The Phase

Every quantum state you have ever encountered carries a hidden degree of freedom. It is not the magnitude — the magnitude gives the probability. It is the phase. The angle. The rotation that no measurement can detect directly, but whose consequences shape every interference pattern ever recorded.

AP09 derives complex amplitudes from the Lorentzian signature of the manifold (AP09 section 3.3). The pre-state lives in a complex Hilbert space ℋ. State vectors in ℋ have complex amplitudes. Every complex amplitude has a magnitude and a phase.

The Born rule (AP09 section 6) gives: P = |ψ|² = ψψ* = Light × Dark. The magnitude determines the probability. The phase does not. Replace ψ with e^{iθ}ψ (where i² = −1) — rotate the phase by any angle θ — and the probability is unchanged:

|e^{iθ}ψ|² = e^{iθ}ψ × e^{−iθ}ψ* = ψψ* = |ψ|²

This is U(1). The group of rotations of a circle. One parameter: θ. One operation: multiplication by e^{iθ}. The simplest continuous symmetry. And you already have it.

This symmetry is not imposed. It is not a choice. It is a consequence of the Born rule, which is a consequence of the involution σ (Axiom S), which is a consequence of the two-sector structure. U(1) is already derived. It lives in the pre-state. It has been there since AP09.

The question is: what does this symmetry look like when read on the manifold?