One Bit of Difference: Moments of Frame Triples in the Albert Algebra

Abstract

Three frames mark the threshold of exceptional generation in the Albert algebra J = H₃(𝕆): any two Euclidean frames generate only a special subalgebra, while an exact triple can generate all of J. For a generic ordered, labelled triple, however, the pairwise overlaps and triple expectations of its nine primitive idempotents determine exactly two F₄-orbits. Writing L for the invariant field, H for the moment field, and δ for the deck involution, [L:H] = 2, H = Lᵟ, and L = H(u) with u² ∈ H: the moments know the square, but cannot choose its root. The ambiguity survives the three natural finite relabellings, and an exact same-moment pair admits no F₄-transporter. It is read by an explicit cubic detector — the norm of a commutator word in the two moving encoders, evaluated on the third — which generates L over H; the corresponding word-degree and multidegree bounds are sharp in their respective grammars. Across K = ℝ, ℂ, ℍ, 𝕆, at the complex function-field generic point, the connected/full-group degrees are (1,2,1,2)/(1,1,1,2): only the octonionic bit survives the full group. 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.