Sym³(Gauss) produces Apéry-like sequences with Dwork supercongruences beyond mod p³

Integer sequences from Sym³(Gauss) with Dwork supercongruences beyond mod p³. The sequence A_n = 27^n [z^n] ₂F₁(1/3,1/3;1;z)³ satisfies an order-2 recurrence and A(mp) ≡ A(m) (mod p⁴) for all tested primes p ≤ 23 and mp ≤ 499; classical Sym² benchmarks A036917 and A002895 reach only mod p³. A second sequence from (1/6,1/3;1) exhibits CM splitting: mod p⁴ iff p ≡ 1 (mod 6). Archive: proofs (PDF), computational report (PDF), Python script (exact integer arithmetic), sequence data (JSON).