Proposition: Equation-ideal, coordinate, and switch terms

For an arbitrary compact interior parameter $\xi$, not necessarily in ${\mathfrak{g}}_p$,

All three displayed tests are smooth and compact. The first is explicitly in $P_m\mathcal{D}$.

Proof. The changed test is

Equation (eq:switchterms) gives $L_{1,g}[K\xi ]=Q_{K\xi }+{\Sigma }_{g,\xi }$. By Lemma lem:Q and $P_m{\mathsf{E}}_{\mathrm{A}}f=f$,

Substitute this identity and separate

Compactness of the gauge support makes $X_{\xi }{\mathsf{E}}_{\mathrm{A}}f$ compact. The other supports are compact because $f$, or the coefficients of $\Sigma$, are compact.