13.1.  From $\mathrm{U}(1)$ to the full Standard Model gauge group

The projected field theory of §12.1 establishes a Lorentzian scalar field theory carrying a topologically quantised $\mathrm{U}(1)$ charge, with a massive radial mode (the Higgs) and a massless Goldstone mode. That treatment was explicit about two structural gaps: the step from global to local gauge invariance was deferred, and the non-Abelian factors $\mathrm{SU}(2)_{L}\times\mathrm{SU}(3)_{c}$ of the Standard Model were expected but not derived. The present chapter closes both gaps and establishes a third result: given the Standard-Model hypercharges, gauge anomaly cancellation is not a numerical coincidence but a structural consistency check that forces $N_{c}=3$, the value PST fixes independently as the dimension of the colour factor $M_{3}(\mathbb{C})$ of the internal algebra.

The internal algebra itself, $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$, is not posited in this section but derived from P1–P3 in §13.7. The substrate’s parity-selected real structure delivers the chirality grading $\gamma_{K}=i\omega_{\mathrm{JW}}$ exactly in the Jordan–Wigner representation (Computation 19), so the chirality projector $f=(I+i\omega)/2$ comes from the substrate itself. Furey 2014 [121] then delivers $A_{F}$ as the commutant of the matter representation on the minimal left ideal $\mathcal{S}=(\mathrm{Cl}(0,6)\otimes\mathbb{C})\,p$, generated by $f$’s canonical rank-1 refinement $p$, Furey’s primitive idempotent. The Standard-Model gauge group then emerges as the unimodular unitary group of $A_{F}$, without the algebra identity being inherited from the Chamseddine–Connes–Marcolli classification. The full derivation is carried out in Computation 6 (gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ as the unimodular unitary group of $A_{F}$; inner-fluctuation count delivering 12 gauge bosons plus the Higgs scalar). Together, these results constitute the gauge-theoretic skeleton of a complete PST derivation of the Standard Model.