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Arithmetic structure in a complete atlas of power-residue runs below 2322^{32}: Rabung profiles, Kummer densities, norm dependencies, and local codes

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Brice Pouly ↗

Version 1 / Oct 19, 2026 / CC BY 4.0

Abstract

We study the arithmetic behind long runs in a complete census of power-residue colourings below 2322^{32}, covering every admissible order 2≤r≤92\le r\le9. The dataset contains 203 280 221203\,280\,221 primes and 677 591 249677\,591\,249 profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length-5353 cubic realizes a three-wise independent binary code whose monochromatic probability is 40964096 times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted.

Provenance statement

The following declaration describes the research and the published source edition: The manuscript was prepared with AI assistance for computation, proof development, literature retrieval, and English writing. Prior primary statistical tests remain closed; no selected template is relabelled as a prospective test, and no universal growth law is claimed. The mathematical examples have explicit proofs or certificates. Brice Pouly is the human author and assumes responsibility for the manuscript, code, data, references and conclusions. Exact arithmetic checks do not replace specialist review of the arguments. The accompanying article archive provides the files, commands and commitments needed for bounded reproduction; full atlas assets are a separate data product identified by the manifest. Repository deposit does not constitute peer review or establish a public disclosure date. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.23339668. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.

References

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026