Current status
Q6a organises gauge structure through equivalence classes of the non-injective projection and proposes $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ as admissibility fixed points. The representation-theoretic bridge must be read with the carrier and factorisation assumptions stated in the corrective O27 chain.
Full-group claim withdrawn. Q6a’s former identification of $\mathrm{SU}(3)$ and $G_{\mathrm{SM}}=\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ relied on O31. O31 version 2.0 withdraws that support. The colour factor, commutation, direct-product structure and complete gauge group are open.
What remains
- The projection-fibre equivalence-class framework and its fixed-point question.
- The proposed phase-sector $\mathrm{U}(1)$ and conjugate-pair $\mathrm{SU}(2)$ readings, subject to their explicit carrier and identification hypotheses.
- A conditional input interface for gauge dynamics: if a compact group and admissible connection are supplied, Q12 studies the corresponding Yang–Mills term.
What is open
No valid result in this chain selects $\mathrm{SU}(3)$ from colour-triplet data or proves that three sectorwise symmetries commute and form the Standard Model direct product. Numerical patterns from O32 do not repair the missing identification theorem.
Reference
Jérôme Beau. Gauge Structure from Admissible Non-Injective Projection.