Two continued fractions of Elimelech et al. (PNAS 2024) in closed form, with proofs

Elimelech et al. (PNAS 121 (2024) e2321440121, arXiv:2308.11829) list three degree-5 polynomial continued fractions with b(n) = -64 n^10 whose limits they give only numerically, noting that these matched no combination of zeta values in their PSLQ tests (rows 4, 5 and 6 of Table 1 in the arXiv version, Table 2 in PNAS). Theorem 1 (version 3): the row 4 limit equals 96 / (372 zeta(5) - pi^5). The proof shows that moments of products of complete elliptic integrals, int_0^1 (4x(1-x))^n (2x-1) K(x)^3 (K(x) - 2K(1-x)) dx, satisfy the recurrence of the continued fraction and form its minimal solution, and uses published evaluations of Zhou and of Wan and Zucker. Theorem 2 (version 4): the row 5 limit equals 8 L(f,1) / (186 zeta(5) L(f,1) - pi^4 L(f,2)) = 2.4517460646287829170478801291688130600833..., where f = eta(2t)^4 eta(4t)^4 is the newform of weight 4 and level 8 (LMFDB 8.4.a.a); equivalently 1/K = 24 lambda(5) - (pi^4/8) L(f,2)/L(f,1). The recurrence is the Mellin dual of the exterior square of the Calabi-Yau operator theta^4 - t(theta + 1/2)^4; the minimal solution is the moment sequence of a bracket of two of its periods (Euler and Abel transforms of products of elliptic integrals) that vanishes to order 2 at t = 1, and its first two moments give the limit. The proof uses published evaluations of Wan and Zucker and of Rogers, Wan and Zucker, and gives the error bound 0 < p_(n-1)/q_(n-1) - K <= 576/(n(n+17)). Both proofs check every finite identity by exact computer algebra (sympy, PARI/GP) with one command each, including negative controls, and each was reviewed by three separate adversarial AI sessions; no human has checked them. Row 6 remains open; an exact identity between the convergents of rows 5 and 6 reduces it to one further growth constant. The archive also contains the three limits to 1300 digits, standard-library verifiers and the list of sources checked as of 2026-10-27, in which we found no earlier closed form or proof for these rows. Searches and verification were carried out with Claude (Anthropic) under the direction of Kerem Özdemir. The proofs were also written with Claude.