Paper 1 — Sign-Parity Classification of Zero-Divisors in Cayley–Dickson Algebras
Independent Researcher, Modena, Italy • ORCID 0009-0000-9804-3452
Abstract
We develop a complete structural theory of zero-divisors in the Cayley-Dickson algebras A_n.
The main results are:
(i) an exact closed-form count |ZD(n)| = 336 × C(n-1,3)_2 for all n ≥ 4, where 336 = 2|PSL(2,7)| encodes the Fano-plane symmetry;
(ii) a sign-parity classification via three invariants satisfying π_A·π_B·π_C = -1;
(iii) a proof that the Born Rule axioms admit a unique solution for n ≤ 3 and no solution for n ≥ 4;
(iv) the spectral decomposition Spec(M_a) = {0,1,2} with multiplicities 1:2:1 for every ZD unit;
(v) a full computational classification of all 460,880 zero-divisor elements in A_4.
All open problems PA1-PA5 are resolved unconditionally.
How to Cite
@misc{levratti2026signparity,
author = {Levratti, Giovanni},
title = {Sign-Parity Classification of Zero-Divisors in Cayley–Dickson Algebras},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.20315751},
url = {https://doi.org/10.5281/zenodo.20315751},
note = {Paper 1 in the series Zero-Divisors
in Cayley--Dickson Algebras, v2}
}