Parity and Asymptotic Laws for Implicit and Reflected Integer Sequences

Ten integer sequences defined by implicit ordinary, reflected, or exponential generating-function equations exhibit parity, modular, and asymptotic laws. We prove sparse parity formulas using integral coefficient projections, Frobenius identities, and the classical Catalan equation. A labelled-partition argument proves simultaneous congruences for two exponential families, while a binary-language correspondence supplies an exact Fibonacci enumeration. For the reflected sequence OEIS A368633, we certify the specific six-shift recurrence proposed by R. J. Mathar using an exact differential operator in an algebraic quotient ring. For OEIS A397591, we establish the previously recorded Cayley-scale asymptotic with its explicit constant by a uniform logarithmic majorant and a summable renewal contraction. We distinguish the earlier sequence definitions, original conjectures, and previously recorded asymptotic expression from the proofs presented here. Separate reproducible exact-arithmetic certificates support finite proof bases and worked examples, and do not replace the all-index arguments.

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