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A Scale-Invariant Entropy Statistic for Distance Distributions

This paper introduces a family of scale-invariant entropy statistics derived from logarithmically aggregated distance distributions of point processes, designed to encode structural features of relative spacing independent of absolute scale, with prime numbers serving as a motivating example.

Original authors: Mohamed Gewily

Published 2026-04-06

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CC BY 4.0

Original authors: Mohamed Gewily

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the "personality" of a crowd of people standing in a line.

Usually, if you look at the distances between them, the numbers depend entirely on how far apart they are standing. If they are all huddled close together, the distances are tiny (like 1 inch). If they are spread out across a football field, the distances are huge (like 100 feet). This makes it hard to compare the two groups: Is the football field crowd more "ordered" or more "chaotic" than the huddled crowd? You can't tell just by looking at the raw numbers because the scale is different.

This paper introduces a clever new tool—a "Scale-Invariant Entropy Statistic"—that solves this problem. Think of it as a magical ruler that changes its own size depending on the crowd, allowing you to compare the pattern of spacing regardless of whether the people are inches or miles apart.

Here is how the method works, broken down into simple steps using analogies:

1. The Problem: The "Zoom" Issue

Imagine you have a photo of a city street. If you zoom in, you see individual bricks. If you zoom out, you see whole buildings. The "distance" between objects changes depending on your zoom level.

The Goal: The author wants to measure the rhythm or structure of the spacing (is it regular like a marching band, or random like a mosh pit?) without caring about the actual size of the steps.

2. The Solution: The "Logarithmic Lens"

To fix the zoom issue, the author uses a Logarithmic Lens.

The Analogy: Imagine a ruler where the numbers don't go 1, 2, 3, 4... but instead go 1, 10, 100, 1000.
On this ruler, a gap of 1 inch and a gap of 100 inches are treated as "neighbors" in terms of their relative size.
By converting all distances into this "logarithmic language," a tiny gap in a small crowd and a huge gap in a large crowd can be compared directly. If the pattern of gaps looks the same on this special ruler, the two crowds have the same "structural DNA."

3. The Process: Turning Spacing into Music

Once the distances are converted to this logarithmic scale, the author turns them into a "soundtrack" to analyze them.

Step A: Grouping (The Bins): They take the logarithmic distances and sort them into buckets (like sorting marbles by size).
Step B: The Spectrum (The Music): They use a mathematical trick (Discrete Harmonic Analysis) to turn these buckets into a frequency spectrum.
- Think of this like a musical equalizer.
- If the spacing is very regular (like a marching band), the music will be a single, pure, low-pitched note (low frequency).
- If the spacing is random (like a mosh pit), the music will be a chaotic mix of all frequencies, sounding like static or white noise.

4. The Result: The "Entropy Score"

Finally, the author calculates a single number called Spectral Entropy.

Low Entropy Score: The "music" is simple and repetitive. The spacing is highly structured and predictable.
High Entropy Score: The "music" is chaotic and complex. The spacing is random and unpredictable.

Why Primes? (The Motivating Example)

The author uses Prime Numbers (2, 3, 5, 7, 11...) as a test case.

Primes are famous for being somewhat random but also having hidden patterns.
The author asks: "If we look at the gaps between prime numbers using our magical logarithmic ruler, do they sound like pure music (ordered) or static noise (random)?"
They compare the "Prime Music" against a "Poisson Model" (which is a mathematical model of pure randomness, like raindrops hitting a roof).

The Big Idea

The paper suggests that this method creates a universal language for spacing.

It doesn't matter if you are looking at stars in a galaxy, trees in a forest, or prime numbers.
By using this "logarithmic entropy score," you can objectively say: "This pattern is more ordered than that one," even if one pattern is microscopic and the other is galactic.

Summary

The author has built a universal ruler that ignores size and focuses only on pattern. By turning distances into a "musical spectrum" and measuring how chaotic that music is, we get a single number that tells us how structured or random a group of points really is. It's a new way to listen to the rhythm of the universe, from the smallest numbers to the largest distances.

1. Problem Statement

The paper addresses the challenge of quantifying the structural features of distance distributions arising from discrete point configurations on the real line (e.g., prime numbers, spatial data) in a way that is independent of absolute scale.

The Challenge: Traditional distance metrics are sensitive to the magnitude of the data. Comparing spacing patterns between small-scale and large-scale configurations is difficult because absolute distances vary drastically.
The Goal: To develop a robust scalar statistic that captures relative spacing information (regularity vs. randomness) while remaining invariant to uniform scaling (multiplying all distances by a constant).
Motivation: The author uses the distribution of prime numbers (specifically prime gaps) as a motivating example, where the spacing between primes grows logarithmically, making scale invariance crucial for meaningful analysis.

2. Methodology

The proposed framework constructs a Scale-Invariant Entropy Statistic through a four-step process applied to a finite configuration of points:

Step 1: Extraction of Truncated Distance Data

For a given set of points (e.g., a set of primes $P$ ), the author defines a truncated distance multiset $D_R(p)$ containing distances $|p - q|$ for all $q$ within a radius $R$ . For a collection of points, these distance measures are aggregated additively.

Step 2: Logarithmic Aggregation at Fixed Resolution

To eliminate scale dependence, distances are mapped to a logarithmic axis.

Binning: The range $[d_{\min}, d_{\max}]$ is divided into $M$ logarithmic bins. The bin edges are defined as $\ell_j = \log d_{\min} + \frac{j}{M}(\log d_{\max} - \log d_{\min})$ .
Probability Vector: The number of distances falling into each bin is counted to form a probability vector $(p_1, \dots, p_M)$ .
Scale Invariance: Multiplying all distances by a constant $c$ shifts the log-axis but preserves the relative bin probabilities (up to boundary discretization effects), ensuring the statistic is scale-invariant.

Step 3: Discrete Harmonic Analysis (Log-Frequency Spectrum)

Instead of analyzing the raw bin counts, the method applies a Discrete Fourier Transform (DFT) to the probability distribution on the log-axis.

Spectrum Definition: The log-frequency spectrum $\hat{\mu}(k)$ is computed using the bin centers $x_j$ as the phase argument:
$\hat{\mu}(k) = \sum_{j=1}^M p_j \exp\left(-2\pi i (k-1) \frac{x_j - x_1}{x_M - x_1}\right)$
Interpretation: Low-frequency concentration indicates regular, slowly varying structures. Broad spectral support indicates irregular or random behavior.

Step 4: Entropy-Based Compression

The spectral information is compressed into a single scalar value: Spectral Entropy.

Normalization: The magnitude of the spectrum is normalized to form weights $w_k = |\hat{\mu}(k)| / \sum |\hat{\mu}(\ell)|$ .
Calculation: The entropy is calculated as $H = -\sum w_k \log w_k$ $H = - \sum w_{k} lo g w_{k}$ .
- Low Entropy: Corresponds to regular, structured configurations (dominated by specific frequencies).
- High Entropy: Corresponds to random, unstructured configurations (flat spectrum).

3. Key Contributions

Novel Statistic: Introduction of a specific entropy statistic derived from the logarithmic frequency spectrum of distance distributions, distinct from standard Shannon entropy of raw gaps.
Fixed Resolution Framework: The method operates at a fixed logarithmic resolution ( $M$ ) rather than relying on asymptotic refinement. This ensures numerical stability and allows for direct comparison across different scales and configurations.
Theoretical Null Models: The paper establishes the behavior of this statistic under a Homogeneous Poisson Point Process (the standard null model for randomness).
- Lemma 7.1 & Proposition 7.2: Proves that for a Poisson process, the logarithmic bin geometry stabilizes asymptotically, and the spectral entropy converges almost surely to a constant $H_{\infty}^{null}(M)$ , independent of the intensity $\lambda$ .
Connection to Prime Number Theory: The framework bridges analytic number theory and statistical physics by applying the statistic to prime gaps.
- Proposition 8.1: Under Cramér's random model (which predicts Poissonian behavior for rescaled prime gaps), the spectral entropy of primes is shown to converge in probability to the Poisson null-model value.

4. Results and Conjectures

The paper does not present new raw numerical results but rather theoretical derivations and conjectures regarding the behavior of the statistic:

Convergence: The entropy statistic for a Poisson process converges to a deterministic limit as the observation radius $R \to \infty$ .
Prime Gap Deviations: The author conjectures that if prime gaps exhibit persistent deviations from Poisson behavior on the logarithmic scale, the spectral entropy of primes will deviate from the Poisson limit ( $\Delta_\infty(p) \neq 0$ ).
Ensemble Tightness: It is conjectured that the distribution of entropy values across ensembles of prime configurations converges to a limiting probability measure, suggesting a statistical limit theory for these observables.

5. Significance and Applicability

Methodological Contribution: The paper positions itself as a methodological tool rather than a source of new raw data. It provides a rigorous way to compare "shape" or "structure" in point processes regardless of their physical scale.
Broad Applicability: While motivated by prime numbers, the framework is applicable to:
- Spatial Statistics: Analyzing point patterns in ecology or urban planning.
- Statistical Physics: Studying disordered systems and energy level spacings.
- Empirical Patterns: Any dataset where multiplicative or scale-free behavior is present.
Interpretability: By fixing the logarithmic resolution, the resulting scalar is interpretable and comparable against stochastic reference models, offering a new lens to detect subtle structural regularities or anomalies in complex data.

In summary, Gewily proposes a robust, scale-invariant metric that transforms distance distributions into a spectral entropy value, offering a powerful tool to distinguish between ordered and random structures in diverse point processes, with prime numbers serving as the primary theoretical testbed.