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

We propose a causal-diamond formulation of semiclassical gravity in which a finite-resolution boundary regulator (Coherency Screen) supplies the minimal edge structure required for a local description in a Wheeler–DeWitt setting. Diamond-local dynamics are defined by an informational variational principle: for each diamond O, the effective cost functional is the relative entropy S_rel(ρ_O || σ_O[g]) between the reduced physical state and a geometric reference family. In the small-diamond modular/KMS regime, a derivative expansion of this cost, implemented via a heat-kernel spectral expansion, yields a local effective action whose leading terms recover the Einstein sector and select a spinorial (Dirac-type) transport structure. A discrete edge-mode counting, together with Newton’s constant G, fixes a characteristic resolution scale M_s ~ 3×10^13 GeV. Treating M_s as the onset of the leading stiffness correction places the high-curvature regime in a plateau universality class, giving a capacity-set scalar amplitude and a tensor target r ~ 10^-3. We further discuss how the same boundary logic constrains the gauge and mass sectors in a spectral-action-compatible formulation, suggesting discrete relations among effective coupling normalizations and a structured organization of charged-lepton scales via geometric accessibility of the boundary algebra. We also outline late-time phenomenological extensions in which finite-resolution boundaries induce a mild running of effective stiffness and horizon-set acceleration scales. Overall, the construction yields a compact set of correlated, falsifiable targets tied to a single microscopic resolution scale.