Derivation Chain

AP10 → three spatial dimensions from three axioms {C, S, B} → three faces of one manifold. Functional identity fixed, geometric assignment free.

Axiom B → the break creates direction as a concept → the manifold orients around the break.

Before collapse: all orientations coexist in superposition (§3.4) → orientation is quantum, not classical.

Non-derivability of {C, S, B} (AP20 §4.2) → no face preferred → global symmetry. Per-event orientation → local symmetry → gauge theory with connection.

Complex Hilbert space (AP09) → orientation state |ε⟩ ∈ ℂ³ (Lemma) → gauge group SU(3) (Proposition 1, not SO(3)).

Isotropy of manifold (AP10) + substrate stiffness (AP15) → macroscopic orientation isotropic → colour neutrality (Proposition 2). Confinement.

B fixes existence, not orientation (AP20 §4) → SU(3) unbroken (Proposition 3) → gluons massless.

Minimum break (Axiom B) + anisotropy costs energy (Proposition 2) → ε is isotropic → electron is SU(3) singlet (Proposition 4).