A full $p^{4r}$ supercongruence tower for a level-three symmetric-cube hypergeometric sequence

Let ${}_2{F}_1(1\mathrm{/}3,1\mathrm{/}3;1;27z{)}^3={\sum }_{n\ge 0}{A}_n{z}^n$. We prove the full prime-power supercongruence tower $A_{m{p}^r}\equiv A_{m{p}^{r-1}}(\mathrm{m}\mathrm{o}\mathrm{d}{p}^{4r})$ for $p\ge 5$, $m\ge 1$, $r\ge 1$. The level-three modular expansion used below was already recorded by Moy; the contribution here is the depth-preserving modulus $p^{4r}$, extending the previously known depth-one modulus $p^4$ case to all prime powers. The proof is modular. After replacing $A_n$ by $B_n=(-1{)}^n{A}_n$, the generating function is realized on $X_0(3)$ by ${\sum }_{n\ge 0}{B}_n{t}^n=\eta (\tau {)}^9\mathrm{/}\eta (3\tau {)}^3$ with $t=\eta (3\tau {)}^{12}\mathrm{/}\eta (\tau {)}^{12}$. The logarithmic derivative $C=({\sum }_{n\ge 0}{B}_n{t}^n)(q\mathrm{/}t)(dt\mathrm{/}dq)$ is the Eisenstein series $3E_{5,{\chi }_0,{\chi }_3}$. Lagrange-Buermann gives $B_m={\mathrm{C}\mathrm{T}}_q(C(q)\mathrm{/}t(q{)}^m)$. The new point is to replace the one-prime Hecke defect by the prime-power defect $T_{p^{s+1}}(C\mathrm{/}{t}^{m{p}^{s+1}})-T_{p^s}(C\mathrm{/}{t}^{m{p}^s})$. Its $i=0$ Hecke layer is the sparse Cartier defect controlling $A_{m{p}^{s+1}}-A_{m{p}^s}$, while all remaining Hecke layers are divisible by the required power of $p$ by induction. A Fricke involution argument on the two-dimensional space $M_5({\mathrm{\Gamma }}_0(3),{\chi }_3)=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{C,tC\}$ then kills the low-order part exactly and the principal part modulo $p^{4(s+1)}$.