Fisher-Rao Geometry as the Common Origin of Cosmological Bounces, Holographic Duality, and Geodesic Regularization: A Derivation from Cencov's Uniqueness Theorem

We prove that the Fisher-Rao metric — the unique Riemannian metric on the statistical manifold of probability distributions, characterised by invariance under sufficient statistics (Cencov 1982) — induces an exact geometric structure on quantum transition amplitudes. For the free Euclidean propagator in R^d, the statistical manifold is isometric to AdSd+1 exactly. From this single fact we derive: (I) the unique geodesic speed-limited Lagrangian is DBI; (II) every trajectory with nonzero angular momentum has a non-singular bounce, universal in the quantum theory; (III) Jackiw-Teitelboim gravity and Hawking-Bekenstein thermodynamics emerge from the Wheeler-DeWitt equation; (IV) AdS/CFT is a theorem, not a conjecture, with Ryu-Takayanagi following from Fisher information geometry. A fifth manifestation: the coherence field of language models obeys the same DBI equation. DOI: 10.5281/zenodo.19015001