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

We present the Variational Curvature Governance (VCG/CAFVCG) protocol for black-hole ringdown audit together with a theoretical Gen-6 Variational Curvature-Transfer Governance (VCTG) extension based on curvature-first conditional-path covariant flow. In the implemented VCG model, black-hole ringdown is treated as a protocol-level observational probe of curvature-response structure rather than solely as linear perturbation on a prescribed Kerr background. Within this modeling language, the core mass-response observable is formulated axiomatically as the second-order scale-flow response of a governance scalar,

m2=M0∂s2log⁡B,m_2 = M_0\partial_s^2\log B,m2=M0∂s2logB,

while an optional hierarchical extension is written as

m≥2=M0∑q=2n(τq∂s)qlog⁡B.m_{\ge2} = M_0 \sum_{q=2}^{n} (\tau_q\partial_s)^q \log B.m≥2=M0q=2∑n(τq∂s)qlogB.

The currently implemented waveform protocol incorporates governance-dependent frequency and damping corrections together with a Structure-Error-Correction (SEC) operation. In the present numerical calculations, SEC is realized by a tanh-type soft-threshold engineering surrogate; it is not identified with the full covariant SEC projection introduced in the Gen-6 theory.

Numerical audits reported in this study compare a GR baseline with VCG (Gen-4) and CAFVCG (Gen-5) templates on selected windows of publicly available LIGO strain data. Under the declared window and preprocessing settings, VCG reduces normalized mismatch by $2.7\%$–$44.5\%$ relative to the GR baseline, while Gen-5 CAFVCG achieves reductions of $3.0\%$–$48.2\%$. SEC-processed residuals exhibit relatively clean periodic geometric structures consistent with the expected form of the second-order mass-response ansatz. These results are protocol-level model-comparison benchmarks; they do not establish that VCG replaces General Relativity, nor do they show that the present numerical implementation already realizes the full Gen-6 dynamics.

The theoretical Gen-6 extension is formulated here as Variational Curvature-Transfer Governance (VCTG). VCTG retains the curvature-first ontology of VCG/CAFVCG, but replaces generic non-local curvature borrowing by a relative-curvature-deviation-ordered conditional-path architecture. In this formulation, GR contains only the passive classical metric governed by the Einstein equation, whereas VCG decomposes the metric structure into a passive metric and an active-governance metric component. SEC acts as a cross-variable correction mechanism among the passive metric, the active-governance metric, curvature deviation, the governance scalar, and the system state. The GR-compatible retraction of VCG is not defined by the vanishing of the total metric evolution. More generally, it is defined by the vanishing of the coupled active-governance residual

Aμνgov:=ϕ′(κ)κ˙+RμνSEC+TμνCAF/VCTG→0.\mathcal{A}_{\mu\nu}^{\rm gov} := \phi'(\kappa)\dot{\kappa} + \mathcal{R}_{\mu\nu}^{\rm SEC} + \mathcal{T}_{\mu\nu}^{\rm CAF/VCTG} \to0.Aμνgov:=ϕ′(κ)κ˙+RμνSEC+TμνCAF/VCTG→0.

Under this condition, the active-governance metric correction becomes dormant,

Δgovgμν→0or(∂tgμν)gov→0.\Delta_{\rm gov}g_{\mu\nu}\to0 \quad \text{or} \quad (\partial_t g_{\mu\nu})_{\rm gov}\to0.Δgovgμν→0or(∂tgμν)gov→0.

Separate vanishing of the curvature-response term, SEC correction, and CAF/VCTG transfer term is only a stronger sufficient realization. The passive GR-compatible metric may still evolve according to the Einstein equation.

A directed VCTG channel $(a\leftarrow b)$ is activated only when the receiving region (Da) (D_a) (Da) exhibits sufficiently higher relative curvature deviation than the supply region (Db) (D_b) (Db). If a more stable admissible region is detected along an active corridor, the supply mechanism may be conditionally re-anchored. Only activated relative-deviation-ordered paths enter the non-local chain variation, after which SEC acts as a covariant proximal projection onto the admissible governance manifold.

Under coercive curvature-energy, smooth conditional gates, bounded CAF kernels, non-amplifying path coverage, positive effective mobility, SEC Lyapunov descent, absorbable geometric and routing remainders, spatial continuation control, and finite external input assumptions, Appendix F gives a conditional covariant non-singularity theorem that elevates the computational stability mechanism of Appendix E to the ideal Gen-6 governance spacetime. The present work therefore supplies a quantitative ringdown benchmark and a mathematically auditable covariant extension for future strong-field and nonlinear-structure tests, rather than an unconditional foundational proof.