Proposition: Explicit companion remainder

Let $\rho$ be a smooth vector field on the slab, spatially compact on every slice, with $\mathrm{supp}\rho \subset \{t\ge \epsilon \}$, and let $P_m\varphi =0$. Then

with $[P_m,g]u=(\square g)u+2({\partial }^{\mu }g){\partial }_{\mu }u$. In particular ${\mathcal{A}}_1(K\rho ;\varphi )=0$ whenever

For $\rho ={\rho }_{\xi }[\gamma ]$, $\xi \in {\mathfrak{g}}_U$, one has $\mathrm{supp}{\rho }_{\xi }\subset J^{+}(\overline{U})\cap M_T$, disjoint from the ramp layer $\{0\le t\le \epsilon \}$; and (eq:companionsupport) holds under the causal plateau condition

Proof. On $\mathrm{supp}\rho$ one has ${\sigma }_{\epsilon }=1$ and ${\sigma }_{\epsilon }^{\prime }=0$, so (eq:legs) gives $\square X_1[K\rho ]=-{\partial }_{\lambda }{a_1}^{\lambda \mu }[K\rho ]=D^{\mu }K\rho =\square {\rho }^{\mu }$; both sides have zero initial data, hence $X_1[K\rho ]=\rho$. Next, X-A's pure-gauge identity for the unswitched first variation, ${\mathsf{Q}}_{K\rho }\psi =D_{\rho }{P}_m\psi -P_m(\rho \cdot \partial \psi )$, gives on the free shell $V_{1,g}[K\rho ]\varphi =-g{P}_m(\rho \cdot \partial \varphi )+({\partial }_{\mu }g){a_1}^{\mu \nu }[K\rho ]{\partial }_{\nu }\varphi$. Since $g\rho \cdot \partial \varphi$ is compactly supported in ${M_T}^{\circ }$, ${E_m}^{\mathrm{R}}{P}_m(g\rho \cdot \partial \varphi )=g\rho \cdot \partial \varphi$, and $g{P}_{m}u=P_m(gu)-[P_m,g]u$. Therefore

Insert into (eq:Aone) with $X_1[K\rho ]\cdot \partial \varphi =\rho \cdot \partial \varphi$. The first integrand of (eq:companion) vanishes outside $\mathrm{supp}f\cap \{g\ne 1\}\cap \mathrm{supp}\rho$; the second, after transposition onto ${E_m}^{\mathrm{A}}f$, vanishes unless some point of $J^{-}(\mathrm{supp}f)\cap \mathrm{supp}\mathrm{d}g$ lies in $\mathrm{supp}\rho$. This proves (eq:companionsupport). For ${\rho }_{\xi }={E_0}^{\mathrm{R}}D{s}_{\xi }$ the source $D{s}_{\xi }$ is compact in $U$, so finite propagation gives $\mathrm{supp}{\rho }_{\xi }\subset J^{+}(\overline{U})\cap M_T$, which lies in $\{t>\epsilon \}$ because $\overline{U}\subset \{t>\epsilon \}$. Under (CP) the compact set $J^{-}(\mathrm{supp}f)\cap J^{+}(\overline{U})\cap M_T$ lies in the open set $P_g$, where $1-g=0$ and $\mathrm{d}g=0$; both sets in (eq:companionsupport) meet $\mathrm{supp}{\rho }_{\xi }$ only inside it.