A formal-power-series proof of the integrality conjecture in OEIS A193550

This AI-assisted research note gives a formal-power-series argument for the integrality assertion recorded as a conjecture in OEIS A193550. For the Stieltjes continued fraction A(x) with weights 1, 1, 9, 9, 25, 25, ..., it constructs the square root B(x)=sqrt((A(x)-1)/x) in Z[[x]] and proves the stronger coefficientwise congruence B(x)=C(x) modulo 4, where C is the Catalan series. The argument reduces the weights modulo 8 and constructs a square root using a division-free integer recurrence. The accompanying Python program uses exact arithmetic to check the square-root identity through degree 600, with a separate continued-fraction computation through degree 40 and a check of the constructive proof through degree 100. AI assistance: ChatGPT generated the mathematical argument, English exposition, and verification program at Peggy Pei-Yu Chao's request. Peggy Pei-Yu Chao initiated the investigation and requested public dissemination. This is an unreviewed research note. Publication priority and independent human verification are not established. No claim of peer-reviewed acceptance, formal proof-assistant verification, or worldwide novelty is made. Original sequence: https://oeis.org/A193550