A Fractional p-Adic Perturbation Bridge for a Supercongruence of Long

Abstract

This preprint proves the modular hypergeometric supercongruence stated by Ling Long as Conjecture 5, equation (14). It establishes the conjecture for every prime at least seven by constructing a direct congruence bridge from Long’s nonintegral hypergeometric datum to a standard Rodriguez–Villegas datum. The proof uses a sharp valuation cutoff arising from denominator resonance, followed by an exact reversal of the truncated sum. The resulting terminating hypergeometric expression is treated through an algebraic extension of the Long–Tu–Yui–Zudilin perturbation identity to the required half-integral parameter, without appealing to analytic continuation from integer parameters. An exact evaluation of the associated Morita p-adic gamma factor then yields the bridge. The known Rodriguez–Villegas supercongruence and the quadratic-twist relation between the relevant weight-four modular forms complete the proof, with the prime seven handled separately. The argument is theoretical and does not depend on finite computation. The accompanying source archive contains the LaTeX source, exact and modular verification scripts, machine-readable outputs, licensing information, and checksum manifests. Independent computations verify the central bridge for more than one thousand primes, together with the reversal identity, reflection signs, and exceptional-prime calculation. The paper addresses the first case in Long’s six-case list. It does not claim the remaining five cases, the full Dwork unit-root tower, or a general fractional perturbation theorem for arbitrary nonintegral hypergeometric data.