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 $1/\pi^2$ due to Jes\'us Guillera. Let $S(p)$ denote the truncated hypergeometric sum at $p-1$ corresponding to Guillera's series involving the factor $\left(-\frac{1}{1024}\right)^n$ and the quadratic polynomial $P(n) = 1640n^2 + 278n + 15$. 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 $\left(\frac{3}{p}\right) = 1$ (i.e., $p \equiv \pm 1 \pmod{12}$), the sum satisfies\[S(p) \equiv 15 \, p^2 \pmod{p^5},\]where the scalar $15 = P(0)$ is identically the constant term of the polynomial. When $\left(\frac{3}{p}\right) = -1$, this symmetry breaks down, though $S(p)$ remains divisible by $p^2$. 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.