Paper C in the Laketek E₈ series. The first two papers derived spacetime and gravity from the split octonion algebra using bilinear operations, where Artin's theorem guarantees associativity. This paper crosses the trilinear threshold: the Jordan triple product, the vertex at which two quantum states meet a gauge field, is unavoidably non-associative, and deterministic prediction of individual outcomes becomes algebraically impossible. Quantum behavior is derived rather than postulated. The central result is the Born Rule Uniqueness Theorem: given any real eight-dimensional non-associative composition algebra of split signature with Lorentz-covariant dynamics, the Born rule P = |q₁|² is the unique probability measure, with the state space, measurement projection, and probability norm all forced by the algebra. The theorem is strictly stronger than Gleason's and applies to any framework that arrives at the split octonions, whether or not it routes through E₈. Additional results include the Tsirelson bound from the Clifford algebra commutator structure, exact non-associative decoherence with no external environment, parity-violation chirality from the Fano plane, and the algebraic origin of the time axis. Part of: "An Algebraic Theory of Everything from E₈₍₈₎" (Papers A through I).
Curtis Laketek (Fri,) studied this question.