Einstein–Matter Coupling from Spectral Stationarity

We separate a conditional variational result from a thermodynamic interpretation that is not yet established. Let SΠʳᵉⁿ[g]=1/2log'(Ag/μ²) be the renormalized projective spectral functional. Assuming a renormalized infrared metric response and an effective matter functional W_Π[g,ψ], stationarity of Γ_Π=SΠʳᵉⁿ-W_Π yields Gμν+Λᵣₑₙgμν=8π GN T(Π)μν at leading derivative order. The coefficient follows directly from the ratio between the matter variation factor $1/2$ and the geometric coefficient (16π GN)⁻¹. We then audit the diagonal heat-kernel quantity u(x;t)=-∂ₜlog K(x,x;t). For a four-dimensional minimal scalar Laplacian it has dimension L⁻² and expansion u=2/t-R/6+O(tR²,t∇²R). Its normalized form b(x;t)=tu(x;t)/2=1-tR/12+⋯ 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 δ Q=T\,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.