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FULL CLOSED FRCFD SYSTEM (WITH VELOCITY SATURATION)

FULL CLOSED FRCFD SYSTEM (WITH VELOCITY SATURATION) 1. Operational Velocity (Field‑Derived) v² = c² · |∇Ψ|² / [ (∂ₜΨ / c)² + |∇Ψ|² ] 0 ≤ v ≤ c • Derived purely from local excitation gradients • No external or frame‑dependent input 2. Saturating Coupling Function κₛₐₜ(v) = κ₀ · tanh(v / c) κₛₐₜ(v) ≈ κ₀ (v / c) for v ≪ c κₛₐₜ(v) → κ₀ as v → c 3. Substrate Field Equation (S) ∂²S/∂t² − c²∇²S + βS³ = κₛₐₜ(v) · SΨ − γ₀ (1 + η S² / Sₘₐₓ²) · ∂S/∂t + σ Θ(T[Ψ] − T₍cᵣᵢₜ₎) · max(0, 1 − S / Sₘₐₓ) 4. Excitation Field Equation (Ψ) ∂²Ψ/∂t² − vΨ² ∇²Ψ + μΨ + λΨ³ = κₛₐₜ(v) · SΨ • Same saturating coupling ensures reciprocal energy exchange • vΨ ≤ c 5. Curvature Regulator (“Snap”) Fᴿ = Θ(T[Ψ] − T₍cᵣᵢₜ₎) · max(0, 1 − S / Sₘₐₓ) T[Ψ] = ∫ (Ψ² + ℓ² |∇Ψ|²) dV 6. Emergent Metric (Velocity‑Dependent) gᵉᶠᶠ_μν = G_μν(S, Ψ, Fᴿ, κₛₐₜ(v)) gᵉᶠᶠ_rr(v) = g_rr⁽⁰⁾ + Δg_rr(tanh(v / c)) 7. Nonlinear Distance Plateau (High‑Velocity Limit) v → c ⇒ κₛₐₜ(v)...
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FRCFD‑SVC: A Saturated Field Theory of Gravitation Without Dark Matter -- Multi‑Domain Validation from Galaxies to Black Holes

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FRCFD‑SVC White Paper FRCFD‑SVC: A Saturated Field Theory of Gravitation Without Dark Matter Multi‑Domain Validation from Galaxies to Black Holes Authors: FRCFD Collaboration Date: April 2026  |  Version: 1.0 (White Paper) Abstract We present a fully closed, nonlinear field theory – FRCFD with Saturating Velocity Coupling (SVC) – that replaces dark matter and modifies general relativity in a controlled, singularity‑free manner. The theory introduces a substrate field S and an excitation field Ψ with reciprocal, velocity‑dependent coupling κ sat (v) = κ₀ tanh(v/c). This coupling naturally saturates at high velocities, leading to flat galaxy rotation curves, a finite black hole shadow, and a velocity‑dependent effective metric g μν eff . We test FRCFD‑SVC across seven independent domains: Galaxy rotation curves – Using the SPARC sample of 175 galaxies, a static approximation (stiffness model) outperforms Navarro‑F...
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FRCFD‑SVC: A Saturated Field Theory of Gravitation Without Dark Matter Multi‑Domain Validation from Galaxies to Black Holes

FRCFD‑SVC: A Saturated Field Theory of Gravitation Without Dark Matter Multi‑Domain Validation from Galaxies to Black Holes Authors: FRCFD Collaboration Date: April 2026 Version: 1.0 (White Paper) Abstract We present a fully closed, nonlinear field theory – FRCFD with Saturating Velocity Coupling (SVC) – that replaces dark matter and modifies general relativity in a controlled, singularity‑free manner. The theory introduces a substrate field S S and an excitation field Ψ Ψ with reciprocal, velocity‑dependent coupling κ sat ( v ) = κ 0 tanh ⁡ ( v / c ) κ sat ​ (v)=κ 0 ​ tanh(v/c). This coupling naturally saturates at high velocities, leading to flat galaxy rotation curves, a finite black hole shadow, and a velocity‑dependent effective metric g μ ν eff g μν eff ​ . We test FRCFD‑SVC across seven independent domains: Galaxy rotation curves – Using the SPARC sample of 175 galaxies, a static approximation (stiffness model) outperforms Navarro‑Frenk‑White (NFW) dark matter halos...
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FULL CLOSED FRCFD SYSTEM (WITH VELOCITY SATURATION)

FULL CLOSED FRCFD SYSTEM (WITH VELOCITY SATURATION) 1. Operational Velocity (Field‑Derived) v² = c² · |∇Ψ|² / [ (∂ₜΨ / c)² + |∇Ψ|² ] Ensures: 0 ≤ v ≤ c Derived purely from local excitation gradients No external or frame‑dependent input required 2. Saturating Coupling Function κ_sat(v) = κ₀ · tanh(v / c) Limits: κ_sat(v) ≈ κ₀ · (v / c) (v ≪ c), κ_sat(v) → κ₀ (v → c) 3. Substrate Field Equation (S) ∂²S/∂t² − c²∇²S + βS³ = κ_sat(v)·SΨ − γ₀ (1 + η S² / S_max²) · ∂S/∂t + σ Θ(T[Ψ] − T_crit) · max(0, 1 − S / S_max) 4. Excitation Field Equation (Ψ) — Reciprocal Coupling ∂²Ψ/∂t² − v_Ψ² ∇²Ψ + μΨ + λΨ³ = κ_sat(v)·SΨ Same saturating coupling ensures energy reciprocity. v_Ψ = propagation speed of excitation field (can be ≤ c). 5. Curvature Regulator (“Snap”) F_R = Θ(T[Ψ] − T_crit) · max(0, 1 − S / S_max) T[Ψ] = ∫ (Ψ² + ℓ² |∇Ψ|²) dV...
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