The Curvature-Entropy Correspondence: Geometric Foundations of Non-Extensive Statistical Mechanics and Applications to Deep Learning

The selection of the appropriate entropy functional for complex systems has long been a subject of axiomatic debate. In this work, we propose a shift from axiomatic to geometric thermodynamics: we demonstrate that the intrinsic geometry of the statistical manifold uniquely determines the entropic functional. We establish two fundamental results with rigorous proofs: (I) The Generalized Chentsov Theorem, proving via escort duality that the q-Fisher metric is the unique Riemannian metric invariant under q-sufficient statistics; and (II) The Curvature-Entropy Correspondence, establishing that constant negative curvature (R < 0) uniquely determines Tsallis entropy Sq, while flat geometry (R = 0) recovers Boltzmann-Gibbs entropy, validated via the Furuhata-Kurose classification theorem for Hessian manifolds. We apply this framework to Deep Learning, proposing geometric corrections to address structural pathologies. We posit that standard architectures suffer from a “Geometric Mismatch”—assuming flat Euclidean geometry (R = 0) for data with hierarchical structure. We derive three geometric modifications: (1) Robustness: the q-Cross Entropy loss, which acts as a bounded potential well, naturally filtering outliers; (2) Long-Range Context: Riemannian Attention mechanisms (q-Softmax) that replace exponential screening with power-law correlations, enabling infinite cor- relation lengths; (3) Dynamics: interpretation of the “Grokking” phenomenon and the Lazy-to-Rich transition as geometric phase transitions. Empirical validation of these proposals is the subject of ongoing work.