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Methods Paper · Preprint Mathematics

Covariance calibration and parity certificates for stable count laws

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Probability and Statistics
Primary subject
Mathematics
Subject subfields
Probability and Statistics
Paper type
Methods Paper
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

We study the normalized covariance deficit of a homogeneous real-stable count law on m categories with total count k. This deficit is a rescaling of an established canonical dependence matrix for two balls drawn without replacement. Its positive semidefiniteness gives a diagonal domination property that controls rare categories and normalization error. We use this structure to construct an upper confidence bound for its operator norm without a minimum category-probability assumption and without knowing the mean. Independent batches of full count observations suffice: the mean stage uses O(m(k − 1)ε^{−2} log(m/δ)/k) observations and the second stage uses O(m(k − 1)²ε^{−2} log(m/δ)). A stable degree-two subclass gives a matching linear category and inverse-square accuracy lower bound up to logarithms. Real-rooted one-dimensional projections then turn a covariance bound and a reference Fourier margin into simultaneous positive-parity certificates. We give a sharp positivity boundary on a binary subclass and stable laws with identical means and covariance but opposite parity. The constants in the confidence bound are conservative, and the observation model is classical labeled counts.

Keywords stable count laws · covariance estimation · rare categories · confidence bounds · parity certificates

Scope of the work

Classical labeled-count observations from a single homogeneous real-stable total-count law, with independent iid mean and pair batches, are required. Real stability is a model assumption, not inferred from the observations. Constants are conservative. Mean and covariance do not determine parity in general, and no quantum hardware observation protocol or production-scale exact-arithmetic performance is claimed.

Reproduction notes

The seven-member package contains the 10-page manuscript, complete LaTeX source, calibrator.py, verify.py, reader instructions, citation metadata and a manifest. Run verify.py in normal and optimized modes and calibrator.py --self-test using the Python standard library. For data input, follow README.md: independent labeled-count histograms and explicit sample-size gates are required. The synthetic fixtures provide exact replay, not a proof of concentration. Compile main.tex twice with a standard LaTeX installation.

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Author disclosures

AI systems assisted research exploration, literature discovery, drafting and internal checking. The author retains responsibility for the manuscript. Internal AI-assisted and computational checks are distinct from external human peer review.

References and related publications

  1. J. Borcea, P . Brändén and T. M. Liggett, Negative dependence and the geometry of polynomials, J. Amer. Math. Soc. 22 (2009), 521–567. arXiv:0707.2340.
  2. I. Diakonikolas, D. M. Kane and A. Stewart, The Fourier transform of Poisson multinomial distributions and its algorithmic applications, arXiv:1511.03592 (2015).
  3. S.-L. Huang, A. Makur, G. W. Wornell and L. Zheng, Universal features for high-dimensional learning and inference, Foundations and Trends in Communications and Information Theory 21 (2024), 1–299. doi:10.1561/0100000107.
  4. C. M. Le, E. Levina and R. Vershynin, Concentration and regularization of random graphs, Random Structures Algorithms 51 (2017), 538–561. arXiv:1506.00669.
  5. P . A. Samuelson, How deviant can you be?, J. Amer. Statist. Assoc. 63 (1968), 1522–1525. doi:10.1080/01621459.1968.10480944.
  6. S. Schubr, Moment certificates for homogeneous stable count laws , author preprint (2026). Author edition 3.
  7. W. Tang and F. Tang, The Poisson binomial distribution—old & new, Statistical Science 38 (2023), 108–119. arXiv:1908.10024.
  8. J. A. Tropp, User-friendly tail bounds for sums of random matrices, Found. Comput. Math. 12 (2012), 389–434. doi:10.1007/s10208-011-9099-z.
  9. X. Xu, W. Wang and S.-L. Huang, On the sample complexity of estimating small singular modes, 2020 IEEE International Symposium on Information Theory . Author manuscript.
  10. N. Anari, K. Liu, S. Oveis Gharan and C. Vinzant. Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid . STOC, 2019. arXiv:1811.01816.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Covariance calibration and parity certificates for stable count laws (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/covariance-calibration-parity-certificates-stable-count-laws/v/1

BibTeX
@techreport{luminary_covariance_calibration_parity_certificates_stable_count_laws_v1,
  title = {Covariance calibration and parity certificates for stable count laws},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Methods Paper},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/covariance-calibration-parity-certificates-stable-count-laws/v/1}
}

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