Proof of the modulo-six and parity conjectures in OEIS A378258, Comments

We prove both conjectures in Paul D. Hanna's OEIS A378258, Comments (3 December 2024): its coefficients avoid residues 1 and 5 modulo 6 beyond degree one, and an odd coefficient occurs exactly at a power-of-two index. The normalized rational solution exists, is unique and has integer coefficients. An integral compositional square root gives parity; reflection modulo 3 gives the sharper even-index residues 3 at powers of two and 0 otherwise. The same construction treats every integer parameter q congruent to 1 modulo 4 replacing 5, and gives companion-series parity and exact dyadic valuations for compositional squares.