Paper 6 — The Factorization Threshold in Cayley–Dickson Integer Algebras

We establish the precise threshold at which the arithmetic of Cayley–Dickson integer algebras breaks down. Let S_n be the canonical integer subalgebra of the n-th Cayley–Dickson algebra over ℝ. We prove that S_n is an integral domain if and only if n ≤ 3, and that S_n is a UFD if and only if n ≤ 1. The two thresholds are distinct:

n ≤ 1 (UFD) ⊊ n ≤ 3 (domain) ⊊ n ≥ 4 (not a domain)

For n ∈ {2,3}, S_n admits exactly 2ⁿ−1 pairwise inequivalent factorisations of 2 (3 for the Lipschitz integers S₂, 7 for the Gravesian octonion integers S₃). The count 2ⁿ−1 = |PG(n−1, 𝔽₂)| connects the arithmetic failure to the same finite-geometric structure governing the zero-divisor count |ZD(n)| = 336·C(n−1, 3)₂.

For n ≥ 4 the failure manifests in two complementary ways: the {0,4,8} Trichotomy shows that no product of two norm-2 elements can give a nonzero norm-2 element; and the right annihilator of every ZD-active element contains non-units of norm 2, which cannot occur in any integral domain.

Computational verification for n = 4: 480 ordered pairs of type (2,2) forming 15 equivalence classes, and 118 ordered pairs of type (4,2) forming 35 additional classes — giving at least 50 inequivalent factorisations of 2 in S₄.

The threshold n = 4 coincides with the endpoint of the classical chain of composition identities for sums of 2ⁿ squares (Hurwitz 1898) and with the failure of norm-multiplicativity.