Status and Dependency

This paper derives the probability structure of actualisation from the axioms. It closes two gaps identified as upstream caps in AP23 (The Single Record).

Gap 1: AP23’s Lemma 1 relies on the claim that reduced density matrices are not record targets — that actualisation targets pure states only.

This was stated as a commitment in AP23’s notation section but not formally derived from the axioms.

Gap 2: AP23’s Proposition 2 (Bell Prediction) and the CHSH quantitative payload use the Born rule. The Born rule was established in AP09 — derived from Axiom S and Gleason’s σ-additivity, conditionally closed at KS-Q.1 on the Hilbert space bridge (KS-Q.7). What was owed was the unconditional derivation: the Gleason route run on the axioms’ own premises, with non-contextuality and orthogonal additivity derived rather than granted.

This paper supplies it.

This paper derives both. Proposition 1 shows that Axiom R’s requirement for definite records forces actualisation to target pure states. Proposition 2 shows that Axiom R and Axiom B jointly imply non-contextual outcome probabilities.

Lemma 1 establishes orthogonal additivity.

Theorem 1 (The Measure) shows that the unique probability measure consistent with these requirements on a complex Hilbert space of dimension ≥ 3 is the Born rule: P(a) = Tr(|ψ⟩⟨ψ| · P_a).

For rank-1 projectors, this gives |⟨a|ψ⟩|². The Born rule is not imported. It is derived.

The dependency chain: Axiom S (Symmetry) → Axiom B (single break) → Axiom R (definite, irreversible record) → AP09 (complex Hilbert space) → AP10 (3 spatial dimensions; physical wavefunctions live in L²(R³), which is separable and infinite-dimensional, so the full physical Hilbert space satisfies dim(H) ≥ 3) → this paper (pure-state targets, non-contextuality, orthogonal additivity, Born rule).

Epistemic status per section. 1 (The Two Gaps): historical. 2 (What Definite Means): derived. 3 (Record Targets Are Pure States): derived — Proposition 1. 4 (Non-Contextuality): derived — Proposition 2. 5 (Orthogonal Additivity): derived — Lemma 1. 6 (The Born Rule): derived — Theorem 1. 7 (What This Closes): derived — retroactive strengthening. 8 (The Deeper Point): synthesis — non-load-bearing.