Notation

|ψ⟩ — a pure state (ray) in Hilbert space H. A single, definite quantum state. An extremal point of the convex set of density operators on H.

ρ = ∑ p_i |ψ_i⟩⟨ψ_i| — a mixed state (density operator). A non-extremal element of the convex set. Admits multiple decompositions into pure states.

P_a — a projector onto the outcome subspace for measurement result a. In the non-degenerate case, P_a = |a⟩⟨a| (rank 1).

In the degenerate case, P_a has rank > 1, projecting onto a subspace of dimension > 1.

Tr(ρ · P_a) — the trace of ρ composed with P_a. The Born rule in its general form.

dim(H) — the dimension of the Hilbert space H.

Frame function — a function μ assigning a non-negative real number to each projection operator in H, such that ∑ μ(P_i) = 1 for every resolution of the identity (every complete set of mutually orthogonal projectors {P_i} with ∑ P_i = I).

A frame function is defined on projectors alone — not on (projector, context) pairs. This is Gleason non-contextuality.

Gleason non-contextuality — the probability μ(P_a) assigned to projector P_a depends only on P_a itself, not on which resolution of the identity (which PVM) P_a belongs to.

This is a property of the probability assignment, not of pre-existing value assignments. It is distinct from Kochen-Specker contextuality, which concerns the impossibility of non-contextual definite-value assignments to all observables simultaneously.

The axioms deny pre-existing values (Axiom R), so KS contextuality is expected; Gleason non-contextuality of outcome probabilities (Proposition 2) is a separate, compatible claim.