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Entropy-Smooth Structures on Topological Manifolds

Amandip Sangha

TL;DR

This work develops an information-theoretic characterization of smooth structures on topological manifolds by replacing charts with small-scale entropy data from local probes and analyzing a quadratic entropy response. It defines entropy-smoothness through a finite, well-behaved family of probes and axioms that recover the classical $C^{\infty}$ atlas, proving an equivalence between the entropy-smooth and standard smooth categories. The framework is stable under perturbations, compatible with products and submanifolds, and provides entropy-based criteria for immersions, submersions, and diffeomorphisms, thereby connecting differentiability to information-theoretic invariants. The results lay the groundwork for broader programs linking entropy, diffusion, and differential geometry, potentially enabling entropy-based reconstructions of Riemannian metrics and curvature. Overall, smooth structure is shown to be an intrinsic information-theoretic phenomenon that can be captured without presupposing differentiability on the underlying space.

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.

Entropy-Smooth Structures on Topological Manifolds

TL;DR

This work develops an information-theoretic characterization of smooth structures on topological manifolds by replacing charts with small-scale entropy data from local probes and analyzing a quadratic entropy response. It defines entropy-smoothness through a finite, well-behaved family of probes and axioms that recover the classical atlas, proving an equivalence between the entropy-smooth and standard smooth categories. The framework is stable under perturbations, compatible with products and submanifolds, and provides entropy-based criteria for immersions, submersions, and diffeomorphisms, thereby connecting differentiability to information-theoretic invariants. The results lay the groundwork for broader programs linking entropy, diffusion, and differential geometry, potentially enabling entropy-based reconstructions of Riemannian metrics and curvature. Overall, smooth structure is shown to be an intrinsic information-theoretic phenomenon that can be captured without presupposing differentiability on the underlying space.

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.

Paper Structure

This paper contains 21 sections, 13 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Axioms for Entropy--Smoothness
  4. Discussion of the Axioms
  5. Correspondence with diffusion semigroups
  6. Correspondence with classical mollifiers
  7. Examples
  8. The real line $\mathbb{R}$.
  9. Euclidean space $\mathbb{R}^n$
  10. The circle $S^1$.
  11. Structural Properties of Entropy--Smooth Manifolds
  12. Existence and Uniqueness
  13. Entropy–smooth maps
  14. Categorical Equivalence
  15. Products
  16. ...and 6 more sections

Key Result

Lemma 3.2

Let $V,W\subset\mathbb{R}^n$ be open sets and let $G:V\to W$ be a homeomorphism. Assume that for every smooth function $h\in C^\infty(W)$, the pullback $h\circ G$ belongs to $C^\infty(V)$. Then $G$ is a smooth map.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2: Entropic tilt
  • Definition 2.3: Quadratic KL response
  • Remark 2.4: What $I_{x,\varepsilon}$ measures
  • Definition 2.5: Entropy-smooth
  • Definition 2.6: Joint entropy coefficient
  • Definition 3.1: Entropy coordinate chart
  • Lemma 3.2: Elementary Smoothness Criterion
  • proof
  • Lemma 3.3: Uniqueness of the smooth structure on $\Omega\subset\mathbb{R}^n$
  • ...and 26 more