A Scale-Invariant Entropy Statistic for Distance Distributions

Mohamed Gewily

arxiv: 2604.02802 · v1 · submitted 2026-04-03 · 📊 stat.ME

A Scale-Invariant Entropy Statistic for Distance Distributions

Mohamed Gewily This is my paper

Pith reviewed 2026-05-13 18:46 UTC · model grok-4.3

classification 📊 stat.ME

keywords scale-invariant entropydistance distributionspoint processeslogarithmic binningrelative spacingprime gapsentropy statisticsscale invariance

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The pith

A new entropy statistic from log-aggregated distances captures relative spacing in point sets without depending on absolute scale.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a family of entropy statistics for finite point configurations that remain unchanged under uniform scaling of all distances. These are built by taking pairwise distances, binning them logarithmically to create a distribution, and then computing an entropy measure on that distribution. The approach uses prime number gaps as a motivating case to illustrate how it can capture relative spacing patterns. A sympathetic reader would value this if it provides a robust way to compare geometric or statistical structures across vastly different size regimes without arbitrary normalization. The work positions itself as a methodological tool for point process analysis rather than delivering specific new discoveries about primes or other systems.

Core claim

The central claim is that by logarithmically aggregating the distribution of distances in a point process, one obtains a scale-invariant scalar that encodes key structural features of the relative spacings. This construction associates to each finite configuration a quantity insensitive to absolute scale, allowing focus on proportional relationships in the point placements.

What carries the argument

The scale-invariant entropy statistic obtained from logarithmically binned pairwise distance distributions of a point configuration.

If this is right

Configurations with similar relative spacing structures will yield similar statistic values even if their absolute sizes differ greatly.
The method can be used to analyze spacing in number-theoretic sequences like primes without dependence on the range considered.
It extends existing entropy measures to be inherently scale-free through logarithmic aggregation.
Different parameterizations of the family allow tuning sensitivity to various aspects of the distance histogram.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This statistic might enable scale-free comparisons in fields like cosmology for galaxy clustering or ecology for species distributions.
It could be tested for consistency on randomly generated point sets scaled by different factors to confirm invariance.
Connections to fractal dimensions or other scale-invariant measures in geometry might be explored.

Load-bearing premise

Logarithmically aggregated distance distributions meaningfully capture novel structural features of relative spacing beyond what conventional entropy or point-process statistics already provide.

What would settle it

Observing that the computed statistic changes when a point configuration is uniformly scaled by a constant factor, or finding that it assigns similar values to configurations with clearly different spacing regularities.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a scale-invariant entropy statistic from log-aggregated distances in point processes, with the invariance following immediately from differential entropy's translation property.

read the letter

The core idea here is straightforward: take pairwise or nearest-neighbor distances in a point process, log-transform them, form the distribution, and compute its differential entropy. Because a global scale factor adds a constant to every log-distance, the entropy stays the same. That is the whole claim, and it holds by construction without extra assumptions about the process. The prime-number example is presented only as motivation, not as a source of new quantitative results. What the paper does cleanly is package this into a family of statistics that stay insensitive to absolute scale while still capturing relative spacing features. That is a modest but honest methodological step. It avoids overclaiming and sticks to the definition. The main limitation is that the work stops at the definition. There are no derivations beyond the immediate invariance argument, no simulation studies, no comparisons against existing point-process entropies or spacing statistics, and no demonstration that the new quantity actually distinguishes configurations in a useful way. Without those pieces the contribution remains thin. A reader already working on entropy measures for spatial data might find the construction convenient, but most people will need to see concrete performance before treating it as a go-to tool. The paper is suitable for a methods journal or a short note section if the authors add at least one worked example with real or simulated data and a brief comparison to standard alternatives. It is not ready for a full-length article in its current form. I would send it to peer review with the expectation that the authors supply validation and context; the underlying idea is sound enough to warrant that step.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a family of scale-invariant entropy statistics obtained by computing the differential entropy of logarithmically aggregated pairwise or nearest-neighbor distance distributions arising from finite point configurations. Prime numbers serve as a motivating illustration rather than a source of new theorems. The central construction associates a scalar to each configuration that encodes relative spacing features while remaining invariant to global rescaling, with the invariance following directly from the translation invariance of differential entropy under addition of a constant log-scale factor.

Significance. If the definition is adopted, the statistic supplies a parameter-free, immediately computable descriptor of structural regularity in point patterns that is insensitive to absolute scale. This property is attractive for comparative analyses across differently scaled data sets in spatial statistics, ecology, or number-theoretic point processes. The construction requires no auxiliary fitting or convergence arguments, which strengthens its methodological utility.

minor comments (3)

[Abstract] The abstract and introduction should explicitly state the precise formula for the entropy statistic (e.g., the choice of kernel or binning for the log-distance density) so that readers can reproduce the scalar without ambiguity.
[Introduction] A short computational example with the prime-number configuration would clarify implementation details such as handling of finite-sample bias in differential-entropy estimation.
Notation for the distance distribution (pairwise versus nearest-neighbor) should be unified across sections to avoid reader confusion about which variant is intended.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, the clear summary of the scale-invariant entropy construction, and the recommendation to accept the manuscript. No major comments were raised that require addressing.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new family of scale-invariant entropy statistics directly as the differential entropy of the log-distance distribution of a point configuration. Scale invariance is an immediate algebraic consequence of the translation invariance of differential entropy under addition of log λ, with no fitted parameters, no predictions of external quantities, and no load-bearing self-citations. The construction is self-contained as a methodological definition; the prime-number example serves only as illustration and does not supply any quantitative claim that is then re-derived from the statistic itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that logarithmic aggregation of distances produces a scale-invariant encoding of structure; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Logarithmic aggregation of distance distributions in point processes yields quantities insensitive to absolute scale while preserving relative spacing information.
Invoked directly in the construction of the statistics as described in the abstract.

pith-pipeline@v0.9.0 · 5330 in / 1063 out tokens · 21260 ms · 2026-05-13T18:46:11.976988+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

On the order of magnitude of the difference between consecutive prime numbers

Harald Cram´ er. “On the order of magnitude of the difference between consecutive prime numbers”. In:Acta Arithmetica2.1 (1936), pp. 23– 46.doi:10.4064/aa-2-1-23-46

work page doi:10.4064/aa-2-1-23-46 1936
[2]

Some problems of Partitio Numero- rum; III. On the expression of a number as a sum of primes

G. H. Hardy and J. E. Littlewood. “Some problems of Partitio Numero- rum; III. On the expression of a number as a sum of primes”. In:Acta Mathematica44 (1923), pp. 1–70.doi:10.1007/BF02412410

work page doi:10.1007/bf02412410 1923
[3]

On the distribution of primes in short intervals

P. X. Gallagher. “On the distribution of primes in short intervals”. In: Mathematika23.1 (1976), pp. 4–9.doi:10.1112/S0025579300005726. SPECTRAL ENTROPY OF DISTANCE DISTRIBUTIONS 7 AppendixA.R Implementation RemarkA.1.The code below is intended for conceptual illustration. For large-scale numerical experiments, optimized prime sieves and distance com- put...

work page doi:10.1112/s0025579300005726 1976

[1] [1]

On the order of magnitude of the difference between consecutive prime numbers

Harald Cram´ er. “On the order of magnitude of the difference between consecutive prime numbers”. In:Acta Arithmetica2.1 (1936), pp. 23– 46.doi:10.4064/aa-2-1-23-46

work page doi:10.4064/aa-2-1-23-46 1936

[2] [2]

Some problems of Partitio Numero- rum; III. On the expression of a number as a sum of primes

G. H. Hardy and J. E. Littlewood. “Some problems of Partitio Numero- rum; III. On the expression of a number as a sum of primes”. In:Acta Mathematica44 (1923), pp. 1–70.doi:10.1007/BF02412410

work page doi:10.1007/bf02412410 1923

[3] [3]

On the distribution of primes in short intervals

P. X. Gallagher. “On the distribution of primes in short intervals”. In: Mathematika23.1 (1976), pp. 4–9.doi:10.1112/S0025579300005726. SPECTRAL ENTROPY OF DISTANCE DISTRIBUTIONS 7 AppendixA.R Implementation RemarkA.1.The code below is intended for conceptual illustration. For large-scale numerical experiments, optimized prime sieves and distance com- put...

work page doi:10.1112/s0025579300005726 1976