The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law

Title: The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law Description This note addresses a persistent conceptual gap in “information physics” programs that derive familiar dynamical laws by extremizing Fisher-information-based functionals: why Fisher information, rather than some other statistical functional? The proposed answer is meta-explanatory. The observer–world interface is modeled as probabilistic, and “objectivity” is formulated as Observer Equivalence (OE): invariance under reparameterization and monotonicity under coarse-graining (information-loss channels). Under these constraints, Chentsov-type characterization results single out the Fisher information metric as the unique local Riemannian geometry on the space of probability distributions (up to an overall scale). Fisher information is thus treated not as an axiom but as an invariance residue forced by OE at the probabilistic interface. To connect geometry to dynamics, the note introduces an explicit postulate—Informational Inertia—motivated as an information-geometric analogue of least-action/geodesic principles: in the absence of forcing, interface states evolve along paths of minimal statistical resistance (geodesics of the OE-fixed geometry), with physical constraints supplying the remaining “potential” structure. A single exemplar is presented: revisiting the Madelung/Reginatto (Hall–Reginatto) route to the Schrödinger equation, where the “quantum potential” arises as the geometric term associated with the Fisher penalty for reshaping the probability density. The overall scale (and thus ℏ) is explicitly acknowledged as empirical input: OE fixes the form of the geometry, not its scale. Keywords observer equivalence; Fisher information; information geometry; Chentsov theorem; coarse-graining; monotone metrics; Schrödinger equation; Madelung transformation; Reginatto; foundations of physics How to cite Ullman, G. “The Invariance of the Interface: Why Fisher Information Reappears in Derivations of Physical Law.” Zenodo. https://doi.org/10.5281/zenodo.18495542 Notes on AI assistance This manuscript was prepared with assistance from large language models for drafting, editing, and LaTeX formatting. The author takes full responsibility for the technical content, claims, and references.