V2 - Final Closed FRCFD System (Velocity Saturation)

Final Closed FRCFD System (Velocity Saturation)

1. Operational Velocity (from Ψ)

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

2. Saturating Coupling Function

κₛₐₜ(v) = κ₀ · tanh(v / c)
κₛₐₜ(v) ≈ κ₀ (v / c) (v ≪ c)
κₛₐₜ(v) → κ₀ (v → c)

3. Substrate Field Equation (S)

□S + βS³ =
κₛₐₜ(v) · SΨ
− γ₀ (1 + η S² / Sₘₐₓ²) · ∂ₜS
+ σ Θ(T[Ψ] − T₍cᵣᵢₜ₎) · max(0, 1 − S / Sₘₐₓ)

□ = ∂ₜ² − c² ∇²

4. Excitation Field Equation (Ψ)

□_Ψ Ψ + μΨ + λΨ³ = κₛₐₜ(v) · SΨ
□_Ψ = ∂ₜ² − v_Ψ² ∇² , v_Ψ ≤ c

5. Curvature Regulator

Fᴿ = Θ(T[Ψ] − T₍cᵣᵢₜ₎) · max(0, 1 − S / Sₘₐₓ)

T[Ψ] = ∫ (Ψ² + ℓ² |∇Ψ|²) dV

6. Emergent Velocity‑Dependent Metric

gᵉᶠᶠ_μν = G_μν(S, Ψ, Fᴿ, κₛₐₜ(v))

gᵉᶠᶠ_rr(v) = g_rr⁽⁰⁾ + Δg_rr · tanh(v / c)

7. Nonlinear Distance Plateau (v → c)

κₛₐₜ(v) → κ₀
gᵉᶠᶠ_rr → g_rr⁽⁰⁾ + Δg_rrˢᵃᵗ
dℓ² = ( g_rr⁽⁰⁾ + Δg_rrˢᵃᵗ ) · dr²

Master FRCFD Equation

L[S, Ψ] = κ₀ tanh(v / c) · SΨ
− γ₀ (1 + η S² / Sₘₐₓ²) ∂ₜS
+ σ Fᴿ