Entropy-Smooth Structures on Topological Manifolds

Abstract

We introduce an information-theoretic framework for smooth structures on topological manifolds, replacing coordinate charts with small-scale entropy data of local probability probes. A concise set of axioms identifies admissible coordinate functions and reconstructs a smooth atlas directly from the quadratic entropy response. We prove that this entropy-smooth structure is equivalent to the classical smooth structure, stable under perturbations, and compatible with products, submanifolds, immersions, and diffeomorphisms. This establishes smoothness as an information-theoretic phenomenon and forms the foundational layer of a broader program linking entropy, diffusion, and differential geometry.