Current status
Q6a defines gauge charges as functionals constant on the projection fibres, and the gauge group as the group of fibre-preserving transformations of the projection. Such a functional factors through the projection, so it is a function of the observable state; its identification with a conserved physical charge is open. $\mathrm{U}(1)$ is the residual phase symmetry: global phase rotations act as a global scalar, which leaves every span-based capacity profile unchanged. On the spinor carrier supplied by O23, whose selection is open, and under the neutral-sector hypothesis of O27, the group acting faithfully on the carrier is $\mathrm{SU}(2)$. This is a consistency statement about a supplied carrier, not a selection: Q6a establishes no uniqueness or minimality of $\mathrm{SU}(2)$. $\mathrm{SU}(3)$ is a structural hypothesis whose identification is open.
What Q6a establishes
- Gauge charges as functions constant on the projection fibres, hence functions of the observable state, and the gauge group as the fibre-preserving transformations.
- The residual phase $\mathrm{U}(1)$ and, on the supplied carrier, $\mathrm{SU}(2)$ as the group acting faithfully on it, which is not a selected group.
- The two continuum hypotheses, which it does not prove, do not determine a connection. On a supplied structure group, representation and connection, the connection generates parallel transport, and the field strength, its curvature, measures the path dependence of transport around infinitesimal loops; flatness gives a path-independent identification of fibres only locally. The Yang–Mills action is a structural motivation, not a derivation from the axioms alone; its uniqueness is the standard result under an explicit truncation.
- A conditional input interface for gauge dynamics: if a compact group and a connection are supplied, Q12 studies the corresponding Yang–Mills term.
What is open
Q6a determines no complete gauge group. No valid result 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 supply the missing identification theorem. The selection of the $\mathrm{SU}(2)$ carrier and the uniqueness or minimality of $\mathrm{SU}(2)$ are open, as are the two continuum hypotheses, the construction of a connection from admissibility structure, the identification of gauge charges with conserved physical charges, and the uniqueness of the Yang–Mills action from the axioms alone.
Interpretive outlook
A structural reading, not a result. Gauge redundancy is read as the symmetry of a fibre that the observables do not resolve. A quantity constant on the fibres is a function of the observable output; a quantity only invariant under the fibre-preserving group is one when that group acts transitively on each fibre, and not in general. Gauge physics needs more: a group acting on the fibre, a rule for comparing fibres at different points, and a dynamics for that rule. Q6a locates where each would enter and derives none of them from non-injectivity together with a continuum description, and for the connection it shows that the continuum hypotheses do not determine it; it is a dictionary and a list of missing inputs, not a derivation of gauge fields.
Reference
Jérôme Beau. Toward Gauge Structure from Non-Injective Projection: Fibre-Invariant Charges, a Supplied SU(2) Carrier, and a Conditional Connection.