The Gauge Structure Sub-Programme

The gauge structure sub-programme asks which internal symmetry groups are forced by the admissibility structure of the non-injective projection $\Pi$, rather than postulated. This note records what that question has and has not established. No derivation of the composite ${G_{\Pi}} = {\mathrm{SU}}(3) \times {\mathrm{SU}}(2) \times {\mathrm{U}}(1)$ is available. O31 withdraws every derivation of a gauge factor and of the Standard Model gauge group as a composite, together with the determinant-one condition, the commutation of the putative factors and the direct-product structure. The ${\mathrm{SU}}(3)$ identification is withdrawn with it: the uniqueness of ${\mathrm{SU}}(3)$ among connected compact subgroups acting irreducibly on $\mathbb{C}^3$ fails on the counterexample ${\mathrm{SO}}(3) \subset {\mathrm{SU}}(3)$; the irreducibility of the admissible transformation group fails on the diagonal torus $T^2$; and the arithmetic criterion $q \equiv 1 \pmod 3$ fails on $\{1, 2, q-3\}$. What remains is one structural sector and one conditional sector, not a group. ${\mathrm{U}}(1)$ is structural: the Hopf phase fibre of $\Pi$ gives charge quantisation $q_{\mathrm{eff}} = w \cdot e$ from $\pi_1(S^1) \cong \mathbb{Z}$, on two premises carried as structural input rather than derived; and on the canonical admissible model no isolated magnetic monopole is topologically admissible. ${\mathrm{SU}}(2)$ is conditional on the spinor carrier supplied by O23: O27 gives unique real-linear factorisation through the admissibility quotient, and the identification of that quotient with ${\mathfrak{su}}(2)$ is supplied rather than derived. Interpretation, not result: the sub-programme's remaining content is a phase sector and a conditional non-abelian sector, and the step that would make them factors of a single gauge group is exactly the step that has been withdrawn.