Information Geometry and the Variational Structure of Physical Dynamics: A Rigorous Foundation

We develop a mathematically rigorous variational principle on statistical manifolds equipped with the Fisher information metric. Starting from seven axioms characterizingdistinguishability between probability distributions, we prove that the Fisher metric is the unique (up to scaling) Riemannian structure satisfying these information-theoreticrequirements. We then demonstrate that geodesic motion on this manifold, subject to domain-specific constraint functionals, recovers the Euler-Lagrange equations of known dynamical systems across physics, evolutionary biology, and machine learning. Our main theoretical contributions include: (i) a complete proof of metric uniqueness via Chentsov’s theorem with minimal assumptions, (ii) rigorous derivations showing consistency with Hamiltonian mechanics, quantum unitary evolution, thermodynamic relaxation, replicator equations, and natural gradient descent, and (iii) novel predictions including quantum decoherence rate bounds and evolutionary speed limits. We emphasize that our framework provides a reformulation revealing structural unity, not a complete derivation of fundamental physical laws. All constraint functionals encoding domain-specific physics must be specified externally. Nevertheless, this geometric perspective yields new testable predictions and unifies previously disparate mathematical structures under a common information-theoretic foundation.