Quadratic critical-value conjectures for mixed seven-Bessel moments

We conjecture evaluations of three mixed seven-Bessel moments as quadratic expressions in critical L-values of a weight-three, level-525 newform. A specified quartic Gauss sum enters two phase-sensitive expressions; a separate squared-modulus expression gives the middle moment. The known Bessel conic combines the phase pair into one complex identity. Directed interval calculations bound the absolute discrepancies by 10⁻³⁰⁰ for the first two moments and 10⁻¹⁹⁰ for the third. Exact Fourier reconstruction, error bounds, and reproducible verification programs accompany the note. The Bessel evaluations remain conjectural. Generative-AI disclosure and author responsibility. The conjectures were 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/ (exact Fourier-coefficient reconstruction and analytic error bounds, the conic and phase algebra, exact critical-value and Bessel-moment intervals with the complete discrepancy audit, Python verification code, and SHA256SUMS for package integrity). Running python -B anc/verify.py assembles the three compact formulas from the stored exact intervals and checks the stated discrepancy bounds, interval nesting, the conic/phase algebra, the finite-field coefficient tables and negative normalization controls using only Python's standard library; python -B verify_package.py checks the SHA-256 manifest and archive hygiene; the optional python -B anc/replay.py (NumPy, Numba, SymPy, mpmath, a C++17 compiler and GMP) regenerates every computational input, including the modular-symbol calculations and the directed Bessel source integrations.