Integrality and Modular Congruences for Self-Referential Exponential Series

Two classes of self-referential exponential power series, originally specified by Paul D. Hanna in the On-Line Encyclopedia of Integer Sequences (OEIS), are investigated by integral bivariate series and coefficientwise congruences. A multivariate Dieudonné--Dwork argument shows that a family whose defining exponent involves $q^n+rA(x)$ has integer coefficients for every pair of integers $q,r\in\mathbb{Z}$. In particular, this resolves, at the level of formal power series, the stated all-integer-parameter integrality conjecture associated with OEIS A163138. We also prove a uniform congruence modulo $4$ between positive and negative feedback, a signed-source congruence, explicit dyadic congruence formulas modulo $4$, and a congruence theorem modulo $8$ for the even coefficients when $q\equiv 2\pmod{4}$ and $r$ is odd. Eight integer sequences are treated as specializations. Exact calculations and separately reproducible examples illustrate, but do not replace, the all-degree proofs. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.