ratio test that eliminates systematics by design.
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Dec 2025
Two Numerical Relations Linking the Fine-Structure Constant to Gravitational Phenomenology
Parameter-free predictions connecting α to gravitational observables: MOND acceleration scale a0=2√α·cH0 and gravitational clock coupling kα=α²/(2π)≈8.5×10−6. Both testable with current optical clock technology.
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Dec 2025
kα and the a² Invariant: A Unified Acceleration Scale from Galaxies to Atomic Clocks
Derives the α-relations from scalar self-coupling structure. Extends predictions to strong-field regimes and provides clock comparison signatures at the 10−5 level across multiple atomic species.
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Oct 2025
Solar-Locked Differential in Ion–Neutral Optical Frequency Ratios
Analysis of published ROCIT frequency ratio data (Yb+/Sr) revealing perihelion-locked modulation: amplitude A=(−1.045±0.078)×10−17 with period matching Earth's orbital eccentricity. Consistent with sector-differential ψ coupling; independent replication encouraged.
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Sep 2025
Parametrized Post-Newtonian Analysis of Density Field Dynamics
Complete PPN expansion in the weak-field, slow-motion limit. All ten PPN parameters match General Relativity at 1PN order: γ=β=1, ξ=α1=α2=α3=ζ1=ζ2=ζ3=ζ4=0. DFD is observationally indistinguishable from GR for all current solar system tests.
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Sep 2025
Strong Fields and Gravitational Waves in Density Field Dynamics
Extension to strong-field regime: photon sphere locations, black hole shadow predictions, and gravitational wave propagation. Tensor wave speed cT=c exactly, satisfying GW170817 constraint. All parameterized post-Einsteinian (ppE) bounds satisfied.
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Sep 2025
Well-Posedness and Boundary Value Problems for the ψ Equation
Rigorous PDE analysis of the DFD field equation. Establishes existence, uniqueness, and regularity of weak solutions in appropriate Sobolev spaces. Proves energy conservation and derives asymptotic boundary conditions.
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Sep 2025
Accidental and Intentional Constraints on EM→ψ Back-Reaction
Laboratory bounds on electromagnetic coupling to ψ from cavity stability measurements. Constrains back-reaction parameter |λ−1|≲3×10−5, demonstrating consistency with precision metrology and identifying future experimental sensitivity targets.
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Sep 2025
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Aug 2025
Density Field Dynamics and the c-Field
Foundational paper establishing the core framework: field equations for ψ, energy-momentum conservation, recovery of Newtonian limit, and classical test predictions. Introduces the optical-refractive interpretation of gravitational phenomenology. Note: Superseded by Unified v3.2 for clock predictions; foundational framework remains current.
No papers match that search.
Open data and code
Everything needed to check the work.
Zenodo · DOI 10.5281/zenodo.19173548
Fine-structure constant: code and data (v1.1)
Lattice Monte Carlo code, the spectral pipeline, and reproducibility scripts for the α derivation.
Open record
v4.0 · zip
Reproducibility package
One command recomputes the headline numbers live in about a minute, each checked against the value printed in the paper.
Download
Zenodo · DOI 10.5281/zenodo.17272596
ROCIT frequency-ratio analysis
Full methods, figures, and scripts for the ion–neutral optical frequency-ratio analysis.
Open record
Explore
Go deeper, your way.
Interactive guide
Gravity is Light
A plain-English, interactive journey through the theory, from the problem with dark matter to the numbers DFD derives.
Start the guide
Reference
Topics index
Every named result, formula, and theorem, with a link to the paper that derives it.
History
Historical lineage
From Fermat and Einstein’s 1911–12 variable-c program to scalar–tensor theories and DFD.
Context
Framework comparison
How DFD compares with General Relativity, MOND, and scalar–tensor alternatives.
Machine-readable
Plain-text corpus
Markdown and plain-text versions of every paper, with an llms.txt index and an RSS feed of releases.
Run an experiment?
DFD is an independent research program open to experimental collaboration: 229Th nuclear clocks, cross-species atomic comparisons, cavity–atom residuals, and galaxy kinematics at high redshift. Send your setup and get a concrete prediction back.
Or email gary@gtacompanies.com