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 provevia Chentsov's theorem that the Fisher metric is the unique (up to a positive constant) Riemannianstructure satisfying these information-theoretic requirements. We then show that geodesic motion on this manifold, subject to domain-specific constraint functionals, recovers the equations of motion governing Hamiltonian mechanics, quantum unitary evolution, thermodynamic relaxation, evolutionary replicator dynamics,and natural gradient descent. Our main contributions are: (i) a complete proof of metric uniqueness from physical axioms; (ii) rigorous consistency demonstrations with explicit error bounds; (iii) seven novel, falsifiable predictions - including quantum decoherence rate scaling with the quantum Fisher information, evolutionary speed limits, and neural network capacity scaling C ~ N0.6 three with preliminary numerical support; and (iv) explicit falsification criteria. We address anticipated criticisms, including the constraint problem, the signature problem, and questions of novelty beyond reformulation. We emphasize throughout that this framework is a reformulation revealing structural unity across disciplines, not a derivation of fundamental constants or a Theory of Everything. All constraint functionals encoding domain-specific physics must be specified externally. The mathematics is rigorous, the predictions are testable, and the question is well-posed. Now nature must answer.