Lemma: Three-term test identity

For compact interior generators,

Every positive-order coefficient of the first term is in $P_m\mathcal{D}$.

Proof. The formal advanced solution of $E[q]s=c_{\epsilon }[q]$ is $s={\mathsf{E}}_{\mathrm{A}}{\mathcal{R}}_q{c}_{\epsilon }={\mathsf{E}}_{\mathrm{A}}{e}_{\epsilon }$. Indeed,

Its coefficientwise advanced uniqueness follows successively from free advanced uniqueness and the compact differential coefficients of $L[q]$.

Writing $s^{\prime }={\mathsf{E}}_{\mathrm{A}}{e}_{\epsilon }[q^{\prime }]$, the defect definitions give

The right-hand side is a compact formal source: the coordinate defect has compact observation-test support, and the switch defect has compact generator coefficient support. Advanced uniqueness yields

Apply $P_m$, and use $P_{m}s=e_{\epsilon }$. Every coefficient of $(F^{\ast }-1)s$ is smooth and supported in the fixed compact generator support.