What question did this study set out to answer?

This research aims to construct a detailed theory of zero-divisors within Cayley-Dickson algebras, exploring their properties and classifications.

May 22, 2026Open Access

Zero Divisor Structure Norm Redistribution and Spectral Theory in Cayley Dickson Algebras

Key Points

This research aims to construct a detailed theory of zero-divisors within Cayley-Dickson algebras, exploring their properties and classifications.
Developed a closed-form count for zero-divisors |ZD(n)| in Cayley-Dickson algebras for n ≥ 4.
Classified zero-divisors based on sign-parity invariants.
Classified all zero-divisor elements computationally in A_4.
Exact count of zero-divisors |ZD(n)| = 336 × C(n-1,3)_2 for n ≥ 4.
A unique solution for Born Rule axioms exists for n ≤ 3 but none for n ≥ 4.
Spectral decomposition shows multiplicities 1:2:1 for every zero-divisor unit.

Abstract

We develop a complete structural theory of zero-divisors in the Cayley-Dickson algebras Aₙ. 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 πA·πB·πC = -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 (Mₐ) = 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 A₄. All open problems PA1-PA5 are resolved unconditionally.

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