Lemma: Advanced solution and compact completed test

For $d\in \mathcal{D}(M)[[\kappa ]]/({\kappa }^3)$, the equation ${\mathsf{E}}_g[q]w=d$ has a unique advanced-type solution,

Proof. Coefficientwise the equations are

Every right side is compact because the interaction coefficients are compact. Applying ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ recursively produces the stated series.

A homogeneous advanced-type solution vanishes on a sufficiently late Cauchy surface; Cauchy uniqueness makes it zero. This proves uniqueness inductively. Finally, $P_m{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}{\mathcal{R}}_{q}d={\mathcal{R}}_{q}d$, since the latter is a compact test.