The Two Gaps

You have rolled dice your entire life. You have never asked why the probability of rolling a six is one-sixth. The die has six faces. The faces are symmetric. The probability follows.

But quantum mechanics is not a die. The probabilities are the squares of amplitudes. Not the amplitudes themselves. Not the cubes. The squares. Why?

The axioms’ account of quantum mechanics rests on two claims about actualisation. First: actualisation targets pure states. Second: the probability of each outcome is given by the Born rule.

Both claims are used throughout the 420 Code. Neither has been derived from the axioms.

Gap 1 (Record targets). AP23 (The Single Record) defines a record target as a pure state and shows that reduced density matrices of entangled systems are not record targets.

This is used in Lemma 1 to prove that local interaction with an entangled subsystem actualises the global state. But the definition is stated as a commitment, not derived from Axiom R.

Why must record targets be pure states? What in the axioms forbids actualisation of a mixed state?

Gap 2 (Born rule). AP09 establishes the complex Hilbert space and derives the Born rule conditionally — KS-Q.1, conditionally closed on the Hilbert space bridge (KS-Q.7).

The CHSH computation in AP23 (6.0.1) uses the Born rule to derive S = 2√2. But the Born rule’s derivation in AP09 remains conditional on the bridge, and its non-contextuality premise was granted rather than derived from {S, B, R, C}.

If the axioms claim to derive physics from four axioms, the probability measure that governs actualisation should follow from those axioms, not be imported alongside them.

This paper closes both gaps. The derivation chain is: Axiom R (definite records) → pure-state targets → non-contextuality → orthogonal additivity → Gleason’s theorem → Born rule. Every physical assumption comes from the axioms.

The only external input is Gleason’s theorem itself, which is a mathematical theorem about Hilbert spaces — not a physical postulate.