A Topological Origin for the Electroweak Sector Deriving SU(2) x U(1) Symmetry and the Lepton Mass Hierarchy

This paper extends the generative V–j–Aj framework to the electroweak sector, deriving the full SU(2)×U(1) symmetry and the lepton mass hierarchy from topological first principles. Following the SU(3) analysis based on Borromean rings, this work identifies the three generations of charged leptons—the electron, muon, and tau—with the first three stable torus knots (3₁, 5₁, 7₁). Their corresponding weak isospin and hypercharge symmetries emerge naturally from the geometry and framing of these knots.

The derivation shows that:

• The SU(2) gauge symmetry arises from the braid group structure of the knot complement and is locally reproduced by the skein algebra’s Goldman bracket relations.

• The U(1) hypercharge emerges from the knot’s intrinsic framing, consistent with the Călugăreanu–White–Fuller relation and the r=10 reader of the Temperley–Lieb algebra (SU(2)₈×U(1)₅).

• A truncation rule (f₉=0) at r=10 enforces the existence of exactly three charged lepton generations, excluding a fourth.

The paper also introduces the Channel-Coupled Mass Law, a new first-principles formula expressing lepton mass as the interaction energy between knot geometry (curvature and torsion) and the elastic response channels of the vacuum. This law unifies the origin of fermion mass with the electroweak coupling structure, constrained by the weak mixing angle (θ_W).

Together, these results present a topological and generative foundation for the electroweak sector, linking SU(2)×U(1) symmetry, charge quantization, and lepton mass hierarchy to the self-consistent geometry of framed torus knots.

Part of the Generative Wholeness series: building on “A Pre-Geometric Origin for Quantum Dynamics” and “A Topological Origin for SU(3) Symmetry.”

Keywords: SU(2)×U(1), electroweak symmetry, lepton generations, torus knots, wholeness, topology, Channel-Coupled Mass Law, Temperley–Lieb algebra.