Shifted Gauss Congruences and Euler–Witt Integrality

We study ordinary generating functions obtained by exponentiating logarithmic transforms of the positive Euler (secant) numbers and integer-parameter Euler-polynomial sequences. Shifted prime-power Gauss congruences, together with the Möbius–Witt criterion, yield integrality for secant-number shifts and for the Euler-polynomial cosine quotients. A uniform integer-frequency quotient lemma supplies a self-contained proof for even parameters; for odd parameters we exhibit a direct reduction to Kallat's earlier shifted-congruence theorem. We give exact counterexamples to unrestricted positivity and to a broader integral-coefficient assertion, and prove modular assertions for two additional OEIS families. Explicit arithmetic examples and reproducible calculations with exact large-integer certificates are supplied for eight OEIS entries. Finite calculations serve only as checks on the analytic arguments. 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.