Part III: Regulated Derivation of Local Information Equilibrium Araki–BKM Geometry, Local Replica Regularization, Spectral Matching, and the Regulated Einstein–Hilbert Area Law

We present a finite-cutoff derivation of the Einstein–Hilbert-sector Local InformationEquilibrium (LIE) area relationδSEH ent = (1 / 4ℏGinfo) δA,for a regulated massless minimally coupled scalar. Version 2 separates two logically independent branches. The information-geometric branch is the Araki–BKM second-variation/firstlaw branch for bounded modular-analytic probes. The geometric branch is the local heatkernel/replica branch that fixes the Einstein–Hilbert coefficient through the universal a1 term and a local factorizing regulator. The BKM susceptibility is not assumed to determine thelocal a1 coefficient. Instead the two branches meet at local equilibrium through the regulatedentanglement first law. The constant zero mode is treated explicitly, the Gaussian replicaidentity is extended to real replica index near n = 1, and the conical spectral approximationis stated with the C1 control needed to interchange the replica derivative with the regulatorlimit. At fixed ultraviolet cutoff,Ginfo = Gscalareff =12π / ℏΛ2within the single-scalar Einstein–Hilbert sector. No ultraviolet continuum limit is taken, andno identification with the observed Newton constant is claimed.