Exact Counts of Sum-and-Carry Functions for Cascaded Addition

For radix B and k successive additions with independently variable incoming carry bits, this paper counts the complete functions retaining the final residue and every outgoing carry. With A_0=1, A_j=j A_(j-1)+1 and t=min(k,B-1), the exact count is B^(k-t)((B-1)A_t+1). A residue-support classification gives a constructive dense encoder and decoder and the exact fixed-length bit minimum. For three 32-bit additions, 36 bits preserve all eight incoming-carry cases. The paper includes proofs, source, complete finite comparisons, and source-derived figure geometry.