Notation

|AB⟩ — the entangled state of two subsystems A and B. A single vector in the tensor product ℋ_A ⊗ ℋ_B.

ℋ_A ⊗ ℋ_B — the tensor product of Hilbert spaces. Combines degrees of freedom algebraically; does not embed them into spacetime.

ρ_A = Tr_B(|AB⟩⟨AB|) — the reduced density matrix of subsystem A. For an entangled state, ρ_A is a mixed state.

This is an improper mixture (a marginal of a global pure state), not a proper mixture (classical ignorance over a definite but unknown pure state).

Record target — a state eligible for actualisation under Axiom R.

A record target is a pure state (a ray in Hilbert space) — a single, definite quantum state from which a record can be written.

A reduced density matrix of an entangled system is not a record target: it is a marginal derived from the global state, not an independent state in its own right.

Actualisation — the writing of a record onto the manifold. The break. Axiom R. Targets pure states.

Pre-state — the complex Hilbert space prior to actualisation (AP09). No spatial structure.

When standard QM writes ψ(x_A, x_B), those coordinates are a representation relative to a record-defined position basis on the manifold; they are not fundamental structure of the pre-state.

Separable state — a state that can be written as |A⟩ ⊗ |B⟩ (a product of independent states). Subsystems can be actualised independently.

Entangled state — a state that cannot be written as |A⟩ ⊗ |B⟩. Non-factorizability forces single actualisation (Lemma 1).