Lemma: The intermediate coordinate weight on this patch

In (eq:K8map), replacing the intermediate factor ${\sigma }_{\epsilon }(s)$ by one leaves the restricted kernel unchanged.

Proof. The source incidence enforces $v=a=z$, with $a^0>\epsilon$. The support of the two retarded kernels enforces

Hence $s^0\ge a^0>\epsilon$, where ${\sigma }_{\epsilon }(s)=1$ and all its derivatives vanish.

Differentiation does not enlarge distributional support. Thus every term containing ${\sigma }_{\epsilon }(s)-1$ or a derivative of that factor has empty support on this patch.