Both halves of Guo–Zudilin's Problem 1: the q-congruence for n ≡ 7 (mod 8) and the closed form for n ≡ 3 (mod 8)

Problem 1(a) of arXiv:1812.11322v2 (Guo-Zudilin, JMAA 475 (2019) 1636-1646), kernel-checked: Phiₙ(q) divides the cleared numerator Nₙ of S(n) = sumₖ₌₀^((n-1)/2) (q;q²)ₖ² (q²;q⁴)ₖ / ((q²;q²)ₖ² (q⁴;q⁴)ₖ) (-q)ᵏ, i.e. S(n) = 0 (mod Phiₙ(q)), at all 14 values n = 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111 (n = 7 mod 8), seven of them composite; deg N₁11 = 12320 with coefficients to 31 decimal digits

The q-congruence Problem 1 of arXiv:1812.11322v2 asks to be FOUND for n = 3 mod 8: S(n) = (q,q²;q⁴)_((3n-1)/8)² (-q;q)_((n-1)/2) q^(-(1-n)²/4) / (q⁴;q⁴)_((3n-1)/4)² (mod Phiₙ(q)) – the source's own Theorem 5 formula with (n-1)/8 -> (3n-1)/8 and (n-1)/4 -> (3n-1)/4, everything else unchanged; kernel-checked as Phiₙ | Eₙ at all 11 values n = 3, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83 (deg E₈3 = 24193) and verified exactly in Z[q] to n = 107

Proof of BOTH halves of Problem 1, for all n: Jackson's q-Clausen identity at a = q⁻³ⁿ, z = -q (which terminates exactly when n = 3 mod 4, where the source's own a = q⁻ⁿ does not) gives sumₖ₌₀^((3n-1)/2) T'ₖ = A' B'; modulo Phiₙ(q) the head is S(n), the middle block (n+1)/2 <= k <= n-1 has Phiₙ-valuation 3 and vanishes, and the tail satisfies T'ₙ₊ᵢ = T'ₙ Tᵢ with regularised value T'ₙ = 1, so the left side is 2 S(n); then B' = 0 by the source's Theorem 3 gives S(n) = 0 for n = 7 mod 8, and A', B' evaluated by the source's Theorems 4 and 3 give the QM2 closed form for n = 3 mod 8 after two elementary Pochhammer reflections

Sharpness, uniform over all four residue classes mod 8: the source's Theorem 2 shows the SAME summand with weight q²ᵏ vanishes modulo Phiₙ(q)² for every n = 3 mod 4, but with weight (-q)ᵏ the Phiₙ-valuation is EXACTLY 1 everywhere – Phiₙ(q)² does not divide Nₙ for n = 7 mod 8 (n = 7..55) nor for n = 5 mod 8 (n = 5..37), nor Eₙ for n = 3 mod 8 (n = 3..35, the QM2 closed form) nor for n = 1 mod 8 (n = 9..33, the source's Theorem 5), nor the Theorem 3 numerator T3(n) for n = 5, 7 mod 8 (n = 5..23)

Engine controls on the import-free Array Int layer: Phi₁, Phi₂, Phi₇, Phi₁2, Phi₁5 against their textbook coefficient vectors; Phi₁05 of degree 48 with exactly two coefficients -2 and every other in 0, +-1 (the classical first cyclotomic polynomial with a coefficient outside 0, +-1); Phiₙ monic on 12 values; deg Nₙ = n² - 1 on 14 values; and qⁿ = 1 (mod Phiₙ) as the e+n shift of the QM2 exponent

Compute-first gate, kernel-checked: the source's Theorem 5 in both published cases (n = 5 mod 8 zero at n = 5, 13, 21, 29, 37, 45; n = 1 mod 8 closed form at n = 9, 17, 25, 33, 41) through exactly the code path rows QM1 and QM2 use, and its Theorem 3 vanishing at n = 5, 7 mod 8 (n = 5..31) together with its failure at n = 1, 3 mod 8

Thirteen refuted claim shapes, false at all 91 (n, claim) pairs checked: Phiₙ(q)² divides none of Nₙ (n = 5, 7 mod 8), Eₙ (n = 1, 3 mod 8) or the Theorem 3 numerator T3(n) (n = 5, 7 mod 8) – 27 pairs; S(n) is not 0 mod Phiₙ(q) for n = 1 mod 8 (n = 9..41) or n = 3 mod 8 (n = 3..43), and T3(n) is not 0 for n = 1, 3 mod 8 (n = 3..27) – 18 pairs, wrong class; and the QM2 closed form with j-1, with j+1, with the source's denominator index (n-1)/4, and with exponent e+1 is false at every one of n = 3, 11, 19, 27, 35 – 20 pairs; plus the 26 non-vacuity pairs of row QC4

Non-vacuity: Phiₙ(q) divides neither the cleared denominator Dₙ = (q²;q²)_((n-1)/2)² (q⁴;q⁴)_((n-1)/2) (14 values of n) nor the closed form's denominator (q⁴;q⁴)_((3n-1)/4)² (12 values), so both are units modulo Phiₙ(q) and the polynomial divisibilities of QM1 and QM2 really are the stated q-congruences