Record Targets Are Pure States

Proposition 1 (Record Target). Let H be the Hilbert space of the pre-state (AP09). The states eligible for actualisation under Axiom R — record targets — are pure states (extremal points of the state space).

Mixed states (non-extremal points) are not record targets.

Proof. Axiom R requires that actualisation writes a definite record. By 2, a definite record corresponds to an extremal post-actualisation state: one that admits no non-trivial convex decomposition. Consider a candidate record target ρ.

If ρ is a mixed state (non-extremal, rank > 1), it admits multiple distinct decompositions into pure states. The record “ρ was actualised” does not uniquely specify the physical state.

It is compatible with infinitely many distinct decompositions, none of which is privileged. This violates Axiom R’s requirement for definiteness. If ρ is a pure state (extremal, rank 1), then ρ = |ψ⟩⟨ψ|.

It admits no non-trivial decomposition. The state is |ψ⟩ and nothing else. The record is definite. Therefore: record targets are pure states. Mixed states are not eligible for actualisation. ∎

Note on improper mixtures. The reduced density matrix of an entangled subsystem (ρ_A = Tr_B(|AB⟩⟨AB|)) is an improper mixture: it arises from tracing out the entangled partner, not from classical ignorance.

Proposition 1 excludes improper mixtures from actualisation for the same reason as proper mixtures — they are non-extremal — but the improper case has an additional reason: the reduced state is not an independent state at all.

It is a marginal of the global entangled state. It has no standalone existence.

This is consistent with AP23’s Lemma 1: local interaction with an entangled subsystem actualises the global pure state |AB⟩, not the reduced state ρ_A.

You have just watched Axiom R — one requirement, definiteness — kill mixed-state actualisation. The universe cannot write a record that says “maybe this, maybe that.” It writes a record that says “this.”

Pure states are the only states that say “this” without ambiguity.