Pith. sign in

REVIEW 2 major objections 5 minor 7 references

Normality Analysis of Current World Record Computations for Catalan's Constant and Arc Length of a Lemniscate with $a=1$

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The first 600 billion digits of Catalan's constant and the lemniscate arc length show digit frequencies consistent with normality in bases 10 and 16.

desk verdict A modest but honest data-check paper: new 600-billion-digit prefixes for Catalan's constant and the lemniscate arc length, analyzed with Trueb's variance test, all consistent with normality. read the letter →

arxiv 1908.08925 v4 pith:FLAUU3LS submitted 2019-08-22 math.GM

Seungmin Kim This is my paper · ORCID
classification math.GM MSC 11-XX62-XX68-XX
keywords Catalan'sconstantlemniscatenormalnumberdigitanalysisfrequencybinomialvariancebase1016
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Normal numbers—real numbers in which every finite block of digits appears with its equal limiting frequency—are conjectured for most familiar constants but proved for almost none. This paper tests that conjecture on the current world-record digit computations: $600{,}000{,}000{,}100$ decimal digits of Catalan's constant, $600{,}000{,}000{,}000$ decimal digits of the arc length of a lemniscate with $a=1$, and the matching hexadecimal strings. Counting every digit block of length 1, 2, and 3 in both bases, the author finds that the actual variance of the block frequencies stays within about $1.8$ standard deviations of the variance a binomial model predicts for a uniformly random digit stream. The conclusion is that the data are consistent with, though not proof of, the conjecture that both constants are normal numbers in bases 10 and 16.

What carries the argument

Central to the argument is the variance of the frequency counts of all $b^k$ digit strings of a fixed length $k$ in base $b$. Under the null hypothesis that every string is equally likely, the count of any particular string among $N$ digit positions follows a binomial distribution with probability $b^{-k}$, giving a predicted variance of roughly $N b^{-k}(1-b^{-k})$. The paper compares the actual variance of the $b^k$ observed frequencies with that predicted variance (including its standard error), and displays the results as histograms with one- and two-standard-deviation bands. This variance comparison, rather than a direct check of each individual frequency, is the appropriate test because individual frequencies fluctuate as $N^{-1/2}$ even for a truly normal number. The approach is adopted from a previous digit-statistics analysis of the first $22.4$ trillion decimal digits of $\pi$.

What would settle it

Recompute a contiguous prefix of each constant with an independent algorithm and implementation—for example, binary-splitting the series $\sum_{n=0}^{\infty}(-1)^n/(2n+1)^2$ for Catalan's constant—and compare digit-by-digit with the archived strings; any mismatch would show the analyzed data are not the true prefix. A second check: run the same variance computation on block lengths 4 and 5 of the archived strings; deviations of several standard deviations there would contradict the paper's claim that the streams behave like a normal number.

Watch

Extended reading notes

Core claim

The paper's central claim is that, at the scale of roughly 600 billion digits, neither constant shows a statistically meaningful departure from the digit-frequency behavior expected of a normal number. For each of the twelve cases—two constants, two bases (10 and 16), and block lengths 1–3—the sample variance of the observed frequencies lies within about $1.8$ standard deviations of the predicted variance. The largest absolute deviation is $1.798\,\sigma$, for two-digit decimal blocks in the lemniscate arc-length string; the hexadecimal results for the same constant deviate by at most $1.080\,\sigma$, and Catalan's constant by at most $1.349\,\sigma$ in either base. The paper states these results as 'persistent to' the conjecture that both constants are normal in bases 10 and 16, explicitly presenting the analysis as statistical evidence rather than a proof.

Load-bearing premise

The argument assumes that the archived 600-billion-digit strings are authentic, correctly computed prefixes of the two constants; if the computation or the archived copies contain any error, the frequency statistics describe a different number and the normality evidence is void.

Editorial extensions

If this is right

  • On the paper's evidence, the normality conjecture for Catalan's constant and the lemniscate arc length survives the largest digit sample yet examined, with no block of length 1–3 showing anomalous frequency variance.
  • Because the method is data-driven and uses archive-accessible digit strings, it can be applied immediately to the world-record prefixes of other constants to screen for early signs of non-normality.
  • At the 600-billion-digit scale, both constants' digit streams match a uniform random model closely enough that block-level statistical tests on these constants should expect near-uniform output.
  • If the data are genuine, these variance numbers furnish reference points for how a normal candidate behaves over 600 billion digits, useful for calibrating future tests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extending the same variance test to four- and five-digit blocks would be a sharper check than anything in the paper, because the binomial predictions remain stable while the number of categories grows to $10^4$ or $10^5$ in base 10.
  • Hexadecimal normality is the more significant agreement for binary structure: a hexadecimal digit is exactly four bits, so uniform hex digits imply uniform bit strings, whereas uniform decimal digits give only indirect information about bits.
  • If the pattern persists, the digit streams of these constants could be used as deterministic pseudo-random sources; normality alone, however, would not certify them for cryptographic use without further randomness tests.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper analyzes digit frequency distributions in the first 600 billion decimal and hexadecimal digits of Catalan's constant and the arc length of a lemniscate with a=1, using data from the author's own y-cruncher world-record computations. For each of the two constants, each base (10 and 16), and each block length 1, 2, and 3, it computes the sample variance of the frequencies of all b^k digit combinations and compares this with the variance expected under a binomial model with success probability b^{-k}. The measured variances are reported to lie within 1.8 standard deviations of the predicted values, and the paper concludes that the data are 'persistent to' (i.e., consistent with) the conjecture that these constants are normal in bases 10 and 16. The paper includes histograms of the frequency distributions and tables of predicted and actual variances.

Significance. If the archived digit strings are indeed the correct prefixes, the paper provides a straightforward, parameter-free statistical check on the normality conjecture for two important constants at an unprecedentedly large scale. The core comparison is simple and transparent, and the conclusion is appropriately cautious, using 'persistent' rather than claiming a proof. The data are made publicly available on the Internet Archive, which supports reproducibility. The main limitations are that only block lengths 1 to 3 are tested, so the evidence for normality is very weak, and that the integrity of the data is not fully established; both issues are addressable in revision. The paper is a useful data-driven sanity check for world-record constant computations.

major comments (2)
  1. [Section 2 and Data Availability Statement] The entire analysis depends on the correctness of the 600-billion-digit files archived at the Internet Archive, but the only verification described is a cross-check of single-digit counts using y-cruncher's Digit Viewer, which is a spot check rather than a validation of the full files. No checksums, independent recomputation results, or repeatable generation scripts are provided, and the acknowledgment mentions verification of the lemniscate record but not of the Catalan record. Please either (a) provide a full verification protocol, such as checksums for both files, an independent recomputation of a substantial portion, or a script that regenerates the data, or (b) explicitly restrict the conclusion to the archived strings so that the reader can judge whether the evidence bears on the constants themselves rather than on possibly corrupted or mislabeled files.
  2. [Section 2, Tables 1 and 2] The paper reports a 'predicted variance and error of frequencies' for each case, for example (1.500±0.707)×10^-13 for decimal length 1, and uses the quoted error to compute the 'Deviation [σ]' column, but it gives no formula for the error on the predicted variance. The quoted values indicate that the error was computed as the standard error of a sample variance from independent normal observations, whereas the b^k block counts are jointly multinomial and are not independent. Please state the exact estimator used for the sample variance, its expectation under the null, and the formula for its standard error (or cite the precise equation in Trueb 2016), and justify the normal approximation for small numbers of cells such as 10. Without this information, the reported 1–2σ thresholds cannot be reproduced or interpreted reliably.
minor comments (5)
  1. [Figure 10 caption] The caption for Figure 10 says 'digits 0–9' for a hexadecimal length-1 histogram, but it should say 'digits 0–F'.
  2. [Figure 11 caption] The caption for Figure 11 says '00–99' for a hexadecimal length-2 histogram, but it should say '00–FF'.
  3. [Figure 12 caption] The caption for Figure 12 says '000–999' for a hexadecimal length-3 histogram, but it should say '000–FFF'.
  4. [Abstract and Discussion] The phrase 'persistent to the conjecture' is nonstandard; 'consistent with the conjecture' would be clearer. The Discussion should also explicitly state that the analysis covers only block lengths 1 through 3 and that this provides only weak evidence for normality, since a number can pass these tests while failing for longer block lengths.
  5. [Section 2] The expected variance of the frequencies is stated to follow from the binomial model, but the paper does not explain how the 'error of the predicted variance' is derived; please add a sentence or a reference to a specific equation in Trueb (2016).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the expected variance is a fixed binomial prediction, and the self-cited digit computations are data provenance rather than a fitted input.

full rationale

The paper's central comparison is between (i) the empirically counted variances of digit frequencies in the archived decimal and hexadecimal strings and (ii) the variances predicted by a binomial model with probability b^{-k} for each length-k block. This predicted variance is a fixed theoretical quantity, not a parameter fitted to the data, so the 'actual variance' reported in Tables 1 and 2 is an independent measurement. The conclusion that the data are 'persistent' with normality is a statistical consistency statement, not a derivation of normality from normality. The only self-referential element is that the digit strings were generated by the author's own y-cruncher computations and archived by the author (Kim 2019a,b), but this is data generation and provenance, not a circular argument: the normality claim is not assumed in order to produce those digits, and the statistical test does not use the normality conjecture as an input. The paper also cross-checks single-digit counts with y-cruncher's Digit Viewer and cites Trueb's established method for the variance calculation. Concerns about the authenticity, completeness, or correctness of the archived files would be data-validity or correctness risks, not circularity, because the analysis would still be a well-defined empirical test of whatever string is actually examined. No self-definitional step, fitted-input-called-prediction step, self-citation load-bearing step, imported uniqueness theorem, or ansatz-smuggling step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the authenticity of the archived digit strings and on the standard binomial model for digit frequencies. No free parameters are fitted to the data, and no new entities are postulated.

assumptions (3)
  • domain assumption The digit strings from y-cruncher are correct representations of Catalan's constant and the lemniscate arc length.
    Section 2 states the computation results were 'calculated and verified using the y-cruncher program'; the analysis assumes the accuracy and authenticity of these digits.
  • standard math Digit frequencies in a normal number follow a binomial distribution around b^{-k}.
    Section 2 derives expected variance using the binomial model, following Trueb (2016).
  • domain assumption The archived data correspond to the first 600 billion digits after the decimal point for each constant.
    The paper assumes the archived data are the prefixes of the constants as stated; no independent verification of digit positions is described.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Normality Analysis of Current World Record Computations for Catalan's Constant and Arc Length of a Lemniscate with $a=1$." pith.science (2026). https://pith.science/paper/FLAUU3LS

@misc{pith2026190808925,
  author       = {Pith},
  title        = {Pith review of: Normality Analysis of Current World Record Computations for Catalan's Constant and Arc Length of a Lemniscate with $a=1$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLAUU3LS}},
  note         = {Machine review of arXiv:1908.08925}
}
abstract

Catalan's constant and the lemniscate constants have been important mathematical constants of interest to the mathematical society, yet various properties are unknown. An important property of significant mathematical constants is whether they are normal numbers. This paper evaluates the normality of decimal and hexadecimal representations of current world record computations of digits for the Catalan's constant (600,000,000,100 decimal digits and 498,289,214,317 hexadecimal digits) and the arc length of a lemniscate with $a=1$ (600,000,000,000 decimal digits and 498,289,214,234 hexadecimal digits). All analyzed frequencies are persistent to the conjecture of Catalan's constant and the arc length of a lemniscate with $a=1$ being a normal number in bases 10 and 16.

Figures

Figures reproduced from arXiv: 1908.08925 by the authors.

Figure 1
Figure 1. Frequencies of all combinations of length 1 (digits 0–9) in the decimal representation of the Catalan’s constant. Entries 100 Mean 02 1.000e− Variance 14 1.939e− Frequency 9.9995 10 10.0005 −3 ×10 Number of Sequences 0 1 2 3 4 5 6 Entries 100 Mean 02 1.000e− Variance 14 1.939e− [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Frequencies of all combinations of length 3 (000–999) in the decimal representation of the Cata￾lan’s constant. Entries 16 Mean 02 6.250e− Variance 13 1.755e− Frequency 62.499 62.5 62.501 −3 ×10 Number of Sequences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Entries 16 Mean 02 6.250e− Variance 13 1.755e− [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Frequencies of all combinations of length 2 (00–FF) in the hexadecimal representation of the Catalan’s constant. Entries 4096 Mean 04 2.441e− Variance 16 5.024e− Frequency 0.2441 0.24415 0.2442 −3 ×10 Number of Sequences 0 10 20 30 40 50 60 70 80 90 Entries 4096 Mean 04 2.441e− Variance 16 5.024e− [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figures from the paper (3 more)
Figure 7
Figure 7. Figure 7: Frequencies of all combinations of length 1 (digits 0–9) in the decimal representation of the arc length of a lemniscate with a = 1. Entries 100 Mean 02 1.000e− Variance 14 2.072e− Frequency 9.9995 10 10.0005 −3 ×10 Number of Sequences 0 1 2 3 4 5 6 Entries 100 Mean 02…
Figure 9
Figure 9. Figure 9: Frequencies of all combinations of length 3 (000–999) in the decimal representation of the arc length of a lemniscate with a = 1. Entries 16 Mean 02 6.250e− Variance 13 1.213e− Frequency 62.499 62.5 62.501 −3 ×10 Number of Sequences 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 …
Figure 11
Figure 11. Figure 11: Frequencies of all combinations of length 2 (00–99) in the hexadecimal representation of the arc length of a lemniscate with a = 1. Entries 4096 Mean 04 2.441e− Variance 16 4.911e− Frequency 0.2441 0.24415 0.2442 −3 ×10 Number of Sequences 0 10 20 30 40 50 60 70 80 90…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

  1. [1]

    Yu. V. Nesterenko. On Catalan’s constant. Proceedings of the Steklov Institute of Mathematics, 292 0 (1): 0 153–170, 2016. doi:10.1134/s0081543816010107

    work page doi:10.1134/s0081543816010107 2016

  2. [2]

    Steven R. Finch. Gauss' Lemniscate Constant, page 420–423. Cambridge University Press, 2003

    work page 2003

  3. [3]

    The Lemniscate Constants, pages 412--417

    John Todd. The Lemniscate Constants, pages 412--417. Springer New York, New York, NY, 2000. ISBN 978-1-4757-3240-5. doi:10.1007/978-1-4757-3240-5_45. URL https://doi.org/10.1007/978-1-4757-3240-5_45

    work page doi:10.1007/978-1-4757-3240-5_45 2000

  4. [4]

    Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi

    Peter Trueb . Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi . arXiv e-prints, art. arXiv:1612.00489, Nov 2016

    arXiv 2016

  5. [7]

    Lemniscate constant, Jul 2019 b

    Seungmin Kim. Lemniscate constant, Jul 2019 b . URL https://web.archive.org/web/20190820034150/https://smkim.sourceforge.io/2019/07/23/lemniscate-constant/

    work page arXiv 2019

  6. [8]

    Portable PyPy distribution for Linux , Apr 2019

    squeaky-pl/portable-pypy . Portable PyPy distribution for Linux , Apr 2019. URL https://github.com/squeaky-pl/portable-pypy

    work page 2019

  7. [9]

    Root — an object oriented data analysis framework

    Rene Brun and Fons Rademakers. Root — an object oriented data analysis framework. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 389 0 (1): 0 81 -- 86, 1997. ISSN 0168-9002. doi:https://doi.org/10.1016/S0168-9002(97)00048-X. URL http://www.sciencedirect.com/science/article/pi...

    doi:10.1016/s0168-9002(97)00048-x 1997

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.