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oai:figshare.com:article/33705022 2026-09-29T00:00:26Z category_29827 item_type_12 month_year_09_2026

An O(1) Bitwise Evaluator for Cayley--Dickson Sign Structure: Ordinary, Split, Dual, and Tensor Constructions maher ben abdessalem (13274766) Algebra and number theory abstract alegbra hypercomplex numbers Cayley-Dickson algebras Computational mathematics <p dir="ltr">The sign of the Cayley--Dickson basis product $e_i \cdot e_j = \pm e_{i \oplus j}$</p><p dir="ltr">coincides with the explicit twist function $\sigma(A,B)$ of Ren and Zhao</p><p dir="ltr">(Theorem~1 for the standard algebras; Theorem~2 for the split relation</p><p dir="ltr">$\sigma_s = \sigma + a_{n-1} b_{n-1}$)~\cite{RenZhao2023}.</p><p dir="ltr">This paper gives an independent treatment of that sign law from the</p><p dir="ltr">computational side. We prove the block sign rules (the ``OPMT sign law'')</p><p dir="ltr">directly from the Cayley--Dickson doubling formula, introduce a holographic</p><p dir="ltr">$O(n)$ descent algorithm whose correctness we prove against these rules, and</p><p dir="ltr">collapse the descent into a strictly table-free $O(1)$ word-RAM evaluator</p><p dir="ltr">using three trailing-zero counts, one maximum comparison, and one population</p><p dir="ltr">count. 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> 2026-09-29T00:00:26Z Text Preprint 10.6084/m9.figshare.33705022.v8 https://figshare.com/articles/preprint/A_Proven_Sign_Law_for_Cayley-Dickson_Algebras_Ordinary_and_Split_Constructions/33705022 CC BY 4.0