Minimum-Degree Obstructions in the Riemann Theta Product: A certified classification and analytic limits
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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.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Riemann theta density, moment matrices, polynomial-square obstructions, minimum degree, computer-assisted proof
- Manuscript date
- 2026-10-18
- Reader reuse
- No additional permission granted
Scope
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.
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.
Reproduction
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.
Publication package corrected to contain publication materials only. The manuscript and edition are unchanged.