First-order repair and the larger-support residual

This is an existential obstruction to extending the theorem to the full compact class. It is not a claim that every generator outside the chosen $U$ produces a nonzero residual. For example, additional generators supported in another part of $\{g=1{\}}^{\circ }$ may also satisfy the cancellation.

The defect is classical, not a loop anomaly. Restricting the gauge domain does not refute the unrestricted obstruction.