Born Rule from (A)∧(B)-Admissible Amplitudes in the PDL Framework

The Projective Dynamic Logo (PDL) framework derives quantum dynamics from combinatorial axioms on finite signed graphs. This paper addresses the measurement problem at Level 1: it proves that the Born rule P₊ = |⟨+θ|ψ⟩|² emerges from the (A) ∧ (B) stability criterion without presupposing probability or the Hilbert space inner product. The derivation proceeds in three steps. First, the coherence indicator χ (τ; σ) ∈ +1, −1 is defined as a purely combinatorial quantity over triangle orientations, with no probabilistic content. Second, the coherence cost εcoh (σ₊; θ, ψ) = |⟨−θ|ψ⟩|² is established from the (A) ∧ (B) apparatus symmetry conditions, with numerical residual 5. 55×10⁻¹⁶ over 20, 000 state-angle pairs. Third, P₊ = 1 − εcoh = |⟨+θ|ψ⟩|² follows by normalisation, algebraically. All five Gleason axioms are verified for this assignment with residuals ≤ 1. 1×10⁻¹⁵, and the Gleason uniqueness theorem — stated in full as a self-contained result — guarantees that this is the unique admissible probability assignment on the Hilbert space of spin-½ states. A structural diagnostic is provided: uniform triangle weights α_τ = 1/NT yield the Heaviside function Θ (cosθ), with maximum discrepancy 1. 00 (exact), confirming that the Born cos² (θ/2) distribution is not trivial and requires the specific geometric weight structure of K₄⊗Gₐpp. This result closes OP5 of D32 at Level 1. Level 2 — the derivation of triangle weights α_τ from PDL axioms without presupposing the spin Hilbert space inner product — remains open.