Dwork-type supercongruences for the truncated series 2F1(r/d, 1−r/d; 1; 1) via a multiplicative q-congruence

For integers d ≥ 2 and 1 ≤ r ≤ d−1 let S_N(r) = Σ_{k<N} (r/d)_k (1−r/d)_k / k!² be the truncation of the hypergeometric series 2F1(r/d, 1−r/d; 1; 1) after N terms. For r/d = 1/2, 1/3, 1/4, 1/6 these are the truncated sums of C(2k,k)²/16^k, C(2k,k)C(3k,k)/27^k, C(2k,k)C(4k,2k)/64^k and C(3k,k)C(6k,3k)/432^k, whose supercongruences were observed by Rodriguez-Villegas and studied by Mortenson, Z.-H. Sun, J.-C. Liu, Guo, Zudilin and others. We prove the Dwork-type supercongruences S_{mp^s}(r) ≡ (−1)^⟨−r/d⟩_p · S_{mp^{s−1}}(r_p) (mod p^{2s}) for every odd prime p not dividing d and all integers m, s ≥ 1, where r_p/d is the Dwork prime of r/d, that is, r_p ≡ r/p (mod d). Equivalently, S_{pn}(r) − (−1)^⟨−r/d⟩_p · S_n(r_p) is divisible by p^{2+2v_p(n)} for every n ≥ 1, and the exponent is best possible. For the four values of Rodriguez-Villegas the two sums coincide, and we obtain the towers S_{mp^s} ≡ χ(p) S_{mp^{s−1}} (mod p^{2s}) with χ(p) = (−1/p), (−3/p), (−2/p), (−1/p) respectively. This extends results of Mortenson and Z.-H. Sun (m = s = 1), J.-C. Liu (s = 1, modulo p²) and H.-X. Ni (m = 1, p ≡ ±1 mod d). It proves the integrality part of a conjecture of Z.-W. Sun (Conjecture 5.4 in arXiv:1610.03384) for every n for which the binomial factor in Sun's normalization is prime to p, and it explains Liu's observation that his congruence does not extend to arbitrary parameters: the companion of S_{pn}(r) is S_n(r_p), with the Dwork prime. The proof goes through a q-analogue. For K_N(r;q) = Σ_{k<N} q^{dk} (q^r;q^d)_k (q^{d−r};q^d)_k / (q^d;q^d)_k² we prove the multiplicative congruence K_{mn}(r;q) ≡ K_n(r;q) · K_m(r_n; q^{n²}) (mod Φ_n(q)²) for every m ≥ 1 and every n ≥ 2 prime to d, where r_n ≡ r/n (mod d). Together with a theorem of Guo, Pan and Zhang (the case m = 1) this gives K_{mn}(r;q) ≡ (−1)^⟨−r/d⟩_n · q^{(n² r_n (d−r_n) − r(d−r))/(2d)} · K_m(r_n; q^{n²}) modulo Φ_n(q)² for all odd n prime to d, and a residue formula at the p-power roots of unity turns these q-congruences into the towers. The key step is an evaluation at roots of unity: a one-parameter deformation f(c) of the truncated sum satisfies a first-order functional equation under c ↦ cζ^d, which forces f(c) to be c^{n−1} times an explicit product. We also show that the theorem of Guo, Pan and Zhang needs a sign correction for even n (an explicit counterexample is n = 2, d = 5, r = 1), locate the source of the discrepancy, and give the corrected sign, checked numerically; only odd n are used in the proofs. Files: the paper (PDF, English), a Spanish translation (PDF), and the verification script verify_dwork2f1.py (Python 3) with its output. The script checks the q-congruences exactly modulo large primes at all primitive n-th roots of unity (1,467 cases of the multiplicative congruence and 963 of the q-towers), the intermediate lemmas (3,596 cases), the classical towers with exact rational arithmetic (4,141 cases, together with the sharpness of the exponent), and the exact cyclotomic valuations. No check fails. Software and AI: the results were obtained with SyntheticMind, a mathematical reasoning engine for conjectures and congruences created, designed and programmed by the author, with the assistance of the general-purpose AI system Claude (Anthropic). The author has checked the arguments and takes full responsibility for the content.