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

This paper presents a comprehensive investigation into the structure and properties of the standard Cayley-Dickson algebras $A_x$ (dimension $2^x$). Our research, driven by computational exploration and pattern recognition, yields both significant quantitative and qualitative results. We establish an explicit formula, $N(x) = \frac{(2^x - 2)(2^x - 4)(2^x - 8)}{16}$, which accurately counts the number of unique zero divisor pairs specifically of the form $(e_i + e_u) * (e_j \pm e_v) = 0$ for $x \geq 4$, as identified and computationally validated by our exploratory methodology. This formula provides a precise quantitative measure for this structurally significant class of zero divisors. Independently, we unveil a detailed structural analysis of the Cayley-Dickson multiplication tables, demonstrating a recursive decomposition into 8x8 blocks (reflecting octonion substructures). We introduce a novel "block type" classification ('x' or 'y') based on the sign of a single indicator element $e_{(8k, 8k+1)}$, which dictates the location (Upper/Lower Triangular Matrix relative to the anti-diagonal) of associated zero divisors within those blocks. This analysis reveals hidden recursive patterns and self-similarity across dimensions. Furthermore, we discuss the conceptual frameworks of an "Observed Pattern Multiplication Table" (OPMT) construction and a sign compression scheme, which served as valuable tools during our investigation. A Python script implementing the standard construction, featuring an exploratory zero divisor finder with computational validation and deduplication, is provided as supplementary material. Our combined findings offer a substantially deeper understanding of Cayley-Dickson algebras and suggest potential applications in related fields. This paper is based on empirical results and lacks most of the time rigorous proofs, so what isn't proven is a conjecture.