Mass as Geometric Governance: VCG Protocol Audits Ringdown and Realizes Background Independence

We introduce the Variational Curvature Governance (VCG) framework, which interprets black-hole ringdown as an observational window into curvature-response spectra, rather than treating it solely as linear perturbations on a fixed Kerr background. We formulate mass as the second-order response of the governance scalar along the scale flow. Furthermore, the model establishes a nested response-order structure: gravity/impedance is driven by the cumulative sum of first- to n-th order scale-flow derivatives; mass is interpreted as an effective geometric response encoded by the cumulative sum of second- to n-th order scale-flow derivatives. Within this framework, mass is axiomatically identified with the second-order response of the governance scalar along the scale flow, m=M_0 ∂_s^2 log⁡B, while the metric evolves dynamically via the explicit governance rate ∂_t g_ij=ϕ^' (κ)⋅κ ̇+"SEC"(Π_S). The VCG governance waveform template explicitly incorporates curvature damping and the governance signal κthrough the time-varying frequency f_t=f_0 g(t)and damping time τ_t.

The Structure-Error-Correction (SEC) operator projects computational trajectories onto the physical sovereignty manifold. In the current numerical implementation, it employs a tanh-type soft-threshold operator as an engineering approximation to systematically dehydrate numerical artifacts. As summarized in Table 1, numerical audits on publicly available LIGO strain data indicate that, under the selected ringdown windows and preprocessing settings, the VCG (Gen-4) template reduces the normalized mismatch by 2.7–44.5% relative to the GR baseline, while Gen-5 CAFVCG further improves the maximum reduction to 48.2%. The SEC-dehydrated residuals exhibit relatively clean periodic geometric structures consistent with the expected form of the second-order mass response. These results provide a quantitative benchmark for geometric-governance-based ringdown modeling in gravitational-wave and nonlinear physical systems, and offer falsifiable targets for O5-era ringdown spectroscopy.

Furthermore, the present results suggest a possible extension toward a higher-level Cross-Area Fusion Variational Curvature Governance (CAFVCG) framework; this direction may be understood not only as a potential AI-architectural extension, but also as a physical-architectural direction grounded in a curvature-first interpretation of geometric dynamics.