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A Scale-Invariant Entropy Statistic for Distance Distributions

Mohamed Gewily

Abstract

We introduce a family of scale-invariant entropy statistics derived from logarithmically aggregated distance distributions of point processes, with prime numbers serving as a motivating example. The construction associates to each finite configuration a scalar quantity encoding structural features of relative spacing while remaining insensitive to absolute scale. This work is intended as a methodological contribution rather than a source of new raw results.

A Scale-Invariant Entropy Statistic for Distance Distributions

Abstract

We introduce a family of scale-invariant entropy statistics derived from logarithmically aggregated distance distributions of point processes, with prime numbers serving as a motivating example. The construction associates to each finite configuration a scalar quantity encoding structural features of relative spacing while remaining insensitive to absolute scale. This work is intended as a methodological contribution rather than a source of new raw results.

Paper Structure

This paper contains 11 sections, 3 theorems, 18 equations.

Table of Contents

  1. Overview and Objectives
  2. Prime Distance Distributions
  3. Logarithmic Aggregation at Fixed Resolution
  4. Log-Frequency Spectrum
  5. Spectral Entropy
  6. Null Models and Interpretation
  7. Poisson Log-Distance Model
  8. Connection to Classical Prime Gap Models
  9. Questions and Conjectures
  10. Broader Applicability and Significance
  11. R Implementation

Key Result

Lemma 7.1

Under the Poisson log-distance model, almost surely Consequently, and the logarithmic bin geometry stabilizes asymptotically. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (19)

  • Definition 2.1: Truncated Prime Distance Measure
  • Definition 2.2: Additivity
  • Remark 2.3
  • Remark 3.1: Scale Invariance
  • Remark 3.2: Resolution Dependence
  • Remark 4.1
  • Definition 5.1: Spectral Entropy
  • Remark 5.2
  • Lemma 7.1: Asymptotic Stabilization of Logarithmic Bins
  • proof : Proof Sketch
  • ...and 9 more