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 relationδSEHent =(1 / 4ℏGscalarind,UV) δA,for a regulated massless minimally coupled scalar under the explicit assumptions statedbelow. The analysis separates two logically independent branches. The information-geometricbranch is the Araki–BKM second-variation/first-law branch for bounded modular-analyticprobes. The geometric branch is the local heat-kernel/replica branch that fixes the Einstein–Hilbert coefficient through the universal a1 term and a local factorizing regulator. The BKMsusceptibility is not assumed to determine the local a1 coefficient. Instead the two branchesmeet at local equilibrium through the regulated entanglement first law. The constant zeromode is treated explicitly, the Gaussian replica trace function is controlled for real n > 1approaching 1, and the conical spectral approximation is stated with the C1 control neededto interchange the replica derivative with the regulator limit. At leading ultraviolet order inthe single-scalar Einstein–Hilbert sector, the induced coefficient satisfies 1 / ℏGscalarind,UV = Λ2 / 12π.The full finite-cutoff Einstein–Hilbert coupling may also contain bare and finite local termsand is not fixed by the leading heat-kernel coefficient alone. No ultraviolet continuum limit istaken, and no identification with the observed Newton constant is claimed.“This preprint is a revised version of a manuscript previously submitted to the Journal of Mathematical Physics.”