Sign-Parity Classification of Zero Divisors in Cayley–Dickson Algebras

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.