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

This work formulates semiclassical gravity within a causal-diamond framework where a finite-resolution boundary provides the edge structure for a local Wheeler--DeWitt description. Because the diffeomorphism-invariant Hilbert space does not factorize, each diamond is equipped with a boundary-completed algebra $\mathfrak{A}_O$, ensuring the operational state $\rho_O$ and the semiclassical reference family $\sigma_O[\Lambda]$ share identical operator content. Dynamics are posed as local statistical inference: the relative-entropy functional $S_{\rm rel}(\rho_O\|\sigma_O[\Lambda])$ quantifies the mismatch between data and reference. This yields the minimal operational axioms defining subsystems, intrinsic clocks, and regulated observables in a finite-resolution, background-independent setting.The topology-locked boundary capacity budget fixes an effective channel multiplicity $N \approx 1.23\times10^{11}$. Calibrating its coherent fraction to Newton's constant determines a matching scale $M_s \approx 3.02\times10^{13}\,{\rm GeV}$. In the modular/KMS regime, the relative-entropy Hessian (Kubo--Mori metric) block-diagonalizes into orthogonal tensor, vector, and scalar sectors. A quasi-local heat-kernel expansion on the fixed $S^3\times S^1$ history manifold maps this near-equilibrium response to a matching-scale EFT, yielding the Einstein--Hilbert tensor structure, Yang--Mills susceptibilities, and leading mass deformations. Vector and scalar responses remain intensive, while the tensor response scales extensively.The reduced scalar-curvature sector fixes the $R^2$ plateau coefficient, equivalently locking the scalaron scale to $M_R=M_s$. The fixed modular protocol and quantized boundary currents give the matching-scale gauge condition $\alpha^{-1}(M_s)=4\pi k$ for normalized current blocks. In the selected Standard Model boundary inventory this yields integer current levels for the strong and electromagnetic sectors, a weak-mixing projection, and a separate static electromagnetic susceptibility on $S^3\times S^1$. The same boundary capacity gives linked plateau targets $n_s\simeq0.965$, $r\simeq0.0038$, and $A_s\simeq2.1\times10^{-9}$, while single-pixel saturation gives electroweak-scale consistency checks for the Higgs, vacuum expectation, and top-sector scales.Translations between causal diamonds act as completely positive trace-preserving (CPTP) updates. The resulting open-modular Walsh filtration algebraically isolates a threefold matter-generation carrier and lepton-mass accessibility invariants. Treating continuum fields as the response of a finite boundary, the framework yields correlated, falsifiable relations for gravitational stiffness, gauge response, plateau cosmology, electroweak saturation, and threefold matter-sector organization from one minimal operational architecture.