What question did this study set out to answer?

The research aims to establish a geometric foundation linking Fisher-Rao geometry with various fundamental physics concepts.

April 6, 2026Open Access

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

Key Points

The research aims to establish a geometric foundation linking Fisher-Rao geometry with various fundamental physics concepts.
Proving that the Fisher-Rao metric induces a geometric structure on quantum transition amplitudes.
Examining the statistical manifold's isometry with AdS_{d+1}.
Deriving several implications including geodesic Lagrangians and non-singular trajectories.
Identified the unique geodesic Lagrangian as DBI.
Confirmed universal non-singular bounces for trajectories with nonzero angular momentum.
Demonstrated emergence of Jackiw-Teitelboim gravity and Hawking-Bekenstein thermodynamics from the Wheeler-DeWitt equation.
Established AdS/CFT as a theorem with implications for Fisher information geometry.

Abstract

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 Rᵈ, the statistical manifold is isometric to AdS₃+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

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