A two-step supercongruence for an Apéry-like sequence

Let $G_n={\sum }_{k=0}^n{4}^k{\left(\genfrac{}{}{0px}{}{2n-2k}{n-k}\right)}^2\left(\genfrac{}{}{0px}{}{2k}{k}\right)$. Zhi-Hong Sun conjectured that, for primes $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, positive odd integers $m$, and $r\ge 2$, the two-step congruence $G_{(m{p}^r-1)\mathrm{/}2}\equiv p^2{G}_{(m{p}^{r-2}-1)\mathrm{/}2}(\mathrm{m}\mathrm{o}\mathrm{d}{p}^{2r-1})$ holds. We prove the stronger valuation statement $G_{(p^{2}M-1)\mathrm{/}2}-p^2{G}_{(M-1)\mathrm{/}2}\in p^{2v_p(M)+3}{\mathbb{Z}}_p$ for every positive odd $M$. The proof converts the sum to a terminating ${}_3{F}_2$, constructs a cancelled digit-transfer operator, and identifies a two-dimensional analytic quotient of its cubic difference operator. The quotient operator has characteristic polynomial $X^2-p^2$; its second trace is evaluated through the Gross–Koblitz formula and Greene’s finite-field Dixon identity. A logarithmic-loss Green inverse converts this spectral identity into an actual integral analytic primitive. The exceptional prime $3$ requires a finite exact PARI/GP certificate, while the infinite tail is bounded symbolically. Thus the computation is finite, reproducible, and separated from the uniform part of the proof.