Gasper's Jacobi-sum conjecture: a proposed proof

We present a proposed analytic-and-algebraic proof of the equivalence between the second-zero Bessel criterion and nonnegativity of all normalized Jacobi partial sums in Gasper's parameter region. The argument uses the published Bessel-boundary geometry of Castillo and Yakubovich. The classical Askey–Steinig–Makai criterion and the required specialization of Bateman's transfer formula are proved in the text. Degrees zero through fifteen are treated by prescribed polynomial squares and explicit rational sign inequalities. For larger degrees, a common integration recurrence, weighted energies, and evaluated source-remainder bounds cover the full spatial interval. The exact sign appendix and coefficient-norm tables form part of the argument. The manuscript is submitted as a proposed proof for scrutiny; independent specialist validation is not asserted. Generative-AI disclosure and author responsibility. All mathematical exploration, proof development, symbolic and numerical calculations, verification-code generation and execution, and manuscript drafting and revision undertaken for this work were carried out by OpenAI's ChatGPT (GPT-6 Pro) and its tools in response to prompts from Jonas Matuzas. The author supplied the prompts and directed the workflow; he has not independently verified the mathematics. Jonas Matuzas takes full responsibility for the mathematical claims, attributions, and final text. Previously published results remain credited to their original authors. Files: the compiled manuscript Gasper_Manuscript_v1.pdf (42 pages) and the publication package Gasper_Publication_Package_v1.zip. The package contains the manuscript; the companion Gasper_Exact_Sign_Appendix_v1.pdf (41 pages) with the complete finite sign factorizations used in Section 4, which is part of the mathematical argument; the self-contained LaTeX sources; SUPPLEMENT_MAP.md (locations and meanings of the mathematical inputs); SOURCE_NOTES.md (bibliographic locations and the limits of source inspection); MANIFEST.sha256 (integrity hashes for all other files); and verification/ with exact Python/SymPy checking programs, rational input data, and the results of the version-1 reconstruction. Running python verification/run_checks.py (Python 3.11 or later; pinned dependencies sympy 1.14.0 and mpmath 1.3.0 in verification/requirements.txt) executes 18 exact steps in a fresh temporary copy and writes verification/results/verification_report.json. A successful run reconstructs the specified exact identities and inequalities; it is not a proof-assistant formalization or an independent verification of every analytical argument and external theorem.