What This Closes

The Born rule is no longer imported. It is derived. This retroactively strengthens every paper in the body of work that uses the Born rule.

AP09 (Complex Hilbert Space and Born Rule). AP09 constructed the complex Hilbert space and derived the Born rule conditionally — closed at KS-Q.1 on the Hilbert space bridge (KS-Q.7), with the σ-additivity premise granted as the measure-theoretic form of record additivity.

Theorem 1 completes that result: non-contextuality and orthogonal additivity are now derived from the axioms, and the Born rule is the unique measure forced by Axiom R on a complex Hilbert space of dimension ≥ 3. The Born rule is no longer conditional on a granted premise.

It is a consequence of the axioms acting on that space.

AP23 (The Single Record). AP23’s two upstream caps are closed. Cap 1: Proposition 1 derives the pure-state record target requirement from Axiom R’s definiteness condition via convex geometry, replacing AP23’s definitional commitment.

AP23’s Lemma 1 (Local Interaction, Global Actualisation) now rests on a derivation, not a definition.

Cap 2: The CHSH computation (AP23 6.0.1) uses the Born rule to derive S = 2√2. Theorem 1 derives the Born rule from {S, B, R, C}.

Therefore S = 2√2 is derived from the axioms. The Bell prediction is now fully axiomatic.

The 420 Code. Every paper that uses probability — AP09 (measurement as actualisation), AP11 (spin statistics), AP15 (charge quantisation), AP16 (electroweak unification), AP19 (SU(3) colour), AP21 (cosmic web) — is retroactively strengthened.

The probability structure was always the Born rule; now it is the Born rule because it is the only possibility.