Unveiling the Structure of Cayley-Dickson Algebras: Zero Divisor Counting and Listing, and a Novel Sign Compression Scheme

This paper presents a comprehensive computational and empirical investigation into the structure and properties of Cayley-Dickson algebras (Aₓ, dimension 2ˣ). We establish and validate an explicit formula, N(x) = (2ˣ - 2)(2ˣ - 4)(2ˣ - 8)/16, which accurately enumerates a specific class of unique zero divisor pairs for x ≥ 4. Complementing this quantitative result, a detailed structural analysis of the multiplication tables reveals a recursive decomposition into 8×8 blocks, reflecting octonion substructures. We introduce a novel ``block type'' classification (`x' or `y') based on an indicator element (e(8k, 8k+1)), which determines zero divisor location and highlights recursive patterns. Conceptual frameworks like an ``Observed Pattern Multiplication Table'' and a sign compression scheme guided this work. A Python script is provided. While OPMT is formally proven and computationally validated, the other many findings are empirical conjectures requiring formal mathematical development, yet they offer a substantially deeper understanding of Cayley-Dickson algebras.