Boolean Primitives as a Constructive Foundation for Fixed-Width Arithmetic

This revised and expanded study investigates whether fixed-width unsigned arithmetic can be constructed from a minimal Boolean substrate rather than assumed as a primitive computational layer. The framework begins with bitwise AND, OR, and NOT together with positional wiring, and develops XOR, multiplexing, carry and borrow propagation, exact addition and subtraction, shift-and-add multiplication, restoring division, and exact perfect-square detection. A central algebraic element is the tripartite support decomposition of two binary operands into their common and mutually exclusive bit regions, providing a unified structural interpretation of several Boolean and arithmetic identities. The theoretical constructions are complemented by exhaustive computational verification over all 65,536 operand pairs in the 8-bit domain, with zero errors for addition, subtraction, multiplication, and the derived tripartite identities; division is verified over all 65,280 valid nonzero-divisor cases. The study also reports negative experimental results where previously conjectured hardware advantages are not supported, clearly separating algebraic validity from architectural performance claims. The resulting framework provides a constructive and reproducible perspective on fixed-width arithmetic based on Boolean primitives and finite bit-level propagation mechanisms.