Published October 30, 2026 | Version v8
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Rate Dormancy and State Retraction in Variational Curvature-Transfer Governance (VCTG): A GR-Compatible Active-Curvature-Governance Framework

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Description

We formulate Variational Curvature-Transfer Governance (VCTG) as a GR-compatible active-curvature-governance extension of general relativity rather than a replacement for GR. At the unidirectional verification stage, the effective metric and active deviation are written as

gμνeff=gμνbase+aμν,qμν:=aμν−aμνeq,g_{\mu\nu}^{\rm eff}=g_{\mu\nu}^{\rm base}+a_{\mu\nu},\qquad q_{\mu\nu}:=a_{\mu\nu}-a_{\mu\nu}^{\rm eq},gμνeff=gμνbase+aμν,qμν:=aμν−aμνeq,

and the active response is governed by the Onsager–Rayleigh-type variational–dissipative equation

τg(Dξq)μν+ℓg2(ΔLq)μν+γgqμν=Fμνgov+Rμνaudit.\tau_g(\mathcal{D}_\xi q)_{\mu\nu}+\ell_g^2(\Delta_L q)_{\mu\nu}+\gamma_g q_{\mu\nu} =\mathcal{F}_{\mu\nu}^{\rm gov}+\mathcal{R}_{\mu\nu}^{\rm audit}.τg(Dξq)μν+ℓg2(ΔLq)μν+γgqμν=Fμνgov+Rμνaudit.

A central clarification is the strict distinction between active-governance rate dormancy and state-level active retraction. Defining

Aμνgov:=Fμνgov+Rμνaudit−ℓg2(ΔLq)μν−γgqμν,\mathcal{A}_{\mu\nu}^{\rm gov} :=\mathcal{F}_{\mu\nu}^{\rm gov}+\mathcal{R}_{\mu\nu}^{\rm audit} -\ell_g^2(\Delta_L q)_{\mu\nu}-\gamma_g q_{\mu\nu},Aμνgov:=Fμνgov+Rμνaudit−ℓg2(ΔLq)μν−γgqμν,

the master equation gives

Aμνgov=τg(Dξq)μν.\mathcal{A}_{\mu\nu}^{\rm gov}=\tau_g(\mathcal{D}_\xi q)_{\mu\nu}.Aμνgov=τg(Dξq)μν.

Consequently, when τg \tau_g τg is nondegenerate and bounded below by a positive constant,

Aμνgov→0\mathcal{A}_{\mu\nu}^{\rm gov}\to 0Aμνgov→0

signifies only active-governance rate dormancy or a quasistationary response; it does not imply

qμν→0.q_{\mu\nu}\to 0.qμν→0.

State-level active retraction at the unidirectional stage requires withdrawal of the active deviation and the effective interface source, whereas GR-compatible bidirectional retraction further requires

qμν→0,Tμνgov→0,Jν→0.q_{\mu\nu}\to 0,\qquad T_{\mu\nu}^{\rm gov}\to 0,\qquad J_\nu\to 0.qμν→0,Tμνgov→0,Jν→0.

We further distinguish active–passive sectorwise closure from macro–micro scale decomposition. Passive and active equation residuals, together with bidirectional coupling terms, are controlled in a positive-definite direct-sum audit space, thereby excluding false total closure generated by cancellation between distinct sectors.

We formulate statistical macro–micro asymptotic decoupling, also referred to here as coarse-grained asymptotic orthogonality. This property does not require instantaneous cross-scale trajectories to decay monotonically in time; it requires fixed-window or equivalently coarse-grained cross-scale correlation statistics, or their overall envelope, to decay under increasing scale separation

ρ‾Mμrate(ϵs)=O(ϵsαr),ρ‾Mμgov(ϵs)=O(ϵsαg),\overline{\rho}_{M\mu}^{\rm rate}(\epsilon_s)=O(\epsilon_s^{\alpha_r}),\qquad \overline{\rho}_{M\mu}^{\rm gov}(\epsilon_s)=O(\epsilon_s^{\alpha_g}),ρMμrate(ϵs)=O(ϵsαr),ρMμgov(ϵs)=O(ϵsαg),

with

αr>0,αg>0,ϵs→0.\alpha_r>0,\qquad\alpha_g>0,\qquad\epsilon_s\to 0.αr>0,αg>0,ϵs→0.

Instantaneous or moving-window correlations may oscillate and may be periodically or intermittently restored during finite-scale activated governance. Conditional path transfer and scale re-anchoring intentionally maintain macro–micro information exchange when governance is required. During withdrawal or dormant phases, persistent governance-induced correlation is again suppressed. Statistical asymptotic decoupling therefore constrains the time-windowed statistical level or overall envelope of activation-modulated correlation rather than forbidding transient correlation recovery.

Five numerical interfaces are presented at unidirectional, reduced-protocol, or phenomenological levels. A static active-metric toy model illustrates a small galaxy-velocity response, while a phenomenological ringdown interface illustrates prescribed frequency, damping, and finite-time path-memory modifications. The low-dimensional damped oscillator first audits micro-diagnostic sensitivity, macro gating, conditional-path activation, and unified dormant retraction. In the low-noise null case, the governed observer returns to the ungoverned baseline within

7.73×10−15,7.73\times 10^{-15},7.73×10−15,

while the effective macroscopic governance rate remains zero. Under growing-noise stress, the RMSE is reduced by 75.7%, and under a localized sensor burst it is reduced by 30.3%.

A dynamically realized two-timescale audit is then added. Each ratio

τM/τμ∈{2,4,8,12,16,24}\tau_M/\tau_\mu\in\{2,4,8,12,16,24\}τM/τμ∈{2,4,8,12,16,24}

corresponds to an independently re-solved dynamics rather than to a different post-processing filter applied to the same trajectory. Across 32 paired random seeds, the governed time-windowed rate statistic yields

αrgov=0.1192,95% CI=[0.1132,0.1255],\alpha_r^{\rm gov}=0.1192,\qquad 95\%\ \mathrm{CI}=[0.1132,0.1255],αrgov=0.1192,95% CI=[0.1132,0.1255],

and the governed state statistic yields

αsgov=0.7906,95% CI=[0.7491,0.8232].\alpha_s^{\rm gov}=0.7906,\qquad 95\%\ \mathrm{CI}=[0.7491,0.8232].αsgov=0.7906,95% CI=[0.7491,0.8232].

Both lower confidence bounds are positive, directly supporting coarse-grained macro–micro asymptotic decoupling in the present low-dimensional dynamic proxy. The no-governance control also exhibits positive exponents, 0.4130 and 1.0905, respectively. The governed-minus-control exponent differences are negative, so the present test does not establish governance-specific acceleration of orthogonality. A more precise interpretation is that scale separation supplies a common asymptotic-decoupling limit, whereas active governance retains more persistent finite-scale exchange without destroying that statistical limit.

In the time-domain waveform-shape audit, the GR-null deviation is 1.77×10−4 1.77\times 10^{-4} 1.77×10−4; under a controlled structural anomaly, the global shape residual improves by 3.12%, the inside-anomaly residual improves by 5.37%, and the gate remains inactive outside the anomaly interval. The fifth interface constructs a quantum-consistent VCTG field–coherence closure candidate for double-slit interference. The null/retraction, no-record, and governance-symmetric-path limits return to the environmental baseline with a maximum visibility deviation of 6.03×10−14 6.03\times 10^{-14} 6.03×10−14. The final weak-record and strong-record visibilities are 0.18833 and 0.047639, respectively, while the reduced density-matrix evolution preserves trace, Hermiticity, and positivity to machine precision.

These results support the protocol-level executability of macro gating, unified dormant retraction, conditional path transfer, activation-modulated statistical asymptotic decoupling, anomaly-localized response, and the reduced field–coherence closure. They do not constitute a tensor-level proof in the complete bidirectional four-dimensional VCTG field system, real gravitational-wave model selection, an independently derived field–record bridge, or a quantum-measurement theorem. Complete bidirectional covariant closure, Bianchi and constraint-propagation audits, full-field statistical asymptotic decoupling, and blind predictions generated by the VCTG–QNM eigenvalue problem remain next-stage tasks.

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Rate Dormancy and State Retraction in Variational Curvature-Transfer Governance (VCTG) A GR-Compatible Active-Curvature-Governance Framework8.pdf

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Additional details

Identifiers

DOI
10.13140/RG.2.2.19587.67364

Dates

Other
2026-07-16
Preprint