From Bohmian Non-Locality to Einsteinian Geodesics: A Correspondence Framework via Araki–BKM Information Geometry, Coarse-Graining, and the Jacobi Metric

We present a mathematically structured framework connecting the non-local Bohmiandynamics living in the 3N-dimensional configuration space to local 4D spacetime geodesics.Our construction uses the Bogoliubov–Kubo–Mori metric induced by the Araki relative entropy. We explicitly derive the general relativistic Jacobi metric from first principles andcompare it with the effective single-particle Jacobi metric obtained from the BKM/Fishergeometry after a controlled coarse-graining of the environmental degrees of freedom. Thecorrespondence is stated as a physical principle rather than as a theorem. We prove a 1Dweak-field reduction and discuss the origin of the light cone, the role of the quantum potential, the violation and restoration of the equivalence principle, and the open analyticalproblems that remain.