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

This paper presents a comprehensive investigation into the structure and properties of Cayley-Dickson algebras, focusing on both quantitative and qualitative aspects. Our central result is an explicit formula for counting zero divisor pairs in the standard Cayley-Dickson algebras of dimension $2^x$, for $x \geq 4$. This formula, expressed as a summation, provides a precise quantitative measure of the emergence of zero divisors in these non-associative algebras. Beyond this quantitative result, we unveil a detailed structural analysis of the Cayley-Dickson multiplication tables, demonstrating a decomposition into recursive 8x8 blocks and a novel "block type" classification based on an indicator element and UTM/LTM properties relative to the antidiagonal. This analysis reveals a hidden order within these complex algebraic structures and provides a systematic method for locating zero divisor pairs. Furthermore, we explore \textit{alternative Cayley-Dickson constructions}, motivated by computational efficiency, and introduce a novel sign compression scheme that captures the recursive sign patterns within the multiplication tables. This scheme highlights the inherent self-similarity of these algebras when grouped by $m \times m$ blocks to dimension $l=n-m$. Our work provides both a deeper theoretical understanding of Cayley-Dickson algebras and potentially more efficient computational methods for working with them. The zero-divisor counting formula is empirically validated, and the structural insights and alternative constructions suggest potential applications in fields leveraging hypercomplex numbers, including theoretical physics, cryptography, and computer science.