A quadratic critical-value conjecture for the fifth Bessel moment

Abstract

We conjecture a quadratic critical-value evaluation of ∫₀^∞ K₀(t)⁵ dt for a weight-three, level-60 newform. We prove that its twisted symmetric-square value at 2 equals √15π² times its Petersson norm, and give exact norm and coefficient-twist identities. These modular theorems identify the companion norm formula with Lim–Tu–Yu's conjecture; they do not prove the Bessel comparison. Chuang's period formulas relate the even and odd determinants. Directed intervals bound the individual-period discrepancy by 10⁻³⁵⁸. Full proofs and reproducible certificates 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, twist and twisted symmetric-square norm 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; python -B verify_package.py checks the SHA-256 manifest and the stated equation status; the optional python -B anc/verify_primary.py (SymPy and mpmath) gives an independent exact reconstruction of the critical-norm coefficients and witnesses. 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. Revision 4 (16 September 2026): Proposition 3.2 proves the purely modular twisted symmetric-square norm identity L(χ₋₄ Sym² f, 2) = √15π² ⟨f,f⟩₆₀ (equation (3.5)), with the intermediate evaluation L(χ₋₄ Sym² f, 3) = π⁴ ⟨f,f⟩₆₀/8; consequently the companion norm conjecture (3.3) and Lim–Tu–Yu's conjecture (3.4) are equivalent. The pure-moment conjecture (1.2) and the Bessel comparisons in (3.3)–(3.4) remain conjectural. The abstract and explanatory paragraphs are revised accordingly; the full analytic argument and rational witnesses are given in anc/proofs/critical_norm_bridge.md and anc/proofs/critical_norm_certificate.md; DLMF is added as reference [11]; the standard-library replay now runs ten jobs. All numerical output records from revision 3 are retained unchanged (discrepancy bounds 10⁻³⁵⁸ for the pure moment and 10⁻³⁵² for each determinant).