Order drop, Hecke descent, and an unconditional mod p⁴ supercongruence for Sym³ hypergeometric coefficients

We prove a supercongruence modulo p to the fourth for the coefficients of the symmetric cube of the hypergeometric function at parameters one third, one third, one. The proof combines a modular identification on X_0(3), an Eisenstein-type congruence for the coefficients of the Hauptmodul derivative, Lagrange–Bürmann coefficient extraction, and a Fricke–Hecke descent argument that yields uniform vanishing of all defects.