Exact State Complexity of Two-Stage Addition

Three fixed radix-B operands define a function of two incoming carry bits through two successive modular additions. We classify the complete response when the residue and both outgoing carries are retained. For every integer B at least two, exactly 5B-4 response functions occur. An explicit normal form, a contiguous state code and constructive representatives prove the count. We determine the multiplicity of every state and the exact distribution of carry-image sizes. The minimum fixed-length description is ceil(log2(5B-4)) bits. Thus three 32-bit operands admit a complete 35-bit response code. Independently recoverable digit stages have a separate product count; four byte stages require exactly 42 bits. The paper includes proofs and executable whole-function verification.