pith. sign in

arxiv: 2604.27772 · v3 · submitted 2026-04-30 · 📊 stat.ME

Single-Observation Uniformity Testing under Increasing Precision via Lacunary Harmonics

Davide Ferrari This is my paper

Pith reviewed 2026-05-08 03:08 UTC · model grok-4.3

classification 📊 stat.ME
keywords uniformity testingsingle observationlacunary harmonicsHadamard gapsdigit expansionFourier analysischi-square limitmultiscale testing
0
0 comments X

The pith

A single high-precision observation suffices to test uniformity on the unit interval by summing trigonometric components at sparse Hadamard-gap frequencies.

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

The paper develops a uniformity test on [0,1] that operates with only one observation provided that observation is recorded with steadily increasing precision. It forms a quadratic statistic by adding sine and cosine terms of the observation across successive digit scales, but only at frequencies separated by Hadamard gaps. Under the null hypothesis of uniformity this statistic converges in distribution to a chi-square law through a lacunary central limit theorem. When the distribution is not uniform the same components are driven by the Fourier content induced by digit-scale transformations of the single point. The construction therefore detects multiscale harmonic departures that remain invisible to ordinary digit-frequency or single-scale tests.

Core claim

Aggregating trigonometric components of the single observation across digit scales at Hadamard-gap frequencies produces a quadratic test statistic whose null limiting distribution is chi-square. Under alternatives the statistic is driven by the coherent accumulation of Fourier coefficients that arise from the digit-scale transformations of the observation, with power increasing as precision grows.

What carries the argument

The quadratic test statistic formed by summing trigonometric components at Hadamard-gap frequencies across multiple digit scales of the single observation, whose null distribution follows from the lacunary central limit theorem.

If this is right

  • Under uniformity the quadratic statistic converges in distribution to chi-square as the number of digit scales increases.
  • The statistic receives contributions only from the Fourier components induced by digit-scale maps of the observation.
  • Power is obtained against alternatives that produce coherent multiscale harmonic structure, even though classical consistency with one observation is impossible.
  • The procedure remains inapplicable to alternatives whose induced Fourier content fails to accumulate across scales.

Where Pith is reading between the lines

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

  • The same lacunary aggregation might be applied to test other properties such as independence when a single high-precision vector is available.
  • Precision here functions as an effective sample size that replaces the usual large-n requirement in Fourier-based testing.
  • The method could be adapted to physical or digital data streams where measurements are delivered at successively finer resolutions.

Load-bearing premise

The observation is recorded with enough and steadily increasing precision that the trigonometric components can accumulate coherently across digit scales without additional regularity on the alternative.

What would settle it

Monte Carlo simulations under the uniform distribution in which the quadratic statistic fails to converge to a chi-square law as the number of included digit scales grows would refute the claimed null convergence.

Figures

Figures reproduced from arXiv: 2604.27772 by Davide Ferrari.

Figure 1
Figure 1. Figure 1: Densities of the lacunary statistic T for precision levels m ∈ {15, 25, 50, 100} under H0 (histograms and dotted kernel density estimates) and under the phase-shift alternative (model (i)) with parameters θ (c) 1,t = 2b cos ϕt , θ (s) 1,t = 2b sin ϕt , ϕt = 2π(t − 1)/b (dashed kernel density estimates). The solid line denotes the asymptotic χ 2 18 null density. Results are based on 1000 post–burn-in Gibbs … view at source ↗
Figure 2
Figure 2. Figure 2: Log p-values versus precision m for truncated numbers Um whose digits are gen￾erated by the irrational rotation xt = x0 + tξ mod 1 with x0 = 0 and ξ ∈ {π, e, √ 2, γ, ζ(3)}. Each curve corresponds to one constant. The horizontal dashed line marks log(0.01), corre￾sponding to the 1% significance level view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative lacunary imbalance √ t Pt r=1 |pb (r) C (d) − 1/10|, t = 1, . . . , 10 for the base-10 log mantissas of the accounting amounts 67,339 (solid line), 275,403,000 (dashed line) and 42,820,000 (dotted line) in the Sino-Forest data. Each panel corresponds to a digits d ∈ {0, . . . , 9}. Under uniformity the curves should fluctuate around the expected cumulative imbalance (b − 1)/b2 = 9/100, while dev… view at source ↗
read the original abstract

A test of uniformity on [0,1] is developed for the setting of a single observation recorded with sufficient precision. Although consistency against general alternatives is not attainable with only one draw in the classical large-sample sense, a multiscale harmonic digit expansion provides a framework for structured inference. By aggregating trigonometric components across digit scales at Hadamard-gap frequencies, a quadratic test statistic is constructed whose null distribution converges to a chi-square law via a lacunary central limit theorem. Under departures from uniformity, the statistic is driven by Fourier components induced by digit-scale transformations of the observation, with detectability depending on their coherent accumulation as precision increases. The resulting procedure detects multiscale harmonic structure that remains invisible to classical digit-frequency methods.

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

2 major / 0 minor

Summary. The manuscript develops a uniformity test on [0,1] for a single observation recorded with increasing precision. It constructs a quadratic test statistic by aggregating trigonometric components across digit scales at Hadamard-gap frequencies in a multiscale harmonic digit expansion. Under the null of uniformity, the statistic converges in distribution to chi-square via an external lacunary central limit theorem. Under alternatives, the statistic is driven by Fourier components induced by digit-scale transformations of the observation, with detectability tied to coherent accumulation as precision grows. The procedure is positioned as detecting multiscale harmonic structure invisible to classical digit-frequency methods.

Significance. If the limiting null distribution can be rigorously justified, the work offers a conceptually novel route to structured inference from one high-precision draw, where classical consistency is impossible. It usefully exploits lacunary harmonics and digit-scale aggregation to target alternatives that standard tests miss. The absence of derivations, error bounds, or verification steps for the key limit, however, keeps the immediate methodological impact modest.

major comments (2)
  1. [Abstract and the derivation of the limiting distribution] The central claim that the quadratic statistic converges to chi-square under the null rests on direct application of a lacunary CLT to the aggregated form. Standard lacunary CLTs guarantee asymptotic normality for linear sums with frequencies satisfying a fixed gap ratio, but the quadratic form requires joint normality of the vector of components together with a diagonal covariance matrix. Digit-scale transformations (fractional parts at successive powers) can modify effective frequencies and induce residual dependence, turning the problem into a triangular-array limit whose gap and orthogonality conditions are not automatically inherited. This verification is missing and is load-bearing for the null-distribution result.
  2. [Numerical illustration and practical implementation] No explicit error bounds, rate of convergence, or simulation evidence is supplied to support the chi-square approximation for finite precision levels. Given that the procedure is intended for a single observation whose precision increases, the practical reliability of the asymptotic reference distribution remains unexamined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. Below we address each major comment point by point.

read point-by-point responses

  1. Referee: [Abstract and the derivation of the limiting distribution] The central claim that the quadratic statistic converges to chi-square under the null rests on direct application of a lacunary CLT to the aggregated form. Standard lacunary CLTs guarantee asymptotic normality for linear sums with frequencies satisfying a fixed gap ratio, but the quadratic form requires joint normality of the vector of components together with a diagonal covariance matrix. Digit-scale transformations (fractional parts at successive powers) can modify effective frequencies and induce residual dependence, turning the problem into a triangular-array limit whose gap and orthogonality conditions are not automatically inherited. This verification is missing and is load-bearing for the null-distribution result.

    Authors: We are grateful for this observation. The manuscript applies a lacunary CLT formulated for triangular arrays, where the Hadamard gap condition holds uniformly across scales, and the digit expansions preserve the necessary orthogonality and gap ratios under the uniform null. This ensures joint normality with asymptotically diagonal covariance, yielding the chi-square limit. To strengthen the presentation, we will add a dedicated subsection deriving these properties explicitly from the cited theorem. revision: partial

  2. Referee: [Numerical illustration and practical implementation] No explicit error bounds, rate of convergence, or simulation evidence is supplied to support the chi-square approximation for finite precision levels. Given that the procedure is intended for a single observation whose precision increases, the practical reliability of the asymptotic reference distribution remains unexamined.

    Authors: We concur that assessing the finite-precision performance is essential. The revised manuscript will incorporate simulation results for the distribution of the test statistic at various precision levels, as well as explicit convergence rates and error bounds obtained from the lacunary CLT. revision: yes

Circularity Check

0 steps flagged

No circularity: external lacunary CLT invoked for limiting distribution

full rationale

The paper constructs a quadratic test statistic by summing squared trigonometric terms at Hadamard-gap frequencies across digit scales and states that its null distribution converges to chi-square by direct application of a lacunary central limit theorem. This theorem is an external mathematical result (not derived or fitted inside the paper), and the construction of the statistic does not reduce to its own inputs by definition, renaming, or self-referential fitting. No load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness claims imported from the authors appear in the derivation chain. The approach remains self-contained against the cited external theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of a known lacunary central limit theorem to the new quadratic form built from digit-scale trigonometric sums; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Lacunary central limit theorem applies to the quadratic form constructed from trigonometric sums at Hadamard-gap frequencies across digit scales
    Invoked to obtain convergence of the null distribution to chi-square as precision increases.

pith-pipeline@v0.9.0 · 5410 in / 1317 out tokens · 87167 ms · 2026-05-08T03:08:50.014126+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    29 R. L. Eubank and V. N. LaRiccia. Asymptotic comparison of Cramér–von Mises and non- parametricfunctionestimationtechniquesfortestinggoodness-of-fit.The Annals of Statis- tics, 20(4):2071–2086,

    work page 2071

  2. [2]

    Reprint of the 1957 Wiley edition. Y. Hong. Consistent testing for serial correlation of unknown form.Econometrica, 64(4): 837–864,

    work page 1957

  3. [3]

    Jiang and T

    T. Jiang and T. Pham. Asymptotic analysis of high-dimensional uniformity tests under heavy-tailed alternatives.arXiv preprint arXiv:2506.00393,

    work page arXiv

  4. [4]

    J. Neyman. Smooth tests for goodness of fit.Scandinavian Actuarial Journal, 1937(3-4): 149–199,

    work page 1937