Einstein–Matter Coupling from Spectral Stationarity

We separate a conditional variational result from a thermodynamic interpretation that is not yet established. Let ${S_{\Pi}}^{\mathrm{ren}}[g]={\tfrac{1}{2}}\log\det'({A_g}/\mu^2)$ be the renormalized projective spectral functional. Assuming a renormalized infrared metric response and an effective matter functional $W_\Pi[g,\psi]$, stationarity of $\Gamma_\Pi={S_{\Pi}}^{\mathrm{ren}}-W_\Pi$ yields $G_{\mu\nu}+\Lambda_{\mathrm{ren}}g_{\mu\nu}=8\pi G_N T^{(\Pi)}_{\mu\nu}$ at leading derivative order. The coefficient follows directly from the ratio between the matter variation factor $1/2$ and the geometric coefficient $(16\pi G_N)^{-1}$. We then audit the diagonal heat-kernel quantity $u(x;t)=-\partial_t\log K(x,x;t)$. For a four-dimensional minimal scalar Laplacian it has dimension $L^{-2}$ and expansion $u=2/t-R/6+O(t\mathcal{R}^2,t\nabla^2R)$. Its normalized form $b(x;t)=tu(x;t)/2=1-tR/12+\cdots$ is a dimensionless spectral response. Neither quantity is a physical temperature, an inverse temperature, or an energy density without an additional calibration map. Consequently no local first law follows from the heat kernel alone. A thermodynamic completion requires an independently defined heat one-form on configuration space, a physical energy notion, and an integrability theorem establishing $\delta Q=T\,{\mathrm{d}} S$. Interpretive status. The conditional content is a variational coupling between a renormalized geometric functional and matter. The companion Gravity paper separates the spectral-cutoff and zeta sectors and identifies the finite Einstein coefficient as a matching datum. Reading that stationarity as equilibrium remains a programme-level interpretation, not a result of this paper.