The Missing Group

The Standard Model of particle physics has gauge group SU(3) × SU(2) × U(1). The axioms have derived two of the three factors.

AP15 derives U(1) from phase freedom. The complex amplitude of ε has an overall phase; local rephasing is unphysical; demanding consistency under local rephasing forces a gauge connection with structure group U(1). This is electromagnetic gauge symmetry.

AP19 derives SU(3) from spatial orientation freedom. The break ε propagates through a 3-dimensional manifold (AP05, AP10); local reorientation of the three spatial axes is unphysical; demanding consistency under local reorientation forces a gauge connection with structure group SU(3). This is colour gauge symmetry.

One factor remains: SU(2). In the Standard Model, this is weak isospin — the internal symmetry that governs the weak nuclear force. It acts only on left-handed fermion fields, organising them into doublets. Right-handed fields are singlets: the weak force does not see them.

The pattern of the previous derivations points to the answer. Each gauge group arises from a physical freedom in how ε is described:

Phase freedom → U(1). Orientation freedom → SU(3). ??? freedom → SU(2).

What degree of freedom has not yet been gauged? The axioms provide one structural feature that has not yet been exploited for gauge purposes: Axiom S. The two-sector structure ℒ/𝒫.

You have heard three notes of a chord. Two are identified — U(1) and SU(3). The third is sounding but unnamed. This paper names it.