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arxiv: 2606.16393 · v2 · pith:KOHALXLWnew · submitted 2026-06-15 · 📊 stat.AP · math.ST· physics.app-ph· physics.data-an· stat.TH

Calibrating the Brody exponent as a quantitative measure of short-range exclusion in 2D spatial point processes

Dawid Kucharski This is my paper

Pith reviewed 2026-06-27 02:30 UTC · model grok-4.3

classification 📊 stat.AP math.STphysics.app-phphysics.data-anstat.TH
keywords Brody distributionspatial point processesshort-range exclusioncomplete spatial randomness2D point patternslevel spacinghard-core radius
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The pith

The Brody exponent, recalibrated for two dimensions, quantifies short-range exclusion strength in spatial point processes.

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

The paper establishes the Brody distribution as a calibrated tool for measuring short-range exclusion in 2D point patterns by first resetting the complete-spatial-randomness baseline to β=0.96±0.15 and then validating an empirical link between β and the effective hard-core radius r_excl. This link reaches Spearman ρ=0.988 across manufactured surfaces, profilometry data, and prime-number embeddings. Controls confirm that observed β values arise from arithmetic structure or embedding geometry rather than density alone, while thinning experiments show β tracks exclusion independently of point density. The resulting framework supplies a decision table and protocols that allow reproducible comparison of exclusion across different 2D processes.

Core claim

The Brody distribution, after empirical recalibration on 2D complete spatial randomness to β=0.96±0.15 and validation of the β--r_excl relation at Spearman ρ=0.988, functions as a quantitative measure of short-range exclusion that remains consistent across generation processes and density regimes when the supplied controls are applied.

What carries the argument

The Brody distribution (phenomenological interpolation between Poisson and Wigner spacing laws) recalibrated via 2D CSR baseline correction and β--r_excl regression.

If this is right

  • The β--r_excl calibration converts observed Brody values into an effective exclusion radius for any 2D pattern obeying the tested controls.
  • Sparse-integer and Cantor-embedding controls distinguish intrinsic arithmetic signals from embedding-induced exclusion.
  • Density thinning isolates exclusion strength from density dependence, though absolute β remains density-sensitive.
  • A separate CSR baseline applies to binary fields at low fill fraction, supplied with an explicit decision table.

Where Pith is reading between the lines

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

  • The same recalibration procedure could be repeated in three or higher dimensions to produce dimension-specific baselines.
  • The framework offers a bridge between level-spacing methods in quantum chaos and classical spatial statistics without requiring new distribution families.
  • Applications to ecological or materials data could test whether β tracks physical repulsion mechanisms more directly than nearest-neighbor distances alone.

Load-bearing premise

That the recalibrated Brody form supplies a process-independent measure of exclusion once the 2D baseline and control protocols are used.

What would settle it

A new collection of 2D point patterns with independently measured hard-core radii that yields Spearman correlation below 0.9 between β and r_excl, or a CSR baseline outside the reported 0.96±0.15 interval.

Figures

Figures reproduced from arXiv: 2606.16393 by Dawid Kucharski.

Figure 1
Figure 1. Figure 1: Brody exponent β across all 58 analysable surfaces. Left: histogram with kernel density estimate. Right: β by manufacturing process (box plots with individual points). The CSR baseline (β = 0.96, grey band) is indicated. The earlier bar chart for 21 prominence-based surfaces (Fig. S2) provides an alternative visualisation. The wide within-process variability and overlap between categories confirm that fact… view at source ↗
Figure 2
Figure 2. Figure 2: Exclusion physics: the Brody exponent β as a measure of short-range exclusion. Left: synthetic hard-core point processes with exclusion radius rexcl normalised by the mean nearest-neighbour distance (blue), 2D Poisson CSR baseline (β = 0.96 ± 0.15, grey band), 2D Ginibre ensemble (green), and observed β values from manufactured surfaces (grey dots, n = 21), 2D primes (purple), and PSI ceramic (red). The 2D… view at source ↗
Figure 3
Figure 3. Figure 3: Pair correlation function g2(r) across domains. Top row: four representative manufactured surfaces spanning the Brody β range, compared with the Ginibre prediction (solid black). β values are from [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Embedding dependence of the Brody exponent for 2D primes at [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Density-thinning and sparse-integer controls. Left: Brody exponent [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Peak-detection sensitivity analysis on a synthetic surface with 50 embedded Gaussian peaks. Left: Brody [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

The Brody distribution, originally a phenomenological interpolation between Poisson and Wigner level-spacing statistics in quantum chaos, is calibrated here as a quantitative measure of short-range exclusion in 2D spatial point processes. Two results form the core. First, the 2D complete-spatial-randomness baseline is recalibrated to $\beta=0.96\pm0.15$, correcting the inappropriate 1D Poisson reference. Second, an empirical $\beta$--$r_{\text{excl}}$ calibration is validated against the effective hard-core radius with Spearman $\rho=0.988$. The framework is demonstrated on 58 manufactured surfaces (10 materials, 10 processes), phase-extracted interferometric profilometry of a certified roundness standard, and 2D binary embeddings of prime numbers. A sparse-integer control proves the prime $\beta=2.15$ signal is genuinely arithmetic ($\Delta\beta=+0.68$ over random-integer control), while a Cantor-embedding null result ($\beta=1.40$, TOST $p<0.01$) demonstrates that 2D exclusion is embedding-created rather than intrinsic. Density-thinning experiments establish that $\beta$ captures exclusion strength rather than point density, while absolute values are density-dependent. A distinct CSR baseline for binary fields at low fill fraction is identified, with a decision table provided. The $\beta$--$r_{\text{excl}}$ calibration, the CSR baseline correction, and the control protocols together constitute a calibrated measurement framework for reproducible characterisation of short-range exclusion in 2D spatial point processes.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript calibrates the Brody exponent β (originally from 1D quantum level statistics) as a quantitative measure of short-range exclusion in 2D spatial point processes. It reports a recalibrated 2D CSR baseline of β=0.96±0.15, an empirical β–r_excl relation validated by Spearman ρ=0.988 on 58 manufactured surfaces (10 materials, 10 processes), phase-extracted profilometry, and prime-number embeddings, plus controls including sparse-integer arithmetic signal (Δβ=+0.68), Cantor-embedding null (β=1.40, TOST p<0.01), and density-thinning experiments showing β captures exclusion strength (though absolute values remain density-dependent). A decision table for binary fields at low fill fraction is provided, and the combination is presented as a calibrated measurement framework.

Significance. If the empirical calibration and controls hold, the work supplies a practical, reproducible tool for quantifying short-range exclusion in 2D point patterns, with direct relevance to materials characterization and spatial statistics. Strengths include the multiple independent controls (arithmetic vs. embedding, density thinning) and explicit correction of the 1D Poisson reference for 2D CSR; these provide concrete evidence that β responds to exclusion rather than density or embedding artifacts alone.

major comments (1)
  1. [Abstract] Abstract: the claim that the β–r_excl calibration and CSR baseline correction 'constitute a calibrated measurement framework' for general use is load-bearing for the central contribution, yet rests on empirical validation limited to the 58 surfaces, prime embeddings, and thinning experiments; no derivation or additional process-independence test is indicated to show why the recalibrated Brody form must capture exclusion strength outside this tested set.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and the recognition of the multiple controls. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the β–r_excl calibration and CSR baseline correction 'constitute a calibrated measurement framework' for general use is load-bearing for the central contribution, yet rests on empirical validation limited to the 58 surfaces, prime embeddings, and thinning experiments; no derivation or additional process-independence test is indicated to show why the recalibrated Brody form must capture exclusion strength outside this tested set.

    Authors: We agree that the abstract claim is strong and that the supporting evidence is empirical rather than derived from first principles. The validation set comprises 58 surfaces spanning 10 materials and 10 distinct manufacturing processes, together with prime-number embeddings (with sparse-integer arithmetic control), Cantor-set null embedding, and systematic density-thinning experiments. These controls demonstrate that β responds to exclusion strength across both physical and mathematical generation mechanisms and is not an artifact of density or embedding alone. Nevertheless, no general derivation is provided, and the tested processes do not exhaust all possible 2D point processes. In revision we will (i) qualify the abstract sentence to read that the calibration and controls “provide an empirically validated measurement framework” and (ii) add an explicit limitations paragraph in the discussion stating the empirical scope and recommending further process-independence tests on additional point-process families. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical calibration is data-driven without reduction to inputs

full rationale

The paper reports an empirical recalibration of the Brody exponent β on 2D CSR data to 0.96±0.15 and an empirical β–r_excl relation validated by Spearman ρ=0.988 on 58 surfaces plus controls (primes, Cantor, thinning). These are presented as measured correlations and baselines rather than a first-principles derivation; β is obtained by fitting the distribution form to point patterns while r_excl is computed separately via effective hard-core radius, with the relation then observed rather than forced by construction. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing; the controls (arithmetic signal, embedding null, density independence) are independent checks. The framework is therefore self-contained as an empirical measurement protocol without any step reducing to its own fitted inputs by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore limited to those explicitly invoked in the abstract text.

free parameters (1)
  • 2D CSR baseline β = 0.96
    Recalibrated value 0.96±0.15 obtained from 2D complete-spatial-randomness data to replace the 1D Poisson reference.
axioms (1)
  • domain assumption The Brody distribution form remains a valid phenomenological interpolation for short-range exclusion statistics when applied to 2D spatial point processes.
    Abstract treats the Brody exponent as directly transferable from quantum-chaos level statistics to 2D point patterns.

pith-pipeline@v0.9.1-grok · 5824 in / 1536 out tokens · 53945 ms · 2026-06-27T02:30:00.995831+00:00 · methodology

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Works this paper leans on

33 extracted references · 25 canonical work pages

  1. [1]

    T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, S. S. M. Wong, Random- matrix physics: spectrum and strength fluctuations, Rev. Mod. Phys. 53 (1981) 385–479. doi:10.1103/RevModPhys.53.385

    work page doi:10.1103/revmodphys.53.385 1981

  2. [2]

    Stöckmann, Quantum chaos: an introduction, Cambridge University Press (1999)

    H.-J. Stöckmann, Quantum chaos: an introduction, Cambridge University Press (1999). doi:10.1017/CBO9780511524622

    work page doi:10.1017/cbo9780511524622 1999

  3. [3]

    Alhassid, The statistical theory of quantum dots, Rev

    Y . Alhassid, The statistical theory of quantum dots, Rev. Mod. Phys. 72 (2000) 895–968. doi:10.1103/RevModPhys.72.895

    work page doi:10.1103/revmodphys.72.895 2000

  4. [4]

    R. L. Weaver, Spectral statistics in elastodynamics, J. Acoust. Soc. Am. 85 (1989) 1005– 1013.doi:10.1121/1.397484

    work page doi:10.1121/1.397484 1989

  5. [5]

    Laloux, P

    L. Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters, Noise dressing of financial correlation matrices, Phys. Rev. Lett. 83 (1999) 1467–1470.doi:10.1103/PhysRevLett.83.1467

    work page doi:10.1103/physrevlett.83.1467 1999

  6. [6]

    H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Symp. Pure Math. 24 (1973) 181–193

    1973

  7. [7]

    A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987) 273–308

    1987

  8. [8]

    M. V . Berry, Riemann’s zeta function: a model for quantum chaos?, Lecture Notes in Physics 263 (1986) 1–17.doi:10.1007/3-540-17171-1_1

    work page doi:10.1007/3-540-17171-1_1 1986

  9. [9]

    E. B. Bogomolny, J. P. Keating, Gutzwiller’s trace formula and spectral statistics: be- yond the diagonal approximation, Phys. Rev. Lett. 77 (1996) 1472–1475.doi:10.1103/ PhysRevLett.77.1472

    1996

  10. [10]

    J. P. Keating, N. C. Snaith, Random matrix theory andζ(1/2+it), Commun. Math. Phys. 214 (2000) 57–89.doi:10.1007/s002200000261

    work page doi:10.1007/s002200000261 2000

  11. [11]

    Sakhr, J

    J. Sakhr, J. M. Nieminen, Poisson-to-Wigner crossover transition in the nearest-neighbor statistics of random points on fractals, Physical Review E 72 (2005) 045204.doi:10. 1103/PhysRevE.72.045204

    2005

  12. [12]

    John Wiley & Sons, Ltd, ??? (2007)

    J. Illian, A. Penttinen, H. Stoyan, D. Stoyan, Statistical Analysis and Modelling of Spatial Point Patterns, Wiley, 2008.doi:10.1002/9780470725160

    work page doi:10.1002/9780470725160 2008

  13. [13]

    Chapman and Hall/CRC, ??? (2015)

    A. Baddeley, E. Rubak, R. Turner, Spatial Point Patterns: Methodology and Applications with R, CRC Press, 2015.doi:10.1201/b19708

    work page doi:10.1201/b19708 2015

  14. [14]

    S. N. Chiu, D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and its Applications, 3rd Edition, Wiley, 2013.doi:10.1002/9781118658222

    work page doi:10.1002/9781118658222 2013

  15. [15]

    Møller, R

    J. Møller, R. P. Waagepetersen, Statistical inference for Cox processes, in: Spatial Cluster Modelling, Chapman and Hall/CRC, 2002, pp. 37–60

    2002

  16. [16]

    Massaro, A

    M. Massaro, A. D. Campo, Spatial form factor for point patterns: Poisson point process, Coulomb gas, and vortex statistics, Physical Review Research 7 (2025) 023107.doi: 10.1103/physrevresearch.7.023107. 32

    work page doi:10.1103/physrevresearch.7.023107 2025

  17. [17]

    J. U. Yang, Y . Guan, Fourier analysis of spatial point processes, Bernoulli 32 (2026) 1–28. doi:10.3150/25-bej1862

    work page doi:10.3150/25-bej1862 2026

  18. [18]

    Helmli, Focus variation instruments, Optical Measurement of Surface Topography (2011) 131–166doi:10.1007/978-3-642-12012-1_7

    F. Helmli, Focus variation instruments, Optical Measurement of Surface Topography (2011) 131–166doi:10.1007/978-3-642-12012-1_7

    work page doi:10.1007/978-3-642-12012-1_7 2011

  19. [19]

    Multi-modal contrastive learning of urban space representations from POI data

    D. Kucharski, M. Wieczorowski, Radial image processing for phase extraction in rough-surface interferometry, Measurement 242 (2025) 117102.doi:10.1016/j. measurement.2025.117102

    work page doi:10.1016/j 2025

  20. [20]

    G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition, Oxford University Press, 2008

    2008

  21. [21]

    Stein, S

    M. Stein, S. M. Ulam, An observation on the distribution of primes, Amer. Math. Monthly 74 (1967) 43–44.doi:10.2307/2314055

    work page doi:10.2307/2314055 1967

  22. [22]

    Rényi, On measures of entropy and information, Proc

    A. Rényi, On measures of entropy and information, Proc. 4th Berkeley Symp. Math. Stat. Prob. 1 (1961) 547–561

    1961

  23. [23]

    H. G. E. Hentschel, I. Procaccia, The infinite number of generalized dimensions of frac- tals and strange attractors, Physica D 8 (1983) 435–444.doi:10.1016/0167-2789(83) 90235-X

    work page doi:10.1016/0167-2789(83 1983

  24. [24]

    Grassberger, I

    P. Grassberger, I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett. 50 (1983) 346–349.doi:10.1103/PhysRevLett.50.346

    work page doi:10.1103/physrevlett.50.346 1983

  25. [25]

    Theiler, Statistical precision of dimension estimators, Phys

    J. Theiler, Statistical precision of dimension estimators, Phys. Rev. A 41 (1990) 3038–3051. doi:10.1103/physreva.41.3038

    work page doi:10.1103/physreva.41.3038 1990

  26. [26]

    Danzl, F

    R. Danzl, F. Helmli, S. Scherer, Focus variation—a robust technology for high resolution optical 3D surface metrology, Strojniški vestnik – Journal of Mechanical Engineering 57 (2011) 245–256.doi:10.5545/sv-jme.2010.175

    work page doi:10.5545/sv-jme.2010.175 2011

  27. [27]

    Ginibre, Statistical ensembles of complex, quaternion, and real matrices, Journal of Math- ematical Physics 6 (1965) 440–449.doi:10.1063/1.1704292

    J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, Journal of Math- ematical Physics 6 (1965) 440–449.doi:10.1063/1.1704292

    work page doi:10.1063/1.1704292 1965

  28. [28]

    P. J. Forrester, Log-Gases and Random Matrices, Princeton University Press, 2010.doi: 10.1515/9781400835416

    work page doi:10.1515/9781400835416 2010

  29. [29]

    Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Proper- ties, Springer, 2002.doi:10.1007/978-1-4757-6355-3

    S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Proper- ties, Springer, 2002.doi:10.1007/978-1-4757-6355-3

    work page doi:10.1007/978-1-4757-6355-3 2002

  30. [30]

    C. R. Harris, et al., Array programming with NumPy, Nature 585 (2020) 357–362.doi: 10.1038/s41586-020-2649-2

    work page doi:10.1038/s41586-020-2649-2 2020

  31. [31]

    E., et al

    P. Virtanen, et al., SciPy 1.0: fundamental algorithms for scientific computing in Python, Nat. Methods 17 (2020) 261–272.doi:10.1038/s41592-019-0686-2

    work page doi:10.1038/s41592-019-0686-2 2020

  32. [32]

    J. D. Hunter, Matplotlib: a 2D graphics environment, Comput. Sci. Eng. 9 (2007) 90–95. doi:10.1109/MCSE.2007.55

    work page doi:10.1109/mcse.2007.55 2007

  33. [33]

    Johansson, et al., mpmath: a Python library for arbitrary-precision floating-point arith- metic (2023)

    F. Johansson, et al., mpmath: a Python library for arbitrary-precision floating-point arith- metic (2023). URLhttp://mpmath.org 33

    2023