A Variational Governance Framework for Positive Mass-Response in Yang–Mills-Type Models within the VCG/CAFVCG Equivalence Class

We formulate a Gen-6 Variational Active Governance (VAG) extension of the VCG/CAFVCG framework for Yang–Mills-type governed mass response. Gen-6 advances the existing cross-regional curvature-governance mechanism by introducing conditionally activated, directed CAF echo channels and a conditional-path re-anchoring rule. For a receiving region DaD_aDa and a candidate supply region DbD_bDb, a directed channel $a\leftarrow b$ is activated only when the receiving region is both above an absolute governance-curvature threshold and sufficiently more curved than the supply region:

Aa←b(t)=Gε0 ⁣(κˉa(t)−κact)Gε1 ⁣(κˉa(t)−κˉb(t)−ϑab).A_{a\leftarrow b}(t) = G_{\varepsilon_0} \!\left( \bar\kappa_a(t)-\kappa_{\rm act} \right) G_{\varepsilon_1} \!\left( \bar\kappa_a(t)-\bar\kappa_b(t)-\vartheta_{ab} \right).Aa←b(t)=Gε0(κˉa(t)−κact)Gε1(κˉa(t)−κˉb(t)−ϑab).

Only such lower-curvature-to-higher-curvature channels contribute non-local CAF terms to the variational metric update. If a still lower-curvature region is detected along an active channel, a smooth re-anchoring gate activates a new admissible directed contribution from the newly identified supply region. The resulting conditional-path CAF budgets are incorporated into a covariant SEC-constrained governance flow,

Lξhij,rgov∈−Mr,ijeff    klδVCP-Gen6δhkl,r−NAr(hr),\mathcal L_\xi h_{ij,r}^{\rm gov} \in - \mathbb M_{r,ij}^{\rm eff\;\;kl} \frac{\delta\mathcal V_{\rm CP\text{-}Gen6}} {\delta h_{kl,r}} - N_{\mathcal A_r}(h_r),Lξhij,rgov∈−Mr,ijeffklδhkl,rδVCP-Gen6−NAr(hr),

where SEC acts as a covariant proximal projection after the conditional variational update has been assembled.

Within the VCG/CAFVCG/VAG governance equivalence class, the core Yang–Mills-type protocol-level mass-response observable is defined by

mgov=M0∂s2log⁡Be(s),m_{\rm gov} = M_0\partial_s^2\log B_e(s),mgov=M0∂s2logBe(s),

with an optional higher-order scale-flow extension

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

Under explicitly stated internal non-degeneracy assumptions, the framework supports a positive protocol-level governed mass response. Numerical audits on simplified $2D/3D/4D$ $U(1)$ lattice models and a $3D$ vorticity toy model provide preliminary consistency evidence for bounded governed dynamics, directional coverage, and positive internal response under the implemented protocols.

At a metric-stationary dormant interface, if activated CAF terms, re-anchoring contributions, SEC corrections, and operator remainders vanish without residual governance modes, the governed formulation conditionally retracts to its baseline classical gravitational and Yang–Mills sectors. The CAF echo admits a model-level quantum-echo interpretation, but no quantum correlator, quantum-channel theorem, or non-Abelian Yang–Mills spectral-gap theorem is derived here. Accordingly, the present work provides a constructive governance mechanism within the VCG/CAFVCG/VAG equivalence class, rather than an unconditional proof of the Yang–Mills mass gap.

Furthermore, this work introduces VCTG, or Variational Curvature-Transfer Governance, as the Gen-6 extension layer to distinguish different observational sectors that originate from the same curvature-deviation governance signal in VCG/CAFVCG. Within this framework, gravity is understood as the local first-order compensation response to curvature deviation; the dark-matter-like effect is interpreted as the accumulation process of first-order curvature responses along regional, historical, or non-local governance paths, manifesting macroscopically as an additional gravity-like residual or effective geometric impedance; and mass corresponds to the second-order response of the governance scalar along the scale flow. By introducing conditional path-transfer mechanisms subject to curvature-ordering and non-amplifying constraints, VCTG enables governed transfer, accumulation, dormancy, and re-anchoring of both first-order curvature responses and second-order governance responses across regions, while preserving the local core definitions of VCG/CAFVCG.

This work does not reject the observational achievements of previous gravitational, dark-matter, or mass-related studies. Instead, it reorganizes these phenomena under a curvature-first governance framework. VCG/CAFVCG interprets gravity and mass as the local first-order response and the second-order scale-flow response of the curvature-governance structure, respectively. VCTG further introduces conditional path transfer, dynamic re-anchoring, non-amplifying constraints, and path accumulation, thereby interpreting dark-matter-like effects as additional gravity-like residuals or effective geometric impedance generated by the accumulation of first-order curvature responses along regional, historical, or non-local governance paths. In this sense, the proposed framework is not in direct conflict with prior theories, but provides a unified, mechanistic, and extensible reinterpretation of existing observational phenomena.