Entropy-Smooth Structures on Topological Manifolds

TL;DR

This paper proposes an information-theoretic approach to defining smooth structures on topological manifolds using local entropy data, establishing an equivalence with classical smooth structures and linking entropy with differential geometry.

Contribution

It introduces a novel entropy-based framework for smooth structures, replacing traditional coordinate charts with entropy responses, and proves their equivalence to classical structures.

Findings

Entropy-smooth structures are equivalent to classical smooth structures.

The framework is stable under perturbations and compatible with manifold operations.

Establishes a foundational link between entropy and differential geometry.

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.