One Bit of Difference: Frame Moments in the Albert Algebra and the First Row of the Magic Square

The generic fibre of the moment map on triples of frames in the Albert algebra — orthogonal triples of primitive idempotents — is a torsor under Z/2: the field of rational invariants is a quadratic extension of the field the moments generate. The ambiguity is stable under every finite relabelling, is resolved by no automorphism (F4 is connected) and, on the free compact locus at a fixed labelling, by no continuous path, and is detected by the cubic norm of one choice-free degree-three Jordan polynomial — within the two grammars studied here, by nothing shallower. Both fibre members are printed exactly, once over Q, provably F4-inequivalent, and once as a Euclidean pair split by conjugating a phase built from the Gaussian primes 2±i. For the unital subalgebras H3(K), K = R, C, H, O — each time at the complexified generic point — the same fibre degree reads 1,2,1,2 across the first row of the Freudenthal–Tits magic square, and 1,1,1,2 under the full automorphism groups: the ambiguity exists only over C and O, and is unresolvable only over O. AI assistance was used extensively in this work: Anthropic's Fable 5 and OpenAI's Sol 5.6 contributed to drafting, symbolic computation, proof search, and cross-checking throughout. The work was human-directed; the central results were additionally audited in separate AI lanes, and many exact computations are corroborated by the replayable certificates in the linked evidence repository. The author takes full responsibility for every statement. The paper's AI-assistance section gives the detailed division of labour.