Infrared Einstein Response from a Renormalized Spectral Entropy Functional

We investigate the covariant metric variation of a minimal spectral entropy functional \( S_\Pi[g] = \frac12 \log \det\nolimits' A_g \) defined by a Laplace-type elliptic operator on a four-dimensional Riemannian manifold. Using heat-kernel methods and zeta regularization, we analyze the renormalized functional and determine the geometric structure of the resulting effective stress tensor. We show that the renormalized spectral variation decomposes into a local two-derivative Einstein tensor term, a cosmological term, local four-derivative quadratic curvature contributions, non-local form factors, and a conformal anomaly piece. The Einstein tensor arises from the counterterm associated with the $a_2$ heat-kernel pole, in direct analogy with induced gravity in the sense of Sakharov. In the weak-curvature regime where all dimensionless curvature invariants satisfy \(R\,\ell_{\mathrm{sp}}^2 \ll 1\), the local two-derivative contribution dominates over higher-derivative and non-local terms. The induced Newton coupling scales parametrically with the spectral cutoff as \(G_N \sim 16\pi^2 \ell_{\mathrm{sp}}^2\), up to scheme-dependent factors. These results show that the infrared-dominant local part of the renormalized spectral entropy variation reproduces the Einstein tensor. We further show that the ultraviolet completion of this infrared sector is uniquely determined, under a coherence extensivity hypothesis, to be of determinantal Born–Infeld type: the action density takes the Eddington-inspired form $\sqrt{-\det(g_{\mu\nu} + \ell_{\mathrm{sp}}^{2}\, R_{\mu\nu})} - \sqrt{-g}$. This mechanism does not rely on a specific microscopic completion, but follows from general properties of the renormalized spectral functional. In this sense the resulting Einstein response appears as the universal two-derivative infrared sector of the effective geometric dynamics, and its Born–Infeld completion as the unique admissible spectral ultraviolet extension. Geometric gravity therefore appears as an effective low-energy response of spectral operator structure rather than as a fundamental dynamical input.