Probabilistic Geometry and the Relativistic Structure of Quantum Motion: From Information Curvature to Entropy–Curvature Duality

We establish a unified probabilistic–geometric formulation of quantum motion that extends infor- mation geometry into relativistic spacetime. The central result is the Nemoto–Curvature Correspon- dence, which identifies the Fisher–Rao scalar curvature RP with the quantum potential QP, thus linking probabilistic curvature to quantum energy density. By embedding the statistical manifold into a Lorentzian spacetime, we derive a covariant metric coupling term that predicts curvature backreaction from statistical fluctuations. A further Entropy–Curvature Duality connects informa- tion curvature with the second derivative of Shannon entropy, offering a new explanation of temporal asymmetry. Numerical analysis for Gaussian states demonstrates curvature-induced quantum dif- fusion, and the model yields testable corrections to spacetime curvature at Planck scales. This framework provides an original and predictive unification between probability, curvature, and en- tropy.