A rational reconstruction of a quintic obstruction for a fixed-center Riemann theta product
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Abstract
We give a complete rational reconstruction of an explicit obstruction to positive definiteness for a fixed-center product of the Riemann theta density. At s_0=(1)/(4)log(13/(4π)), the signed derivative functional evaluated on the square of P(X)=X^5-264X^3+10952X lies in the exact rational interval [-1412370791.837791749, -1412370791.837791748]. The proof constructs derivatives by formal power series, retains twelve theta summands, bounds the entire remaining series, and encloses exponentials by range-reduced alternating series. A printed table of rational moment intervals proves the negative contraction. The odd derivative Hankel block is indefinite, and the six-point translation Gram has a negative direction at every sufficiently small positive spacing.
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- Quintic_Theta_Obstruction_Independent_Reconstruction.pdf80.4 KiB · pdf
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cb056cc728903efe37abb6d6cc6b37f50d67422763d268dfc53bd515c46c1348 - Revised_publication_package.zip95.3 KiB · package
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Riemann theta density, positive definite functions, Hankel matrices, rational interval arithmetic
- Manuscript date
- 2026-10-16
- Reader reuse
- No additional permission granted
Scope
A distinct rational reconstruction of the fixed-center quintic theta-product obstruction, including twelve retained summands, a complete tail bound, a printed rational moment contraction and small-spacing finite translation-Gram indefiniteness.
Disclosures
Generative AI tools, including OpenAI's Codex, assisted with mathematical exploration and review, preparation of verification code, literature searches, and manuscript drafting and formatting. The author takes responsibility for the manuscript and supplementary materials, including all mathematical claims, computations, and references.
Reproduction
The paper has 6 pages. The editable-source archive contains 8 files. Compile main.tex with proof.tex. Standard-library supplement/verify_reconstruction.py freshly reconstructs the rational moments, complete tail and printed contraction.