This repository contains the source of Einstein--Matter Coupling from Spectral Stationarity: A Conditional Variational Result and the Open Thermodynamic Bridge.
This note separates a conditional variational identity from a thermodynamic reading that is not established.
Let
[ S_\Pi^{\mathrm{ren}}[g] = \frac12\log\det'!\left(A_g/\mu^2\right) ]
be the renormalized projective spectral functional, and let (W_\Pi[g,\psi]) be an independently defined effective matter functional. If the leading infrared metric response of (S_\Pi^{\mathrm{ren}}) contains specified renormalized Einstein and cosmological terms, 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 local derivative order. The coefficient is the ratio
[ \frac{1/2}{(16\pi G_N)^{-1}}=8\pi G_N. ]
This is a conditional joint geometric--matter variational statement. The companion Gravity paper separates the spectral-cutoff and zeta sectors and treats the finite Einstein coefficient as a matching datum. It is not, by itself, a thermodynamic equilibrium theorem.
For the minimal scalar Laplacian in four dimensions,
[ u(x;t)=-\partial_t\log K(x,x;t) =\frac2t-\frac16R(x)+O(t\mathcal R^2,t\nabla^2R). ]
Because ([t]=L^2), the spectral rate has dimension ([u]=L^{-2}). Its normalized form is
[ b(x;t)=\frac{tu(x;t)}2 =1-\frac{tR(x)}{12}+O(t^2\mathcal R^2,t^2\nabla^2R). ]
The quantity (b) is a dimensionless curvature response. Neither (u) nor (b) is identified with a physical temperature, inverse temperature, or energy density.
A thermodynamic completion requires: an independently defined heat one-form, a Lorentzian energy notion, an operational temperature calibration, and an integrability theorem establishing (\delta Q=T,\mathrm dS).
- The diagonal heat kernel defines a local spectral rate and a dimensionless normalized curvature response.
- Joint stationarity of the geometric and matter functionals yields the infrared Einstein--matter equation under explicit renormalization hypotheses.
- Once the renormalized Einstein coefficient is supplied, the factor (8\pi G_N) follows from a ratio of variation coefficients.
- A thermodynamic interpretation remains open and is not used in the proof.
Spectral determinant, heat kernel, induced gravity, Einstein equation, matter coupling, variational principle, thermodynamic integrability
J. Beau, Einstein--Matter Coupling from Spectral Stationarity: A Conditional Variational Result and the Open Thermodynamic Bridge, 2026.
Portions of the development benefited from iterative interactions with large language models used as analytical assistants. All claims, interpretations, and final formulations remain the author's responsibility.