Conditions for an Infrared Einstein Sector from Spectral Geometry

We determine the conditions under which a four-dimensional Laplace-type operator can support a local infrared Einstein sector. The analysis separates two regularization statements that must not be conflated. A physical proper-time cutoff $\Lambda=\ell_{\mathrm{sp}}^{-1}$ produces power-sensitive local terms $a_0\Lambda^4$ and $a_2\Lambda^2$, while zeta regularization produces finite and logarithmic scale dependence governed in four dimensions by $a_4=\zeta_A(0)$. For $S_\Pi=+\tfrac12\log\det' A_g$, one minimal scalar contributes $\Delta c_{\mathrm{EH}}=-\Lambda^2/[12(4\pi)^2]$ to the Einstein–Hilbert coefficient. The finite renormalized value, its sign, and the observed Newton constant require a matching condition and the complete operator content; they are not predicted by this determinant alone. Once these coefficients are supplied, the metric variation decomposes into Einstein, cosmological, higher-derivative, and non-local sectors. The Einstein term dominates locally when the renormalized coefficient ratio satisfies $R L_4^2\ll1$. Under the separate coherence-extensivity hypothesis, the determinantal Born–Infeld density is an admissible tensorial completion of a supplied Einstein infrared term, but the structural conditions do not select it uniquely: an explicit one-parameter family of distinct admissible completions shares its infrared limit and saturation behaviour. Interpretive status. Spectral geometry fixes the allowed local structures and their cutoff sensitivities, but it does not yet derive the positive finite gravitational coupling. The result identifies the matching problem that must be solved before gravity can be claimed as an emergent spectral prediction.