Paper 3 — Sign Structure, Automorphism Groups, and the ZD Zeta Function of Cayley–Dickson Algebras

We study the structure of zero-divisors in the Cayley–Dickson algebras Aₙ (n ≥ 4) from three directions: orbit classification, generating functions, and Fano geometry.

Orbit classification. The automorphism group Aut(Aₙ) ≅ (ℤ/2ℤ)ⁿ ⋊ GL(3,𝔽₂) acts on ZD(n) with four orbit sizes {2, 14, 84, 336}, determined by the projective geometry of S(z) ⊆ PG(2,𝔽₂). Closed-form count formulas hold for all n: O₃₃₆(n) = 8^(n-4), O₂(n) = |ZD(n-3)|/2, O₁₄(n) = (|ZD(n-2)| − |ZD(n-3)|)/2. Verified through n=7 by GPU computation.

ZD zeta function. The generating function Z(x) = Σ |ZD(n)| xⁿ = 336x⁴ / [(1−x)(1−2x)(1−4x)(1−8x)] satisfies the functional equation Z(x) = 64x⁴ Z(1/8x) and the Weil identity Z(x) = 336x⁴ · Z_Weil(PG(3,𝔽₂), x). The four poles correspond to the four Hurwitz algebras ℝ, ℂ, ℍ, 𝕆 with dimensions 1, 2, 4, 8. The closed form |ZD(n)| = (2ⁿ−2)(2ⁿ−4)(2ⁿ−8)/4 follows by partial fractions.

Fano bijection. For ZD*(n) = {z ∈ ZD(n) : S(z) ≠ ∅}, the map f(z) = span{π(aᵢ), π(aⱼ), π(bᵢ)} defines an Aut(Aₙ)-equivariant map onto G(3, n−1; 𝔽₂) with uniform fibres of size 336. The Weil identity becomes geometric: ZD(n) is a rank-336 bundle over G(3,n−1;𝔽₂) whose Schubert stratification recovers the four Frobenius eigenvalues of H²ᵏ(PG(3,𝔽₂)).

The non-split exact sequence 1 → (ℤ/2ℤ)ⁿ → Aut(Aₙ) → GL(3,𝔽₂) → 1 is established for all n ≥ 4.