Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients

We prove that the symmetric-cube coefficients $A_n=(-27{)}^n[z^n]{}_2{F}_1(1\mathrm{/}3,1\mathrm{/}3;1;z{)}^3$ satisfy the supercongruence $A(mp)\equiv A(m)p^4$ for every prime $p\ge 5$ and every $m\ge 1$. The proof rests on three ingredients: (i) the modular identification $F(t(\tau ))=\eta (\tau {)}^9\mathrm{/}\eta (3\tau {)}^3$ with $t(\tau )=\eta (3\tau {)}^{12}\mathrm{/}\eta (\tau {)}^{12}$, whose logarithmic derivative is the weight-5 Eisenstein series $C(q)=3E_5({\chi }_0,{\chi }_3)$ on ${\mathrm{\Gamma }}_0(3)$; (ii) exact congruences $c_{m{p}^r}\equiv c_{m{p}^{r-1}}{p}^{4r}$ for the coefficients of $C$, combined with a Lagrange-Burmann extraction; and (iii) a Hecke descent on weakly holomorphic forms, where the defect is expanded in the two-dimensional space of weight-5 forms on ${\mathrm{\Gamma }}_0(3)$ with character ${\chi }_3$, spanned by $C$ and $tC$, via a cusp-adapted basis, with the second cusp handled by the Fricke involution $W_3$. As an independent result, we show that the Mao-Tian cubic recurrence drops from order 3 to order 2 at the specialization $(1\mathrm{/}3,1\mathrm{/}3,1)$.