Six Hypergeometric Supercongruences of Long: Proofs and a Correction

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Chinese explanation / 中文解读

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Original abstract

This paper resolves the six hypergeometric supercongruence cases proposed by Ling Long. Five of the six congruences are proved in their originally stated form for every prime at least seven. The remaining case is shown to be false as printed, and a corrected congruence is established. The discrepancy is structural rather than numerical: the modular form appearing in the original statement is a quadratic twist of the form required by the hypergeometric data. For the first three cases, the proof develops a third-order comparison between truncated classical hypergeometric sums and finite p-adic hypergeometric Frobenius traces. The key step is to retain the full contributions of the double and triple poles in the relevant rational functions, producing weighted companion identities that account for the correction terms required at cubic p-adic precision. An exact duality and half-period transformation further relates two of the finite hypergeometric traces. For the remaining three cases, a perturbation argument for rigid Calabi–Yau hypergeometric data is extended from integral perturbations to arbitrary p-adic perturbation parameters. A quadratic interpolation argument then permits the required central specialization. The resulting classification is complete: Long's cases (11) through (15) hold as stated, while case (16) becomes valid after replacing the originally printed modular form by the correct quadratic-twist representative. The paper also identifies the precise twist obstruction explaining when the original version of case (16) can nevertheless hold at individual primes. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.

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