What “Definite” Means

Axiom R says: the break writes a definite, irreversible record. The record is a fact about what happened. It answers the question completely. There is no residual ambiguity about what the record says.

What does “definite” mean in the language of Hilbert space?

The space of quantum states has a convex structure. Any probabilistic mixture of two states is also a state.

Pure states are the extremal points of this convex set: they cannot be written as a non-trivial mixture of other states.

Mixed states are non-extremal: they sit inside the convex body and can be decomposed into mixtures of pure states in multiple distinct ways.

A definite record corresponds to an extremal post-actualisation state — one with no residual convex ambiguity. If the post-actualisation state is extremal (pure), there is exactly one state consistent with the record.

If the post-actualisation state is non-extremal (mixed), the same density operator ρ can be decomposed as ρ = ∑ p_i |ψ_i⟩⟨ψ_i| = ∑ q_j |φ_j⟩⟨φ_j| with {|ψ_i⟩} ≠ {|φ_j⟩}.

The record “ρ was actualised” is compatible with multiple distinct underlying realities.

Even if the record attempted to specify a particular decomposition, there is no fact of the matter about which decomposition is correct — the same ρ is compatible with infinitely many choices.

A definite record must correspond to a single physical description, not a class of descriptions. Definiteness requires extremality. In Hilbert space, extremal states are pure states.

You have signed a contract. The signature is definite — it is yours, not a superposition of signatures. It admits no ambiguity about who signed.

That is what Axiom R demands of every record: one answer, no residual uncertainty, no alternative decomposition. A mixed state is a contract with multiple possible signatories. The universe does not write ambiguous contracts.