Corollary: On the carrier

For $\xi \in {\mathfrak{g}}_U$, the functional ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$ belongs to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and in ${\mathcal{E}}_H$

Under (CP) the first class vanishes. The second is the class of a compact microcausal degree-two kernel in $(b,\varphi )$; it vanishes on configurations with $D^{\ast }b=0$ on $U$ and is not asserted to vanish in ${\mathcal{E}}_H$.

Proof. Shifting the metric slots of a compact microcausal kernel by the smooth configuration $K\xi$ smears slots against a smooth function and preserves compactness and the cone bound (XII-A), so ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ once ${{\Phi }^{(2)}}_{\mathrm{D}}\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ (Theorem thm:admission). On ${\mathcal{C}}_{\mathrm{sol}}$, $P_m\varphi =0$ and (eq:shellrow) hold, so Theorem thm:plateau(a) is an identity of restrictions to the shell, hence of classes. The kernel of the $b$-row is $\beta$'s kernel $B_{f,g}$ (XII-D, $\mathrm{WF}\subset Z_2$) acted on by differential operators and localized in $U$, hence compact with $\mathrm{WF}\subset Z_2$. Under (CP) the companion vanishes on every configuration with $P_m\varphi =0$, in particular on ${\mathcal{C}}_{\mathrm{sol}}$.