What Is Established

Before this paper, the corpus has derived the following.

SU(2). From Axiom S. The two sectors give Z₂. Z₂ = π₁(SO(3)) in N = 3 spatial dimensions (AP10). The double cover is SU(2) → SO(3) (AP11 section 2). This is the gauge group of the weak force.

One Z₂, one classification, two descriptions. The sector involution σ is the unique non-trivial binary classification in the pre-state: paired vs unpaired. AP20 proves: the abstract structure IS the manifold (EH). When this classification expresses itself on the manifold in N = 3, it IS π₁(SO(3)) = Z₂ — tensors vs spinors. AP11 proves the correspondence. There is no connecting theorem needed between a “sector Z₂” and a “topological Z₂.” They are the same classification described at two levels. The axioms already say it.

U(1)_Y. From the phase freedom of the complex Hilbert space (AP09), expressed on the local manifold as the U(1)_Y connection B_μ (AP15 section 3). Before the break, this is U(1)_Y — the hypercharge gauge group. After the break, a specific combination of U(1)_Y and SU(2) survives as U(1)_em, whose connection is A_μ (see section 3.1). This is the gauge group of electromagnetism.

The fermion. The unpaired element ε, no σ-image, half-integer spin, the electron (AP09 section 3.2).

The boson. Paired elements, σ-image exists, integer spin, the photon (AP09 section 3.1, AP15 section 7).

SU(2) and U(1)_Y are both derived. They coexist in the pre-state. SU(2) from the sector structure. U(1)_Y from the phase structure. They are independent symmetries of the same Hilbert space. Together they form SU(2) × U(1)_Y. The electroweak symmetry group.

This paper derives what Axiom B does to it.