Overview
This article separates a conditional variational result from a thermodynamic interpretation that is not yet established. Let $S_\Pi^{\mathrm{ren}}[g]=\tfrac12\log\det'(A_g/\mu^2)$ 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 factor $8\pi G_N$ is the ratio between the matter variation factor $\tfrac12$ and the geometric coefficient $(16\pi G_N)^{-1}$ — a consequence of the conventions, not an independent prediction of the coupling.
Core contributions
- Conditional variational coupling: given a renormalized infrared metric response and a matter functional, stationarity of $\Gamma_\Pi$ yields the Einstein equation with a cosmological term. The result is proved given the supplied coefficients.
- Dimensional audit of the heat-kernel quantity: $u(x;t)=-\partial_t\log K(x,x;t)$ carries dimension $L^{-2}$ — an eigenvalue scale — with $u=2/t-R/6+O(t\mathcal{R}^2,t\nabla^2R)$.
- A dimensionless spectral response: the normalized form $b(x;t)=t\,u(x;t)/2=1-tR/12+\cdots$ is well defined and covariant.
- No temperature, no first law: neither $u$ nor $b$ is a temperature, an inverse temperature, or an energy density without an additional calibration map. Consequently no local first law follows from the heat kernel alone.
- The bridge is named, not crossed: a thermodynamic completion requires an independently defined heat one-form, a physical energy notion, and an integrability theorem.
Why $u$ is not a temperature
For a Laplacian $A_g\sim L^{-2}$, the heat parameter is a diffusion parameter with $t\sim L^2$, so $u=-\partial_t\log K\sim L^{-2}$ — the same dimension as the scalar curvature $R$. In natural units a physical temperature, like an energy, carries dimension $L^{-1}$.
The quantity $u$ is therefore an eigenvalue scale, not a thermal one. Reading it as a temperature is not a matter of a missing factor or a mislabelled exponent: it is an identification that the heat kernel does not license. An expression of the form $\delta S_\Pi=\int\beta(x)^{-1}\delta E_\Pi$ does not survive this audit.
The open thermodynamic bridge
Establishing a genuine thermodynamics requires, in order: an independently defined heat one-form $\xi_{\mathrm{sp}}$ on configuration space; a physical energy notion; and an integrability theorem showing that $\ker\xi_{\mathrm{sp}}$ is flat, hence that $\delta Q=T\,\mathrm{d}S$ holds locally.
This is the classical criterion of Carathéodory, revisited by Jauch and given a modern geometric form by Bryan W. Roberts, who shows that heat defines a gauge connection on a line bundle over work configurations: vanishing curvature is equivalent to the local existence of entropy and temperature functions, while global equilibrium may still fail through a thermal analogue of geometric phase. That criterion turns a terminological reservation into a precise mathematical test — one this paper does not yet pass, and does not claim to.
Relation to the Gravity paper
The renormalized infrared metric response assumed here is supplied by the companion Gravity paper, which separates the spectral-cutoff and zeta sectors and identifies the finite Einstein coefficient $c_{\mathrm{EH}}^{\mathrm{ren}}$ as a matching datum rather than a prediction. The conditional character of the present result follows directly: what is supplied there is assumed here.
No horizon structure, Rindler wedge, or Raychaudhuri equation is required. The derivation is entirely variational and spectral.
Interpretive status
The conditional content is a variational coupling between a renormalized geometric functional and matter. Reading that stationarity as equilibrium remains a programme-level interpretation, not a result of this paper: the word carries thermodynamic commitments the construction does not yet earn.
This is a deliberate narrowing. A weaker claim that survives scrutiny is worth more to the programme than a stronger one that does not, and naming the missing link precisely is what makes it addressable.
References
Jérôme Beau. Einstein–Matter Coupling from Spectral Stationarity: A Conditional Variational Result and the Open Thermodynamic Bridge. Preprint. 10.5281/zenodo.18825655
B. W. Roberts. Heat as a Gauge Connection. arXiv:2503.08753