What Is Now Derived

Axiom S → Z₂ → SU(2) (via π₁(SO(3)) in N = 3, AP10; double cover AP11) → weak gauge group.

AP09 + AP15 → complex amplitudes → phase freedom → U(1)_Y → hypercharge gauge group.

SU(2) × U(1)_Y → the full internal symmetry of the pre-state on the manifold → electroweak symmetry group.

Axiom B → ε ∈ ℒ, no σ-image in 𝒫 → sectors no longer interchangeable → SU(2) broken → U(1)_em survives → SU(2) × U(1)_Y → U(1)_em. Electroweak symmetry breaking.

The Higgs field = ε → nonzero ground state (ν(ε) = 1) → the break cannot be zero. The condition for existence.

Three broken generators → W⁺, W⁻, Z⁰ acquire mass (coupling to ε) → massive weak bosons.

One unbroken generator → photon stays massless (U(1)_em exact, Born rule exact) → massless photon. You hold the kill switches.

Self-coupling of ε (Yukawa: fermionic excitation × scalar order parameter) → electron mass m_e → minimum viable splinter.

α_em = leakage at the electromagnetic face → structurally determined, empirically known → the architecture’s empirical contract (STRUCTURAL, not DERIVED — see section 5).