Proof pieces for a Mellin-Barnes closed form of a pass sum

A pass sum from Kim and Thorne (1991), written in a closed form through a Mellin-Barnes representation, with written proofs of the pieces an earlier draft left open, and the integer values where the logarithmic terms vanish: a chain given by the Pell-type equation y^2 - 3b^2 = -2. Checked by exact arithmetic and high-precision numerics; not yet read by a mathematician. The proofs and the checking programs were drafted with an AI instrument (Anthropic's Claude) at the author's direction and checked in AI verification passes; no mathematician has reviewed them yet.