Congruence proofs for two self-consistent continued fractions: OEIS A338632 and A158119

This AI-assisted research note gives formal-power-series proofs of the three congruence assertions listed as conjectures in OEIS A338632 and of the parity assertion in A158119 using its recorded continued-fraction characterization. Normalizing the denominators converts each equation to A(x)=F(x/A(x)^2). Reduction of the weights modulo 4, 3, or 2 gives short algebraic equations and exact coefficient classifications. For A338632, it proves a(n)=2 modulo 4 for n>1 and both modulo-3 classifications involving A191107 and the positive terms of A186776. For the continued-fraction characterization of A158119, it proves that a(n) is odd exactly when n=0 or n is a power of 2. The A158119 argument uses the characterization stated in OEIS; its alternative bisection definitions are not separately rederived. The accompanying Python program uses exact integer arithmetic to compute both sequences through degree 256, checks the listed initial coefficients and all stated residues, compares an independent nested-denominator calculation through degree 32, and runs 24 fixed-seed general-weight trials through degree 20. All checks passed. AI assistance: ChatGPT generated the mathematical arguments, 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, global novelty, and independent human verification have not been established. No claim of peer-reviewed acceptance or proof-assistant verification is made. The source entries still labeled these assertions as conjectures in the copies retrieved on 13 October 2026. Original sequence entries: https://oeis.org/A338632 and https://oeis.org/A158119