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Methods Paper · Preprint Mathematics

Covariance confidence bounds and parity certificates for stable count laws

Scott Schubr

Luminary Oddities

  • Mathematics
  • Probability and Statistics
Primary subject
Mathematics
Paper type
Methods Paper
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · 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

Unknown-mean covariance confidence bounds without a minimum category probability for homogeneous real-stable labeled count laws, with linear-category lower bounds and sufficient positive-parity certificates.

Reproduction notes

10-page manuscript and 8 source-package members. Compile main.tex. Standard-library Python runs verify.py, also under -O, and calibrator.py --self-test. To calibrate observations, run calibrator.py input.json with the documented two independent count-histogram batches; certification requires the homogeneous real-stable model and both sample-size gates.

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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

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A stable scholarly record

Cite this edition

Scott Schubr. (2026). Covariance confidence bounds and parity certificates for stable count laws (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/covariance-calibration-parity-certificates-stable-count-laws/v/2

BibTeX
@techreport{luminary_covariance_calibration_parity_certificates_stable_count_laws_v2,
  title = {Covariance confidence bounds and parity certificates for stable count laws},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Methods Paper},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/covariance-calibration-parity-certificates-stable-count-laws/v/2}
}

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