Ordered Carry Frontiers for Parallel Addition

For m fixed offsets added to the same radix-B word, the joint carry vector after i digits has exactly one plus the number of distinct nonzero offset residues modulo B^i possible values. The states form a chain and attain the sharp bound min(m+1,B^i). At nonterminal cuts these states are exactly the residual classes needed to preserve every future output digit. In the binary case, stable partition and monotone sweeps construct an exact counter for arbitrary input and output bit masks in O(wm) operations on wide integers. The algorithm also returns the least satisfying input. The paper gives proofs, exact source, independent whole-input comparisons, and a source-derived Blender figure.