Binomial-Sum Congruences, Parameter Rigidity, and Exponential Integrality

We study three related mechanisms for binomial sums sharing an Apéry-type kernel: a prime-window cancellation theorem modulo the third power of a prime, a complete parameter-equivalence classification modulo four, and Gauss congruences governing the integrality of two exponential generating products. A polynomial interpolation argument gives simultaneous divisibility statements for two shifted binomial sums. Binary carries yield a necessary and sufficient condition for two members of a Zudilin–Apéry deformation to agree modulo four at every index. A unit-factorial argument proves both ordinary and shifted Gauss congruences at every prime and hence integrality of associated exponential series. We distinguish these results from the previously established cubic congruences of Coster. 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.