Lemma: Microcausal linear kernels have zero charge

If $\ell \in {\mathcal{S}}_1$ has no nonzero nonspacelike wavefront covector, then $\mathfrak{b}(\ell )=0$.

Proof. Write $\ell =m+q_a$. Both $\ell$ and $m$ avoid nonspacelike wavefront covectors, so their difference $q_a$ does too. The supplied smooth response $b={G_{\mathrm{A}}}^{\mathrm{chart}}a$ is smooth near $\Sigma$. Lemma lem:surfaceWF forces its trace to vanish. The intrinsic trace formula gives the conclusion.