Curvature–Dissipation Dynamics in High-Dimensional Statistical Systems

We derive a fundamental evolution equation for the geometry of statistical manifolds under non-equilibrium dynamics. For a parametric family of probability densities evolving by a Wasserstein gradient flow, the Fisher metric g₈₉ obeys ∂ₜ g₈₉ = -2R₈₉ + 2D₈₉, where R₈₉ is the intrinsic Ricci curvature of the statistical manifold and D₈₉ is a positive-semidefinite dissipation tensor measuring Fisher-weighted entropy production. The law follows from Otto calculus, the Bakry–Émery Γ₂ identity, and a Gauss–Codazzi reduction under explicit assumptions (geodesic embedding, constant mobility). For translation families the equation yields exact scaling relations: curvature grows as R ∼ N and the metric as g ∼ N^1/2 with system size N. The competition between curvature-driven contraction and entropy-driven expansion produces a rigidity transition at a critical size Nc. Above Nc the manifold becomes locked and deviations from the constraint manifold are exponentially suppressed. The critical exponent ν ≈ 0. 62 is consistent with the 2D Ising universality class, indicating shared scaling structure rather than microscopic equivalence. This work establishes a general theoretical framework linking information geometry, stochastic thermodynamics, and critical phenomena. The work is positioned within statistical physics and does not involve artificial intelligence systems or applications. This is the author’s preprint version of a manuscript under journal review (2026).