Irreducibility and Divisor Laws in the Gravesian Integer Octonions

An integer octonion z in the Gravesian order Z^8 — whole-number coordinates — is irreducible if and only if N(z) is an ordinary prime, or N(z) = 2^k with k >= 2 and the norm-2 divisor count σ(z) = 0. This Strong Irreducibility Theorem classifies reducibility in the non-maximal ring that differs from the seven octavian maximal orders only at the prime 2. Aside from prime-norm elements, every composite-norm irreducible is a hole: a point invisible to norm-2 divisors, forming asymptotically 49/64 of each large power-of-two shell. The proof rests on a congruence reduction: divisibility by a norm-m element is a linear condition modulo m, so divisor counts depend only on residues. Three divisor laws emerge: the Parity-Fano Theorem (σ in {0,16,48,112}, governed by the extended Hamming code [8,4,4]); the Odd Divisor Theorem (exactly 16 norm-p divisors per side for odd p); and the Norm-4 Divisor Theorem (19 residue profiles mod 4). Holes factor in maximal orders; product visibility follows a closed Fano product formula. Under the 1344-element automorphism group, hole arithmetic preserves species access in fixed 2:1:1 proportions.