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

Abstract: We prove the level-8 Apery-limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence. As a corollary we prove the Ramanujan Machine continued-fraction identity PCF((2n+1)(3n^2+3n+1), -n^6) = 8/(7 zeta(3)). The argument uses the level-8 eta-product parametrization, a Wronskian identity, Eichler integrals, and Mellin-Barnes extraction of the period polynomial. The value (7/32) zeta(3) was identified by Almkvist-van Straten-Zudilin (2008) and stated by Golyshev (arXiv:0908.1458); this paper gives the first complete proof. The companion level-6 result was proved in arXiv:2604.06239.