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

Posted: August 23, 2026
Server: Zenodo
DOI: 10.5281/zenodo.22051873

Abstract

This paper presents a comprehensive computational and empirical investigation into the structure and properties of Cayley-Dickson algebras ( $A_x$ , dimension $2^x$ ). We establish and validate an explicit formula, $N(x) = \frac{(2^x - 2)(2^x - 4)(2^x - 8)}{16}$ , which accurately enumerates a specific class of unique zero divisor pairs for $x \geq 4$ .

Complementing this quantitative result, a detailed structural analysis of the multiplication tables reveals a recursive decomposition into $8\times8$ 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.

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