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

This paper is the first in a three-part series establishing a rigorous, finite-cutoff connectionbetween quantum information geometry and semiclassical gravity. Here, we lay the foundational algebraic framework, in which the Araki–Bogoliubov–Kubo–Mori (BKM) metric ofa local von Neumann algebra is formulated as a regulated gravitational susceptibility. Toovercome the unboundedness of the continuum stress-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 modularanalytic class required by the Araki perturbation framework; the corresponding constructionand its covariance properties are established in Part II. The Lorentzian spacetime signatureis obtained from the KMS condition supplemented by a Wick rotation postulate, appliedexplicitly at this regulated level. Furthermore, we exhibit a Kähler structure on the regulated transverse-traceless sector, providing a compatible geometric normalization in whichthe quantum of action (ℏ) and the gravitational scale (Ginfo) enter through the combinationCarea := ℏGinfo. Finally, we contextualize the regulated Einstein–Hilbert-sector LIE relationderived in Part III under explicit assumptions, which anchors the dimensionless informationgeometry to physical area via the information-geometric gravitational coupling Ginfo. Bycleanly separating robust finite-cutoff theorems from open continuum-limit conjectures, thisfoundational part sets the stage for the spectral and replica-geometric derivations in thesubsequent parts.