A quadratic critical-value conjecture for the fifth Bessel moment

We conjecture a quadratic critical-value evaluation of ∫₀^∞ K₀(t)⁵ dt for a weight-three, level-60 newform. We prove exact Petersson-norm and coefficient-twist identities for this form, and distinguish these modular theorems from the conjectural Bessel comparison. The even and odd determinants of Lim–Tu–Yu are related by an exact consequence of Chuang's period formulas. Directed interval computations give an absolute discrepancy below 10⁻³⁵⁸ for the individual-period conjecture. Proof certificates and numerical verification code accompany the note. Generative-AI disclosure and author responsibility. The conjecture was obtained with OpenAI's ChatGPT-6-pro, which was also used for mathematical exploration, drafting, and development of the verification programs. The author is responsible for the final manuscript. 2020 Mathematics Subject Classification: Primary 11F67; Secondary 33C10, 11F11. Files: the compiled PDF (4 pages) and the source archive containing main.tex together with the ancillary directory anc/ (full proofs of the norm and twist identities, exact modular certificates, Python interval-verification code, and SHA256SUMS for package integrity). Running python -B anc/verify.py regenerates the numerical enclosures and checks the modular proof witnesses using only Python's standard library. Revision 3 (15 September 2026): Broadhurst's Paris lecture slides of 7 June 2017 are now cited directly, with page and section pointers and credit for the joint work with D. P. Roberts; the introduction distinguishes the specific M(0,5,0) candidate from the established Bessel–L-value programme; Section 3 clarifies the two parity conventions. All mathematical formulas, proof arguments, verifier programs and numerical output records from revision 2 are unchanged.