Proposition: Compact gauge-fixed residual subgroup

The condition

has only the zero solution. Hence invariance tested only against compact gauge-fixed residual parameters does not certify the full compact geometric gauge identity.

Proof. Compact spacetime support lies to the future of a Cauchy surface on which both Cauchy data vanish. Uniqueness for a normally hyperbolic operator gives $\xi =0$. This uses the checked globally hyperbolic slab and the normally hyperbolic Cauchy theorem BGP. Testing an identity only at the zero parameter is vacuous.