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) → κ₀
gᵉᶠᶠ_rr → g_rr⁽⁰⁾ + Δg_rrˢᵃᵗ
dℓ² = [ g_rr⁽⁰⁾ + Δg_rrˢᵃᵗ ] · dr²
🔷 FINAL STRUCTURE SUMMARY
- Velocity saturation (κₛₐₜ)
- Curvature saturation (Sₘₐₓ)
- Nonlinear dissipation (η)
- Threshold regulator (Fᴿ)
- Local operational velocity from Ψ
- Reciprocal coupling S ↔ Ψ
🔷 PHYSICAL INTERPRETATION
- Mass = finite‑response drag between S and Ψ
- Velocity → saturation, not divergence
- High‑energy motion enters a plateau regime
- Singularities replaced by bounded saturated states
🔷 STATUS
- Closed nonlinear field system
- Internally consistent
- Simulation‑ready
- Compatible with SPARC, pulsars, GW tests
