Quadratic Derivations and Uniform Modular Periods for Six Inverse-Function Sequences

We study the integral derivation $D(s)=c^a$, $D(c)=sc^{a-1}$ on the quadratic algebra $\mathbb Z[s,c]/(c^2-s^2-1)$, with even-derivative columns of inverse-function flows as integer-sequence observables. We prove a uniform eventual-period theorem modulo every positive integer for all even exponents, including a separate vanishing-prime analysis. The same framework gives full-index consequences for five OEIS inverse-function sequences. For the odd exponent $a=5$ we establish prime-power eventual vanishing away from $2$ and $3$, a two-adic period lifting and a three-adic period lifting, settling both precise source conjectures for A281428. The arguments use integral differential operators, finite free rings, restricted characteristic-$p$ derivations, and the Chinese remainder theorem. We distinguish the original source definitions and recurrences from the congruence theorems established here.

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