Scope
This sub-programme asks which internal symmetries, if any, are selected by admissible non-injective projection. It contains a phase sector, a conditional non-abelian sector, charge and chirality constructions, and the principal bundle of a supplied compact group; the derivation of Yang–Mills equations on that bundle belongs to Presentation Note 8.
Current status. The sub-programme identifies no gauge group. $\mathrm{U}(1)$ is structural on premises carried as input, and $\mathrm{SU}(2)$ is conditional on a supplied carrier. O31 withdraws every derivation of $\mathrm{SU}(3)$ and of $G_{\mathrm{SM}}=\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$, colour-triplet co-admissibility and $\mathrm{SU}(3)$ uniqueness included; colour, commutation, direct-product structure and the complete gauge group are open. Q12 and Q13 take the gauge group as a supplied compact group, and the note takes no metric coefficient from Presentation Note 2.
Current chain
- Q6a defines gauge charges as fibre-constant functions of the observable state and the gauge group as the fibre-preserving transformations of the projection; $\mathrm{U}(1)$ is the residual phase symmetry for a span-based projection and $\mathrm{SU}(2)$ is read under a supplied carrier. It does not use O31, and its two continuum hypotheses do not determine a connection, which is supplied.
- O27 supplies the canonical representation action $\mathfrak{su}(2)\to\operatorname{End}(V_\rho)$ only after the conditional carrier and identification are supplied; the relevant equivariant intertwiner is unique only up to a scalar. It does not construct a vector lift.
- Q12 derives the local Yang–Mills equation from vertical spectral variation for a supplied compact gauge group and connection. It does not select the group.
- O32 reports finite-$q$ numerical measurements; they establish no pointwise co-admissibility and identify no group.
Constituent papers
Open problems
- Construct a canonical causal carrier and the required inter-sector identifications.
- Identify a colour representation and compact group without importing the withdrawn O31 claims.
- Prove determinant, commutation and direct-product properties if a multi-factor group is proposed.
- Derive the weak-current $V-A$ structure and extend the monopole analysis beyond the canonical model.