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

The sign of the Cayley--Dickson basis product $e_i \cdot e_j = \pm e_{i \oplus j}$coincides with the explicit twist function $\sigma(A,B)$ of Ren and Zhao(Theorem~1 for the standard algebras; Theorem~2 for the split relation$\sigma_s = \sigma + a_{n-1} b_{n-1}$)~\cite{RenZhao2023}.This paper gives an independent treatment of that sign law from thecomputational side. We prove the block sign rules (the ``OPMT sign law'')directly from the Cayley--Dickson doubling formula, introduce a holographic$O(n)$ descent algorithm whose correctness we prove against these rules, andcollapse the descent into a strictly table-free $O(1)$ word-RAM evaluatorusing three trailing-zero counts, one maximum comparison, and one populationcount. We prove that the evaluator computes exactly the twist functionof~\cite{RenZhao2023}, and that our split variant implements theirTheorem~2. The structural-break descent, the constant-time collapse, and theequivalence theorem are new; the sign function itself is dueto~\cite{RenZhao2023}, building on Albuquerque--Majid~\cite{AlbuquerqueMajid1999}and the Cayley--Dickson process of Schafer~\cite{Schafer1954}.We extend the framework to the dual family$A_n[\varepsilon]/(\varepsilon^2)$, provingproving proving $A_n[\varepsilon]/(\varepsilon^2) \cong A_n \otimes_{\mathbb{R}} \mathbb{R}[\varepsilon]/(\varepsilon^2)$, and to tensor products via a signcomposition principle. We further present an empirically validated countingformula for a class of zero-divisor pairs, stated explicitly as a conjecture.Three implementations are provided and cross-verified: a full table builder,and $O(n)$ and $O(1)$ single-product evaluators.