Research Article · Preprint Mathematics
Sharp collision criteria for frozen color histograms
- Primary subject
- Mathematics
- Subject subfields
- Probability and Statistics · Combinatorics and Graph Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
Let a fixed probability law select an n-element subset of a finite ground set, and color every ground element independently once, with prescribed positive probabilities q_1,…,q_m. We compare the conditional histogram law after freezing the coloring with Mult(n,q). Under upper negative-cylinder inequalities, the normalized marginal collision sum τ=n^(-1)Σ_ePr(e∈ S)^2 determines expected chi-square convergence: the distance tends to zero if and only if (m-1)τ does, and is asymptotic to (m-1)τ in that regime. The statements are uniform over positive color probabilities, without a minimum-probability assumption. For fixed positive q and fixed m≥2, expected total variation tends to zero if and only if τ does. The necessary half follows from an explicit covariance statistic and holds for every fixed-size subset law. The proof combines the classical multinomial overlap kernel with negative-cylinder factorial moments. An explicit diffuse non-real-stable family separates the hypotheses from real stability; spanning-tree and fixed matroid-base-rounding laws give applications.
Keywords multinomial approximation · negative dependence · frozen coloring · collision probability · total variation · matroid rounding
Scope of the work
Under upper negative-cylinder inequalities, (m−1)τ characterizes expected chi-square convergence uniformly over positive color probabilities; fixed positive colors give a τ criterion for expected total variation, with a necessary covariance bound for every fixed-size subset law.
Reproduction notes
The 9-page manuscript and complete editable LaTeX sources are supplied in this 7-member package. README.md gives the build and reproduction commands. Reproducibility code: collision_controls.py, overlap_controls.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.
The manuscript
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Author disclosures
AI assistance was used in mathematical exploration, drafting, calculation, and checking. The author is responsible for the statements, attribution, and final manuscript.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Sharp collision criteria for frozen color histograms (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-collision-criteria-frozen-color-histograms/v/2
BibTeX
@techreport{luminary_sharp_collision_criteria_frozen_color_histograms_v2,
title = {Sharp collision criteria for frozen color histograms},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/sharp-collision-criteria-frozen-color-histograms/v/2}
}Loading discussion…