On a Quintic Supercongruence Arising from a Calabi-Yau Threefold and Guillera's Series

We report on a remarkable quintic supercongruence associated with a rigid Calabi--Yau threefold and a 2003 Ramanujan-type series for 1/π² due to Jes\'us Guillera. Let $S(p)$ denote the truncated hypergeometric sum at $p-1$ corresponding to Guillera's series involving the factor (-1/1024)ⁿ and the quadratic polynomial P(n) = 1640n² + 278n + 15. Based on exact rational computations and p-adic modular analysis, we conjecture that for all primes $p > 3$ for which the base quadratic field splits, namely (3/p) = 1 (i.e., p ≡ ± 1 12), the sum satisfies\[S(p) ≡ 15 \, p^2 {p^5},\]where the scalar $15 = P(0)$ is identically the constant term of the polynomial. When (3/p) = -1, this symmetry breaks down, though $S(p)$ remains divisible by p². We provide exact computational evidence for all primes $p < 100$ and discuss potential approaches to a proof via the Wilf--Zeilberger (WZ) method and p-adic hypergeometric cycles.