Phase 0.9 Final Report: Spectral Scaling Laws and Substrate Saturation in FRCFD Clean copy
Phase 0.9 Final Report: Spectral Scaling Laws and Substrate Saturation in FRCFD
Technical Abstract
This document formalizes the results of the Phase 0.9 Scaling Sweep, a rigorous metrological characterization of the fundamental frequency redshift observed in the Finite-Response Coupled Field Dynamics (FRCFD) framework. By mapping the spectral response across a mass range of 60 M⊙ to 520 M⊙, we have established a calibrated Scaling Law defining the non-linear "lugging" effect of the substrate under varying gravitational stress. The analysis confirms a model-dependent divergence at high mass scales, providing a specific, falsifiable observational envelope for future gravitational wave ringdown studies.
I. The Master Scaling Equation
The transition from a linear response to a saturated state is governed by the substrate's finite capacity to propagate field perturbations. At LIGO-scale baseline (M = 60), the model predicts a conservative redshift (~4.6%). As mass density increases, the frequency shift Δf exhibits non-linear decay characteristic of a high-viscosity medium:
∂²Ψ/∂t² − v² ∇²Ψ + μΨ + λ|Ψ|²Ψ = κ S Ψ
Where Ψ represents the coupled field potential, v is the propagation velocity within the substrate, and κ is the coupling constant defining congestion per unit mass S. The observed frequency f₀ is a direct function of the temporal latency induced by κ S Ψ, producing measurable "clock-slowing" during the ringdown phase.
II. Empirical Rung Analysis
The Phase 0.9 sweep employed two distinct escapements—the Rational and Power-Law models—to bracket substrate stiffness uncertainty. Three regimes were identified:
- Linear Regime (M = 60–80): Envelope remains tight (Width ≈ 1.3%). Redshifts 4.6–6.8%. Choice of model is second-order to the existence of shift.
- Transitional Regime (M = 240): Model sensitivity emerges. Rational exhibits -12.4%, Power-Law -22.8%, opening a 10.4% divergence envelope.
- Extreme/Asymptotic Regime (M = 520): Saturation Brake observed. Rational plateaus at -17.35%, Power-Law reaches -32.69%, defining the high-stress envelope.
III. Tabulated Spectral Records
| Mass (M⊙) | Model | f₀ (Hz) | Shift (%) |
|---|---|---|---|
| 60 | Rational | 238.6 | -4.64% |
| 80 | Rational | 236.4 | -5.51% |
| 80 | Power-Law | 233.1 | -6.83% |
| 240 | Rational | 219.1 | -12.43% |
| 240 | Power-Law | 193.1 | -22.82% |
| 520 | Rational | 206.8 | -17.35% |
| 520 | Power-Law | 168.4 | -32.69% |
IV. Physical Interpretation
The high-mass divergence demonstrates the substrate as a Finite-Response Substrate (FRS). As mass approaches the congestion threshold, the λ|Ψ|²Ψ term provides counter-pressure to infinite temporal slowing. The Rational branch plateaus, representing a physically stable limit. The Power-Law branch softens locally, producing deeper redshift signatures potentially observable in intermediate-mass black holes (IMBH). This completes the Phase 0.9 framework and prepares the system for Phase 1.0: observational verification.
