Proposition: A large-support witness

There are compactly supported gauge parameters after ${\Sigma }_0$ for which (eq:defect) is nonzero, including parameters that are constant on a neighbourhood of $\mathrm{supp}g$.

Proof. Choose $\chi \in {C_c}^{\infty }(M_{+}^{\circ })$ with $\chi =1$ on a neighbourhood of $\mathrm{supp}g$, and let

for a nonzero constant vector $c$. Then $K\xi =0$ on a neighbourhood of $\mathrm{supp}g$, hence

The switched scalar correction has zero variation, while the coordinate correction changes by $\xi$. Therefore

Choose free Cauchy data such that $c^{\mu }{\partial }_{\mu }{\varphi }_0$ is nonzero at some point of $U$. Such data exist, for example by taking an appropriate real Klein–Gordon plane wave and restricting its smooth Cauchy data. A test function supported sufficiently close to that point gives a nonzero smeared variation.

This $\xi$ is not supported in $U$: its nontrivial cutoff to zero takes place outside the region on which it is constant, and that region contains $\mathrm{supp}g$.