Conditions for an Infrared Einstein Sector from Spectral Geometry

We investigate a minimal operator-theoretic mechanism by which a Newtonian interaction emerges from the variation of a projective entropy functional. For a positive elliptic Laplace-type operator $A$ in three spatial dimensions, we define $S_\Pi[A] = \tfrac12 \log \det{}' A$ and show that, in the weak-perturbation regime, its first variation reduces to the resolvent trace $\delta S_\Pi = \tfrac12 \mathrm{Tr}(A^{-1}\delta A)$. The spatial structure of the entropy response is therefore governed entirely by the Green kernel of $A$. In three dimensions, the Green function of a Laplace-type operator exhibits a universal $1/r$ decay. Consequently, the entropy variation necessarily inherits a Newtonian long-range profile, independently of any assumed force law. We demonstrate this mechanism numerically using a discrete three-dimensional Laplacian with localized symmetric perturbations.The simulations confirm three key structural features:(i) the validity of the first-order resolvent expansion,(ii) the dimensionally determined $1/r$ Green profile, and(iii) the existence of a finite spatial coherence threshold forgenerating a macroscopic entropy response. Compact perturbations alter the spectrum locally without inducing a long-range effect, whereas sufficiently extended perturbations produce a coherent Newtonian amplitude shift. The analysis provides operator-theoretic evidence that a Newtonian potential can arise as a resolvent-mediated consequence of projective entropy variation in three spatial dimensions. The framework is purely elliptic and restricted to the weak-field regime, and does not address relativistic dynamics.