Einstein–Matter Coupling from Spectral Stationarity

We separate a conditional variational result from a thermodynamic interpretation that is not yet established. Let S_{}^reng=1{2}' (Ag/²) be the renormalized projective spectral functional. Assuming a renormalized infrared metric response and an effective matter functional W_, , stationarity of _=S_{}^ren-W_ yields G_+ₑ₄₍g_=8 GN T^ () _ at leading derivative order. The coefficient follows directly from the ratio between the matter variation factor 1/2 and the geometric coefficient (16 GN) ^-1. We then audit the diagonal heat-kernel quantity u (x;t) =-ₜ 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, t²R). 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 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.