Summary of Derivation

Axiom S → two sectors, involution σ = complex conjugation → Born rule P = ψψ* → phase rotation e^{iθ} leaves |ψ|² invariant → U(1) symmetry.

Axiom R + Axiom C + G → records written, manifold emerges with locality (AP20) → the manifold is local, the pre-state is global.

Global phase on local manifold → connection Aμ encodes phase coherence across separated points → U(1) gauge field. [DERIVED — ε charged (Thm 1) → non-trivial bundle (Thm 2)]

Curvature of connection → Fμν = ∂μAν − ∂νAμ → electromagnetic field tensor.

Axiom B → ε = electron = source of connection → charge.

Discrete break → quantised charge. [DERIVED — Theorem 5: U(1) compact → integer representations (Peter–Weyl) → ε = minimum (Axiom B) → all charges integer multiples of e. Gap E: CLOSED]

Axiom R (conservation) + U(1) + D = 4 (AP10) → Maxwell action unique up to coupling constant → Maxwell’s equations.

Born rule exact (KS-5 CLOSED) → U(1) unbroken → photon massless, spin 1, propagates at c.

Pre-state pure (S = 0) → phase coherence partitions into field (connection curvature on M) + entanglement (correlations in ℋ) → ε-record-events transfer coherence from entanglement to field, irreversibly (Axiom R), rate-bounded (Axiom C), receipt k_B T ln 2 (AP06). [DERIVED for partition + transfer + irreversibility + rate-bound; STRUCTURAL for the conservation reading (KS-31 LIVE)]