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.

• The paper introduces a novel scale-invariant entropy statistic by using logarithmic binning and spectral analysis to quantify distance distributions.
• It rigorously demonstrates asymptotic convergence under both homogeneous Poisson processes and Cramér’s prime gap model, establishing statistical universality.
• The framework is broadly applicable to various point processes, enabling effective hypothesis testing and comparison in diverse scientific fields.

Scale-Invariant Entropy for Distance Distributions: Methodological Framework and Analysis

Introduction

The paper "A Scale-Invariant Entropy Statistic for Distance Distributions" (2604.02802) introduces a systematic framework for quantifying structural features in discrete point configurations on the real line, with an emphasis on distance distributions. The methodology leverages scale-invariant representations via logarithmic aggregation, spectral analysis, and entropy compression. While the motivating example is the prime numbers, the approach is generalizable to arbitrary point processes exhibiting multiplicative or scale-free structure. Key contributions include a well-defined scalar statistic based on entropy of log-frequency spectra and rigorous analysis of asymptotic properties under Poisson and Cramér models.

Methodological Construction

The central observable is constructed through four sequential steps:

1. Extraction of Truncated Distance Data: Given a finite configuration of points $\mathbf p$ and truncation radius $R$, the truncated distance multiset $D_R(\mathbf p)$ collects the absolute differences between points within radius $R$.
2. Logarithmic Aggregation at Fixed Resolution: Distances are aggregated in $M$ logarithmically spaced bins, defining normalized probabilities $p_j$. This step enforces scale invariance; multiplication of all distances by a positive constant yields identical bin probabilities, up to boundary discretization.
3. Harmonic Analysis on Log-Distance Coordinate: For bins with centers $x_j$, the discrete log-frequency spectrum $\widehat{\mu}(k)$ is computed using a Fourier-type sum over normalized probabilities. This spectrum encodes spatial regularity and randomness in the distance distribution.
4. Entropy-Based Compression: Spectral entropy $H$ is calculated as $-\sum w_k \log w_k$, where weights $R$0 are normalized magnitudes of Fourier coefficients. Linear magnitudes moderate sensitivity to single dominant frequencies.

This composite statistic is robust with respect to absolute scale, numerically stable due to fixed binning, and interpretable via comparison to null reference models.

Null Models and Asymptotic Behavior

The paper details rigorous analysis of the spectral entropy under two principal null models:

• Homogeneous Poisson Point Process: Under logarithmic transformation, the Poisson process becomes inhomogeneous. The log-bin geometry stabilizes asymptotically, and the spectral entropy converges almost surely and in $R$1 to a constant, independent of intensity $R$2.
• Cramér’s Model for Prime Gaps: After rescaling by local mean gap, rescaled prime gaps converge in distribution to a homogeneous Poisson process. Logarithmic aggregation removes scale effects, leading to entropy convergence with probability to the Poisson null-model value.

Strong claims are made regarding entropy convergence, with proof sketches based on bin stabilization and application of strong law of large numbers to bin counts. These results establish universality classes for spectral entropy with respect to random models.

Deviations and Conjectures

Building on reference models, the paper posits several conjectures regarding prime configurations:

• Asymptotic Stability: For fixed prime $R$3 and resolution $R$4, spectral entropy stabilizes as $R$5, bounded by a vanishing function $R$6.
• Non-Poissonian Deviations: Persistent deviation $R$7 from Poisson null entropy signals structural departures of prime gaps from random behavior on logarithmic scale.
• Ensemble-Level Tightness: Empirical entropy distributions $R$8 for multisets of fixed size $R$9 are tight and converge, after centering by the null-model entropy, to a limiting probability measure.

These conjectures outline a statistical limit theory for entropy observables in structured point processes and indicate potential emergence of systematic deviations in prime number distributions.

Broader Implications and Generalizability

The methodological framework transcends analytic number theory and primes, applying to generic point processes and configurations where absolute scale varies or is lacking intrinsic meaning. The statistic is applicable in spatial statistics, statistical physics, and analysis of disordered systems. Working at fixed logarithmic resolution is natural in many empirical settings, supporting comparison across diverse regimes.

The scalar nature of the entropy makes it a versatile tool for benchmarking observed point distributions against theoretical models, potentially facilitating hypothesis testing or parameter estimation in stochastic geometry and related fields.

Conclusion

The paper presents a coherent framework for constructing scale-invariant entropy statistics from distance distributions of point processes. Rigorous analysis supports the robustness and asymptotic universal behavior under random models, with conjectures about structural deviations in prime gaps and ensemble-level limit distributions. The approach is generalizable, numerically tractable, and forms a promising foundation for statistical analysis of complex discrete configurations in both theoretical and applied contexts.