Residual gauges and boundary discipline

There are nonzero homogeneous $Q$-solutions with general spacelike-compact Cauchy data. They are not compactly supported spacetime transformations of the frozen group.

Likewise, the retarded solution of $Q\xi =-\mathcal{F}z$ need not be compact in time. It does not prove that every configuration has a gauge-fixed representative inside XX-A's compact spacetime group. The gauge-fixed action is defined independently of that unproved reachability claim.

Compact admitted diffeomorphisms preserve the asymptotic rod labels. Their pullbacks preserve the timelike character of a classical scalar gradient and its chosen branch locally around the background. The gauge condition does not itself enforce a pointwise positive quantum lapse.