A Determination and Proof Scheme for the Unique Decomposition of the Main Consecutive-Block Object in Conjectures Related to Ulam Words

This paper studies the unique decomposition problem for the main consecutive-block object in conjectures related to Ulam words, namely words of the form 0^a 12^k 0^b with a,b,k ≥ 1. The argument is organized into four connected steps: exhaustive classification of candidate cuts, uniform reduction of internal cuts to one-sided words, combinatorial counting of internally effective parameters, and a final total-value reading argument. Within the scope of the consecutive-block main object treated here, the paper proves that the object belongs to the set of Ulam words if and only if the total length a+b of the two zero-blocks satisfies the corresponding congruence condition modulo 2^k. The main contribution of the paper is to rewrite the original local cut-by-cut problem as a structured proof chain involving cut classification, one-sided reduction, combinatorial counting, and global uniqueness reading.