Overview
This article develops a local thermodynamic interpretation of the projective spectral entropy functional $S_\Pi[g]=\tfrac12\log\det' A_g$, defined by a minimal Laplace-type operator on a four-dimensional Riemannian manifold. The diagonal heat kernel $K(x,x;t)$ defines a local spectral energy density $u(x;t)=-\partial_t\log K(x,x;t)$ whose Seeley–DeWitt expansion yields a local multiplier field $\beta(x)^{-1}=\beta_0^{-1}-\tfrac16 R(x)+O(\nabla^2R)$.
In the infrared-dominant regime $R\,\ell_\chi^2\ll1$, the metric variation of the renormalized spectral entropy and the metric variation of the projected matter functional are related by a compatibility identity $\delta S_\Pi=\int\beta(x)^{-1}\delta E_\Pi(x)$. Imposing a constrained extremum of $S_\Pi$ at fixed projected matter content yields $G_{\mu\nu}=8\pi G_N T_{\mu\nu}$.
Core contributions
- Local spectral energy density: $u(x;t)=-\partial_t\log K(x,x;t)$ defined from diagonal heat-kernel data.
- Curvature-corrected multiplier field: the $a_2=\tfrac16R$ coefficient controls the leading geometric correction to $\beta(x)^{-1}$.
- Spectral first law as compatibility condition: $\delta S_\Pi=\int\beta^{-1}\delta E_\Pi$ holds at leading infrared order if and only if Einstein’s equation is satisfied.
- Spectral equilibrium principle: extremizing $S_\Pi$ at fixed projected matter content yields $G_{\mu\nu}=8\pi G_N T_{\mu\nu}$.
- Normalization of $8\pi G_N$: the factor arises uniquely from the geometric normalization $(16\pi G_N)^{-1}$ and the stress-tensor definition factor $1/2$.
Local multiplier field
Evaluating the spectral energy density at a fixed admissibility scale $t_*$ gives $\beta(x)^{-1}=2/t_*-\tfrac16R(x)+O(\nabla^2R)$. The universal term $\beta_0^{-1}=2/t_*$ depends on the spectral resolution scale and is not a physical temperature. The curvature term is covariant and entirely controlled by the Seeley–DeWitt hierarchy.
Choosing spectral units where $\beta_0^{-1}=1$ corresponds to fixing $t_*=2$. This is a gauge choice in spectral resolution and does not affect the induced Newton coupling.
Spectral equilibrium
The constrained functional $\mathcal{F}[g,\psi]=S_\Pi[g]-\beta_*^{-1}W_\Pi[g,\psi]$ defines a spectral analogue of free energy. Stationarity under compactly supported metric variations gives $\tfrac{1}{16\pi G_N}G_{\mu\nu}=\tfrac{\beta_*^{-1}}{2}T_{\mu\nu}$. For the canonical choice $\beta_*^{-1}=1$, one recovers Einstein’s field equation.
No horizon structure, Rindler wedge, or Raychaudhuri equation is required. The derivation is entirely variational and spectral.
Relation to induced gravity
The geometric coefficient $(16\pi G_N)^{-1}$ originates from the $a_2$ pole subtraction in the zeta-regularized determinant. In the weak-curvature regime $R\,\ell_\chi^2\ll1$, higher Seeley–DeWitt coefficients contribute only suppressed corrections, and the Einstein term dominates the infrared response.
Relation to the Cosmochrony program
This article is formulated as a self-contained spectral-geometry analysis. It connects to the broader Cosmochrony framework through the admissibility scale $t_*$ and the induced-gravity normalization, but it is intended to be evaluable independently of the full pre-geometric program.
References
Jérôme Beau. Local Spectral Thermodynamics and the Einstein Equation as a Spectral Equilibrium Condition. Preprint. 10.5281/zenodo.18825656