Proposition: Local support reduction

On $A\times B\times Z$,

Moreover $K_6,K_8,K_9$ have support in $\{a=z\}$. The mixed derivatives of $B_{10},B_{11}$ are identically zero.

These are distributional support statements, not evaluations of singular kernels at $(u_0,x_0,u_0)$.

Proof. In $B_1$, the scalar and one metric slot share the inner vertex. Selecting the other metric slot for contraction forces $b=z$; selecting the metric at the scalar vertex forces $a=b$. Thus

The same slot-location argument applies to $B_3$, giving the same support inclusion.

All three observable slots in $B_2$ share one vertex, so

Since $\overline{A}\cap \overline{B}=\varnothing$ and $\overline{Z}\subset A$, these supports miss $A\times B\times Z$. Differentiating a distribution does not enlarge its support, so the argument covers all derivative placements.

Both metric slots of $B_6$ share their local vertex. Both metric slots of $B_8,B_9$ share the source vertex, including when the source differentiates either slot. Hence their mixed kernels are supported on $a=z$.

The last assertion follows either directly from the absence of a metric argument or from (eq:allmixed).