Relative-Entropy Variational Principle for Semiclassical Gravity with Finite-Resolution Boundaries

We propose a causal-diamond formulation of semiclassical gravity where a finite-resolution boundary regulator (Coherency Screen) supplies the edge structure for a local Wheeler--DeWitt description. Dynamics are defined by an informational principle: for each diamond \( O \), the action is the relative entropy \( S_{\mathrm{rel}}(\rho_O\|\sigma_O[\lambda]) \) between the physical state and a reference family on a fixed algebra. In the modular/KMS regime, the vacuum is at entanglement equilibrium; the leading dynamics become a linear-response problem governed by the Hessian of relative entropy (Kubo--Mori metric). This Hessian organizes deformations into tensor, vector and scalar sectors, yielding Einstein stiffness, Yang--Mills susceptibilities and mass gaps. The resulting local EFT is organized by a heat-kernel expansion (identifying the leading \( R^2 \) operator) and is compatible with a spinorial transport structure. Edge-mode counting and Newton's constant \( G \) fix the resolution scale at \( M_s \sim 3\times 10^{13}\,\mathrm{GeV} \). Identifying \( M_s \) with stiffness saturation places the high-curvature regime in a plateau universality class, predicting a tensor-to-scalar ratio \( r \sim 10^{-3} \). We further discuss how this boundary logic constrains gauge and mass sectors, suggesting discrete coupling relations and a geometric hierarchy for charged leptons. The construction yields correlated, falsifiable targets tied to a single scale.