The Electroweak Sector as Geometry: One Chiral Generation from the Canonical Complexes of the Spatial Sphere

Paper 3 completed the strong sector of the Standard Model as geometry — colour and three families from the 〖CP〗^1×〖CP〗^2 internal factors — and named the electroweak sector as its successor. This paper works within the block B=S^3×〖CP〗^1×〖CP〗^2, adds no new geometry, and reports what representation, index, and CR spectral theory establish about the electroweak sector, graded explicitly into what is exact, what is conditional, and what is open.Exact. The spatial three-sphere is the link of the Kähler cone C^2 and therefore carries a canonical strictly pseudoconvex CR structure — no choice beyond the complex structure of the cone, the input Paper 3 priced as Commitment HY. That structure breaks the isometry SO(4) to SU(2)_L×U(1)_R and distinguishes five canonical carrier bundles, fixed by a target-independent admission rule (§2.2) and adjudicated by a single graded CR–Dirac operator (§2.3). Under one band criterion the chiral content of their low band is exactly the representation content, charge structure, and handedness grading of one Standard-Model generation. The two low-band extras are removed by canonical mechanism: a weak-singlet pair by the quaternionic self-conjugation only the spin complex carries — one vector-like Dirac fermion — and the charge-three sector by the anomaly ledger, its colour-singlet channels forced to Y=0 by the local equations and then excluded by the Witten Z_2 anomaly at odd multiplicity and an invariant pseudoreal pairing at even multiplicity, with the required internal dressing in any case absent from the block’s realised sectors. Given the carrier assignment, the Standard-Model hypercharges are the unique anomaly-free point of the finite geometric menu; the block’s two abelian charges admit exactly one gaugeable direction; and the global gauge group is computed by exact kernel enumeration to be (SU(3)×SU(2)×U(1))/Z_6, the Standard Model’s global form.Conditional. The interpretation of these exact structures as a local four-dimensional electroweak gauge theory — including the identification of cone chirality (Dolbeault-degree parity, which places the doublets Q, L at one handedness and the singlets u^c, d^c, e^c at the other: the V–A pattern) with emergent physical handedness — passes through the bridge principle, Commitment SI (§3.6), whose mechanism, group, and module obligations are discharged here and whose dynamical obligations are not.Open. Gauge dynamics, the weak coupling, a Lorentz-invariant local effective theory, the electroweak scale, and symmetry breaking are not claimed. The honest summary: the canonical CR geometry reproduces the representation content, charge structure, and geometric handedness grading of one Standard-Model generation; its interpretation as the physical electroweak sector remains conditional on Commitment SI. The hypotheses and commitments of Papers 1–3 remain in force, and the route and audit record of the working masters is preserved in Appendix A.