Why the edge condition is not changed

The gauge vector ${\xi }_{\ast }=\psi {\partial }_1$ is supported in the smaller observation region $O⋐U$. Thus it satisfies the original edge and initial-data conditions with room to spare. The witness does not exploit a gauge vector meeting $dg$, $\Sigma$, or the temporal cutoff.

Removing ${\xi }_{\ast }$ after the calculation would change the frozen problem. No such removal is made.