An O(1) Bitwise Evaluator for Cayley--Dickson Sign Structure: Ordinary, Split, Dual, and Tensor Constructions

We prove that the evaluator computes exactly the twist function</p><p dir="ltr">of~\cite{RenZhao2023}, and that our split variant implements their</p><p dir="ltr">Theorem~2. The structural-break descent, the constant-time collapse, and the</p><p dir="ltr">equivalence theorem are new; the sign function itself is due</p><p dir="ltr">to~\cite{RenZhao2023}, building on Albuquerque--Majid~\cite{AlbuquerqueMajid1999}</p><p dir="ltr">and the Cayley--Dickson process of Schafer~\cite{Schafer1954}.</p><p dir="ltr">We extend the framework to the dual family</p><p dir="ltr">$A_n[\varepsilon]/(\varepsilon^2)$, proving</p><p dir="ltr">proving proving $A_n[\varepsilon]/(\varepsilon^2) \cong A_n \otimes_{\mathbb{R}} \mathbb{R}[\varepsilon]/(\varepsilon^2)$, and to tensor products via a sign</p><p dir="ltr">composition principle.

We further present an empirically validated counting</p><p dir="ltr">formula for a class of zero-divisor pairs, stated explicitly as a conjecture.</p><p dir="ltr">Three implementations are provided and cross-verified: a full table builder,</p><p dir="ltr">and $O(n)$ and $O(1)$ single-product evaluators.</p>