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 Π, rather than postulated. This note records what that question has and has not established. No derivation of the composite GΠ = SU(3) × SU(2) × 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 The SU(3) identification is withdrawn with it: the uniqueness of SU(3) among connected compact subgroups acting irreducibly on C³ fails on the counterexample SO(3) ⊂ SU(3); the irreducibility of the admissible transformation group fails on the diagonal torus T²; and the arithmetic criterion q ≡ 1 3 fails on \1, 2, q-3\. What remains is one structural sector and one conditional sector, not a group. U(1) is structural: the Hopf phase fibre of Π gives charge quantisation qeff = w · e from π₁(S¹) Z, on two premises carried as structural input rather than derived; and on the canonical admissible model no isolated magnetic monopole is topologically admissible. 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 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non-abelian sector, and the step that would make them factors of a single gauge group is exactly the step that has been withdrawn.