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Research Article · Preprint Mathematics

Minimum-Degree Obstructions in the Riemann Theta Product: A certified classification and analytic limits

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Number Theory and Special Functions
Primary subject
Mathematics
Subject subfields
Number Theory and Special Functions
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link

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Overview

Abstract

We classify the least degree of a polynomial whose square has negative signed-moment value for the normalized fixed-center Riemann theta product. On 3 ≤ x ≤ 8 the least degree is finite, has eleven transitions, and takes the successive values 5, 6, 7, 9, 10, 9, 10, 11, 13, 14, 13, 12. We isolate every transition and determine the degree at equality. Jacobi symmetry extends the classification to π²/8 ≤ x ≤ 8. For each fixed degree, the scaled moment matrix tends to a Gaussian moment matrix as x → ∞; hence every fixed-degree test becomes strictly positive at both center extremes. The single-summand theta model has a positive degree-five matrix for all x ≥ 3, whereas two summands already produce the quintic obstruction at x = 3.64. The proof includes the complete infinite tails, a directed Taylor cover, independent exact-rational root checks, and a self-contained reproduction supplement.

Keywords Riemann theta density · moment matrices · polynomial-square obstructions · minimum degree · computer-assisted proof

Scope of the work

This is a distinct follow-up to “Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem,” edition 2 (October 18, 2026). Its first quintic transition and common fixed-center moment/Schur/theta-tail methods are prior work, explicitly credited. The additional contribution is the complete least-degree classification on [3,8], ten additional transitions, equality cases including degree six at the old first root, nonmonotonic and skipped degrees, Gaussian fixed-degree escape and finite-summand comparison. Jacobi reflection extends the stated classification to [π²/8,8]. This is not an all-center or global minimum-degree classification and implies no Riemann Hypothesis result. Independent AI reviews are not external human peer review or proof-assistant verification; worldwide priority remains unverified. The existing local-transition paper and its editions are preserved as separate publications.

Reproduction notes

The sealed corrected October 18 package includes the manuscript and PDF, a self-contained source-only supplement, a 277-cell directed cover, eleven transition windows and eleven independent exact-rational root checks, equality/mechanism checks, manuscript-binding and failure controls, revision/preservation records and the checksum manifest. The package records a successful fresh eight-stage source-only replay before publication intake. Intake verified the sealed file identities without replaying or changing that research. The publication-relationship record details the overlap and new results relative to the earlier local-transition paper. Original release files and archives remain preserved.

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Author disclosures

AI tools assisted mathematical development, verification code, computation and manuscript preparation. The author takes responsibility for the final manuscript. Independent AI reviews are not external human peer review or proof-assistant verification. Company affiliation is retained from the manuscript.

References and related publications

  1. Scott Schubr. Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem (Version 2). Luminary Research Archive, October 18, 2026.

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Cite this edition

Scott Schubr. (2026). Minimum-Degree Obstructions in the Riemann Theta Product: A certified classification and analytic limits (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/minimum-degree-obstructions-riemann-theta-product/v/1

BibTeX
@techreport{luminary_minimum_degree_obstructions_riemann_theta_product_v1,
  title = {Minimum-Degree Obstructions in the Riemann Theta Product: A certified classification and analytic limits},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/minimum-degree-obstructions-riemann-theta-product/v/1}
}

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