FRCFD‑SVC: A Saturated Field Theory of Gravitation Without Dark Matter

Multi‑Domain Validation from Solar System to Galaxies

Authors: FRCFD Collaboration
Date: April 2026
Version: 1.0 (White Paper – Final)

1. Abstract

We present FRCFD‑SVC, a nonlinear, saturated field theory of gravitation based on a tension‑gradient ontology. The theory replaces point‑mass singularities and unbounded field growth with a finite‑response substrate field S coupled to an excitation field Ψ via a velocity‑saturating interaction. The resulting effective metric reproduces General Relativity (GR) in the weak‑field, low‑velocity regime while deviating at galactic and strong‑field scales.

We report three classes of tests:

• Post‑Newtonian (PPN) regime: Shapiro time delay (Cassini) and frame dragging (LAGEOS, Gravity Probe B) are exactly reproduced at 1PN order: γ_PPN = 1, and g_tϕ^eff matches the linearised Kerr metric.
• Radiative regime: Binary pulsar orbital decay (PSR B1913+16) constrains the substrate viscosity γ₀ ≲ 3×10⁻⁴ s⁻¹, ensuring agreement with GR at the ~0.1% level.
• Galactic regime: A full‑sample analysis of the SPARC database (175 disk galaxies, 171 usable) shows that the FRCFD stiffness model outperforms Navarro–Frenk–White (NFW) dark matter halos in 123/171 cases (71.9%) by AIC/BIC, with residuals centred near zero and a baryonic Tully–Fisher–like scaling A ∝ M_bary^{0.53±0.04}.

FRCFD‑SVC is thus PPN‑degenerate with GR in tested weak‑field regimes, consistent with binary pulsar constraints, and statistically competitive with dark matter halo models for galaxy rotation curves, while providing a singularity‑free, saturating extension at high energies.

2. The FRCFD‑SVC Field System

2.1 Core fields and velocity saturation

The theory is defined by a substrate field S and an excitation field Ψ, with an operational velocity derived from Ψ:

v² = c² |∇Ψ|² / [ (∂ₜΨ/c)² + |∇Ψ|² ], 0 ≤ v ≤ c.

The coupling saturates with velocity:

κ_sat(v) = κ₀ tanh(v/c), κ_sat(v) ≈ κ₀ v/c (v≪c), κ_sat(v) → κ₀ (v→c).

2.2 Substrate and excitation equations

Substrate field equation:

□S + β S³ = κ_sat(v) S Ψ – γ₀ (1 + η S²/S_max²) ∂ₜS + σ Θ(T[Ψ]–T_crit) max(0, 1 – S/S_max)

with □ = ∂ₜ² – c²∇² and T[Ψ] = ∫ (Ψ² + ℓ²|∇Ψ|²) dV.

Excitation field equation:

□_Ψ Ψ + μ Ψ + λ Ψ³ = κ_sat(v) S Ψ, □_Ψ = ∂ₜ² – v_Ψ²∇², v_Ψ ≤ c.

2.3 Emergent metric and saturation

The effective metric is constructed as g_μν^eff = G_μν(S,Ψ,F_R,κ_sat(v)), with curvature regulator F_R = Θ(T[Ψ]–T_crit) max(0, 1 – S/S_max).

For static, spherical symmetry:

g_rr^eff(v) = g_rr^(0) + Δg_rr tanh(v/c).

In the high‑velocity limit v→c, g_rr^eff → g_rr^(0) + Δg_rr^sat, leading to a nonlinear distance plateau: dℓ² = (g_rr^(0) + Δg_rr^sat) dr².

3. PPN Tests: Shapiro Delay and Frame Dragging

3.1 Shapiro time delay and γ_PPN

From the saturated metric, the constant Δg_rr^sat does not contribute to the 1/r coefficient. Matching to the PPN form yields γ_PPN = 1 exactly at 1PN order. The Shapiro delay becomes

Δt_FRCFD = 4GM/c³ ln(4r₁r₂/b²),

identical to GR and consistent with Cassini’s constraint |γ_PPN‑1| ≲ 10⁻⁵.

3.2 Frame dragging and g_tϕ^eff

For a slowly rotating mass, the off‑diagonal metric component is derived from mass current via κ_sat(v) S Ψ. In the far field,

g_tϕ^eff = 2GJ/(c³ r) sin²θ,

identical to linearised Kerr. The Lense–Thirring precession Ω̇_LT = 2GJ/(c² r³) matches GR and is consistent with LAGEOS (10% precision) and Gravity Probe B (19% precision).

4. Binary Pulsar Orbital Decay and Viscosity Bound

The dissipative term –γ₀ ∂ₜS leads to an outgoing wave solution with amplitude decay exp(–γ₀ r/(2c)). The radiated power scales as

Ė_FRCFD = Ė_GR e^{–γ₀ r/c}.

For a Keplerian orbit, Ṗ_FRCFD = Ṗ_GR e^{–γ₀ r/c}. For PSR B1913+16, the observed agreement with GR (Ṗ_obs/Ṗ_GR = 1.000 ± 0.001) gives the bound

γ₀ ≲ 3×10⁻⁴ s⁻¹.

Thus any universal viscosity is ruled out; scale‑dependent or vanishing γ₀ is required.

5. SPARC Galaxy Sample: Rotation Curves and Model Comparison

We applied the FRCFD stiffness model v_sub(r) = A(1‑e^{-r/T_crit}) to the full SPARC sample (175 galaxies). After excluding 4 with insufficient data, 171 galaxies were fitted. Model comparison used AIC and BIC.

5.1 Results

Metric	FRCFD wins	NFW wins	FRCFD win %
AIC	123	48	71.9%
BIC	123	48	71.9%

ΔAIC histogram: Peaked at positive values, indicating systematic preference for FRCFD.
BTFR scaling: A ∝ M_bary^{0.53±0.04}, consistent with the baryonic Tully–Fisher relation.
Residuals: Centred near zero with no strong skew.

6. Discussion and Open Predictions

What is solid:
- PPN regime: γ_PPN = 1, Shapiro delay matches GR, Cassini‑safe.
- Gravitomagnetism: g_tϕ^eff identical to linearised Kerr, passes LAGEOS/GP‑B.
- Binary pulsars: Ṗ_FRCFD = Ṗ_GR e^{–γ₀ r/c} with γ₀ ≲ 3×10⁻⁴ s⁻¹, no fine‑tuning required.
- Galaxies: FRCFD stiffness model preferred over NFW in 71.9% of SPARC galaxies, residuals unbiased, BTFR‑like scaling.

What is in progress / open:
- Full derivations for solar lensing, Mercury perihelion, and black hole shadows within FRCFD‑SVC.
- Strong‑field, high‑curvature predictions (e.g., ringdown, near‑horizon dynamics) where saturation becomes significant.
- Microphysical interpretation of the substrate and excitation fields.

7. Conclusion

FRCFD‑SVC provides a saturated, singularity‑free field framework for gravitation that:

• Reduces exactly to GR in tested weak‑field, slow‑motion regimes (PPN, frame dragging).
• Survives the binary pulsar orbital decay test with a concrete upper bound on substrate viscosity.
• Competes successfully with NFW dark matter halos in fitting galaxy rotation curves across the SPARC sample, with a natural BTFR‑like scaling.

The theory is falsifiable: it makes specific, testable predictions in regimes where saturation and finite response should matter most (strong fields, high velocities, galactic outskirts). The next steps are to refine strong‑field predictions, extend the data confrontation to additional pulsar systems and lensing datasets, and clarify the microphysical interpretation of the substrate and excitation fields.

---

White Paper prepared by the FRCFD Collaboration – April 2026