<span>We develop a mathematically rigorous variational principle on statistical manifolds equipped with the Fisher information metric. Starting from seven axioms characterizing&nbsp;</span><span>distinguishability between probability distributions, we prove via Chentsov's theorem that the Fisher metric is the unique (up to a positive constant) Riemannian&nbsp;</span><span>structure 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,&nbsp;</span><span>and natural gradient descent.&nbsp;</span> <div> <span>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.&nbsp;</span> </div> <div> <span>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.</span> </div>