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

We develop a local thermodynamic interpretation of the projective spectral entropy functional S_{}g = 12' 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 K (x, x;t), whose Seeley–DeWitt expansion yields a local spectral multiplier field (x) ^-1 = ₀^-1 - 16R (x) + O (² R). Here ₀^-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 S_{} and the metric variation of the projected matter functional are related through the local multiplier field, yielding a constraint S_ = (x) ^-1\, E_ (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 S_{} at fixed projected matter content. The resulting Euler–Lagrange equation is Einstein's field equation G_ = 8 G\, T_, with the factor 8 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.