Part I: A Finite-Cutoff Araki–BKM Framework for Gravitational Susceptibility

This paper is the first in a three-part series developing a controlled finite-cutoff connection between quantum information geometry and semiclassical gravity under explicitly stated assumptions. Here, we lay the foundational algebraic framework, in which the Araki– Bogoliubov–Kubo–Mori (BKM) metric of a local von Neumann algebra is formulated as a regulated gravitational susceptibility. To overcome the unboundedness of the continuumstress-tensor, we evaluate the BKM geometry on a bounded, modular-analytic regulated operator constructed via Gaussian smoothing. The regulator is constructed so that, at finite cutoff, VM,ε belongs to the bounded modular-analytic class required by the Araki perturbation framework; the corresponding construction and its covariance properties are established in Part II. The Lorentzian spacetime signature is obtained from the KMS condition supplemented by a Wick rotation postulate, applied explicitly at this regulated level. Furthermore, we exhibit a Kähler structure on the regulated transverse-traceless sector, providing a compatible geometric normalization in which the quantum of action (ℏ) and the gravitational scale (Ginfo) enter through the combination Carea := ℏGinfo. Finally, we contextualize the regulated Einstein–Hilbert-sector LIE relation derived in Part III under explicit assumptions, which anchors the dimensionless information geometry to physical area via the informationgeometric gravitational coupling Ginfo. By cleanly separating robust finite-cutoff theorems from open continuum-limit conjectures, this foundational part sets the stage for the spectraland replica-geometric derivations in the subsequent parts.“This preprint is a revised version of a manuscript previously submitted to the Journal of Mathematical Physics.”