An Explicit Quintic Obstruction for the Riemann Theta Kernel: An Independent Rational Reconstruction
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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 s₀ = ¼ log(13/(4π)), the signed derivative functional evaluated on the square of P(X) = X⁵ − 264X³ + 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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Edition files
- Quintic_Theta_Obstruction_Independent_Reconstruction.pdf78.0 KiB · pdf
SHA-256
284e76e417e26d21a5f00a8e891f7895324d226cc2a9fe8bbaeb87ff5d0a3a6c - main.tex3.1 KiB · source
SHA-256
95a4c40ac17e6876d0c2190a32e9f8ba39399ef78c1347f47313caf641cf742e - proof.tex11.1 KiB · source
SHA-256
de4690d7ecfdacc08171b2fc77c7dc77a7f4d0bdeb707bce1385c0e6ffd9b9a2 - Quintic_Theta_Obstruction_Independent_Reconstruction_Publication_Package.zip92.6 KiB · package
SHA-256
5a170e7956550a07cd33df74afd726689beb3c88985ee59bf2e6eff73fd552e5 - SHA256SUMS.json1.1 KiB · manifest
SHA-256
8b5e898fdc2b3ad2c2f53ac860f549bb2283b8d54f296f29c9db42ea5ab4d7c5
Available editions
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
This companion independently reconstructs the fixed-center obstruction in the original edition: https://luminaryoddities.com/researcharchive/explicit-quintic-obstruction-riemann-theta-kernel . The original publication and its edition 1 remain unchanged.
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 complete publication package contains main.tex and its required proof.tex, the PDF, a Python-standard-library verifier, certificate.json, moment_table.json, and checksum manifest. A fresh local run reconstructed the intervals and matched the retained certificate. Download the complete package to reproduce all dependencies.