The split-prime p^(4r) supercongruence tower for 64^n [z^n] 2F1(1/4,1/4;1;z)^3: a proof of the conjecture in OEIS A397369

Let a(n) = 64^n [z^n] 2F1(1/4,1/4;1;z)^3 (OEIS A397369: 1, 12, 348, 13264, 577500, ...). In 2026 A. Shvets conjectured in the entry that a(m p^r) ≡ a(m p^(r−1)) (mod p^(4r)) for every prime p ≡ 1 (mod 4) and all m, r ≥ 1, by analogy with his proof of the full p^(4r) tower for the level-three sequence 27^n [z^n] 2F1(1/3,1/3;1;z)^3 (arXiv:2606.15462). We prove this conjecture, in the stronger form of a congruence between whole q-series. We also prove the companion statement of the entry: 5 a(p) ≡ 188 (mod p) for every prime p ≡ 3 (mod 4), so that at those primes the congruence fails already modulo p. The proof realizes Σ a(n) x^n = θ^6 on Γ0(4), where θ is the Jacobi theta function and x = η(τ)^24 η(4τ)^24 / η(2τ)^48 is a Hauptmodul for X0(4)^+, and works with the weight-5 forms f_N = C/x^N, where C = θ^6 Dx/x is an Eisenstein series; then a(m) is the constant term of f_m. As in the level-three case, the sparse Cartier defects U^(s+1) f_(mp^(s+1)) − U^s f_(mp^s) are compared with their prime-power Hecke lifts T_(p^(s+1)) f_(mp^(s+1)) − T_(p^s) f_(mp^s) by induction on s. The new difficulty is that x is invariant under the Fricke involution W4, so the forms f_N have poles at both cusps ∞ and 0 and the Fricke step of the level-three proof has no analogue. It is replaced by the Fricke-fixed cusp 1/2, where x has a double pole: the Hecke operators T_(p^a) divide orders at 1/2 by at most p^a, and on the W4-eigenspace of C these orders lie in 3/2 + 2Z, which forces the low coefficients to vanish exactly. The eigenspace is preserved by T_p exactly when χ−4(p) = 1; this is where the hypothesis p ≡ 1 (mod 4) enters, and it explains the failure for p ≡ 3 (mod 4). The note was obtained with the author's methodology for open statements in the OEIS (subprocesses with time budgets, a log of obstacles with an algorithm added per obstacle, scripts that test the variants of a formula, literature search, compiled code for the blocking computations, independent verification and review by a second AI system), including the soak test (the numerical check is extended in range rather than repeated, it is run against negative controls that must fail, and the heavy part is rewritten from scratch in compiled code). The work was done with the assistance of an AI system under the author's direction; the methodology section of the note describes the process. An independent review by a second AI system, with its own code, checked the theorem in all 97894 cases with m p^r ≤ 100000 and concluded "proved, with minor fixes"; the fixes are incorporated. The verification programs are included. The Python scripts chk1.py to chk5.py (with base.py; they use python-flint for exact q-series and mpmath for high-precision numerics) check the modular realization, the Fricke eigenvalues, the expansions at the cusp 1/2, the Hecke operators on the forms C/x^N, the induction step and the series congruences, together with the controls for p ≡ 3 (mod 4). The independent Rust program (super4check.zip) computes a(n) exactly for n ≤ 12000 and, independently and without the recurrence, modulo p^K for all odd primes p ≤ 2500. It checks the theorem in all 10452 cases with m p^r ≤ 12000, the statement for p ≡ 3 (mod 4), and negative controls (the exponent 4r + 1 fails in 7060 cases, and the congruence fails for p ≡ 3 (mod 4)). Both PDFs (English and Spanish) are the same note.