data Proof in proof_case_6_4.md. Exact two-way reduction (sympy-verified identities): (x+1)⋯(x+6)=t(t+4)(t+6) with t=x^2+7x+6, and (y+1)⋯(y+4)+1=u^2 with u=y^2+5y+5, giving u^2=t^3+10t^2+24t+1 with side conditions 4t+25 and 4u+5 perfect squares. The curve is LMFDB 10388.b1 (shift t=X-3, sympy-verified; conductor 2^2·7^2·53, rank 2, trivial torsion). Integral-point completeness proved by SageMath 10.7 (case64_sage.sage): gens(proof=True) = [(-4,1),(-6,1)], certified rank bounds (2,2), integral_points (elliptic-logarithm method, provably complete given a full Mordell-Weil basis) = 14 points, matching the LMFDB list and an independent direct search over -10<=t<=10^7; direct search of the original equation to x<=2·10^6 also finds only (1,6). Filter table in the proof shows only t=14 has 4t+25 square with x>=1. Referee corroboration: agrees with the prior published resolution by Hajdu-Pintér 2000 (see claim 3), and with an independent referee Sage run on the minimal model 10388.b1.