Lemma: Channel separation by output locations

Retain the interaction vertex in $Y$ and the output metric variable in $Z$. The scalar–scalar channel has zero kernel germ on this patch.

Proof. Its residual metric variable equals the interaction vertex. Its kernel consequently has support in $z=y$, including any field derivatives. Since $\overline{Y}\cap \overline{Z}=\varnothing$, the restriction is zero.