The level-8 Apéry-limit and a proof of the Ramanujan Machine conjecture Z1

We prove lim B_n⁽⁸⁾/s_n = (7/32)ζ(3) for the level-8 Apéry-like sequence s_n = Σ C(n,k)²C(2k,n)² and its rational companion B_n⁽⁸⁾, and deduce the Ramanujan Machine conjecture Z1: PCF((2n+1)(3n²+3n+1), −n⁶) = 8/(7ζ(3)). The value was identified by Almkvist–van Straten–Zudilin (2008) and stated by Golyshev (arXiv:0908.1458); this is the first complete proof. Method: eta-product parametrization, Eichler integral, Mellin–Barnes period polynomial, singularity transfer. The companion result Z2 was proved in arXiv:2604.06239. https://arxiv.org/abs/2604.14219