Local Spectral Thermodynamics and the Einstein Equation as a Spectral Equilibrium Condition

We develop a local thermodynamic interpretation of the projective spectral entropy functional SΠ[g] = 1/2log' Ag, extending the covariant framework established in a companion paper. The diagonal heat kernel $K(x,x;t)$ defines a local spectral energy density u(x;t) = -∂ₜ log K(x,x;t), whose Seeley–DeWitt expansion yields a local spectral multiplier field β(x)⁻¹ = β₀⁻¹ - 1/6R(x) + O(∇² R). Here β₀⁻¹ = 2/t_* is a universal UV contribution fixed by the spectral admissibility scale t_*, while the curvature correction is geometric and covariant. We then formulate a local spectral first law as a compatibility theorem: in the infrared-dominant regime, the metric variation of SΠ and the metric variation of the projected matter functional are related through the local multiplier field, yielding a constraint δSΠ = ∫ β(x)⁻¹\,δ E_Π(x). This is not a definition but a structural consequence of the Seeley–DeWitt hierarchy and the induced-gravity normalization established in the companion paper. We then impose a spectral equilibrium principle: admissible geometries extremize SΠ at fixed projected matter content. The resulting Euler–Lagrange equation is Einstein's field equation Gμν = 8π G\, Tμν, with the factor 8π G arising purely from the normalization conventions of the geometric and matter sides. The derivation requires no horizon structure, no Rindler wedge, no Raychaudhuri theorem, and no assumption of local Lorentz invariance at the Planck scale. Einstein's equation emerges as a spectral extremum condition within the effective infrared geometry.