Lemma: Corrected completed-kernel support

The first completed test

has a compact kernel $D_1(x,z)$ satisfying

Its local completion terms are supported on $x=z$. Its coordinate term has $x\in K_f$ and $z\in C_f$.

Proof. The coordinate term has the factor $f(x)$ and a retarded coordinate kernel with $z\in J^{-}(x)$ and $t_z\ge 0$. The local terms have factors $g(x)$, derivatives of the smooth ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f(x)$, and derivatives of $\delta (x-z)$. Since $\mathrm{supp}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f\subset J^{-}(K_f)$, their first slot lies in $\mathrm{supp}g\cap J^{-}(K_f)$ and their second slot equals it. All these sets are compact.