The Gauge–Gravity Stratification Sub-Programme

The gauge–gravity stratification sub-programme of the Cosmochrony corpus addresses a single central question: why does the same admissible spectral functional produce Einstein gravity and Yang–Mills gauge dynamics at different spectral orders? The answer replaces the classical question ``what symmetry unifies gravity and gauge?'' with ``at what Seeley–DeWitt order does the admissible projection respond?''. Starting from the joint spectral entropy functional ${S_\Pi}[g,A] = \tfrac{1}{2}\log\det' A_{g,A}$, the sub-programme establishes two things. Q12 gives the Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ from the $a_4$ vertical variation at fixed base metric, the gauge component $\Omega^{\mathrm{gauge}}_{\mu\nu} = F_{\mu\nu}$ being isolated from the total bundle curvature. Q13 gives a conditional Einstein–Yang–Mills system from the joint variation of a single matched renormalized local action: the metric variation of the same $\mathrm{tr}_\rho(F^2)$ term supplies the Yang–Mills energy–momentum tensor, and since $\mathrm{tr}_\rho(F^2) = {I_\rho} F^a_{\mu\nu}F^{a\,\mu\nu}$, canonical matching fixes ${c_{\mathrm{EH}}} = 1/(16\pi{G_{\mathrm{N}}})$ and ${c_{F}}{I_\rho} = 1/(4{g_{\mathrm{YM}}}^2)$, so that $8\pi{G_{\mathrm{N}}} = 1/(2{c_{\mathrm{EH}}})$ depends on the Einstein coefficient alone. No coupling is predicted: ${G_{\mathrm{N}}}$, ${g_{\mathrm{YM}}}$ and the cosmological coefficient are independent renormalized matching data, and no numerical value for ${G_{\mathrm{N}}}{g_{\mathrm{YM}}}^2$ follows. The $a_6$ sector is given only as a representative spanning list of invariants, modulo integrations by parts and Bianchi identities, with matching coefficients. Statements about $\Lambda^4$, $\Lambda^2$ and $\log\Lambda$ behaviour belong specifically to a proper-time cutoff and are not scheme-independent; the derivations are Euclidean. Interpretive status. The stratification $a_2 \to \text{gravity}$, $a_4 \to \text{gauge}$ is offered as an organising reading, not a theorem.