Part II: Regulated BKM–Euclidean Hessian and Finite-Cutoff Gravitational Susceptibility

We establish a finite-cutoff relation between quantum information geometry and semiclassical gravity through the Araki–Bogoliubov–Kubo–Mori susceptibility. For each finite regulator (Λ, M, ε) and with the corresponding spectral/heat-kernel regularization of the Euclidean determinant, the regulated Araki–BKM Hessian admits the stated comparison with the Euclidean metric Hessian, up to local contact terms. We construct a local, bounded, modular-analytic regulator VM,ε by Gaussian smoothing along the modular flow, allowing the bounded Araki perturbation theorem to apply rigorously. The Euclidean Hessian decomposes into bubble and local contact terms via the Seeley–DeWitt heat-kernel expansion. The continuum limit remains open and requires renormalization and endpoint estimates. Thespecific canonical BKM–modular construction considered here does not by itself fix an absolute scale; a Local Information Equilibrium relation introduces the information-geometric coupling Ginfo and defines the area scale Carea := ℏGinfo (in c = 1 units). A Kähler structure on the transverse-traceless sector determines the physical scaling of the symplectic form by Ginfo. The analysis separates finite-cutoff theorems from open conjectures.