Remark: What sits where

The preparation jet of ${\sigma }_{\epsilon }$ (order $2,1,0$ for $C_2,M_2,J_3$) sits in $P_A(y)$, hence in $a_{\lambda }$. The residual metric jet of $h$ sits in ${L^{pq}}_{ij}(w;h)$, with $|r|=2-|p|-|q|$ (XVI-K, Proposition prop:hess(1)). The switch ${\chi }_{\mathrm{I}}(w)$ is undifferentiated (XVI-K, Proposition prop:hess(5)). No silent integration by parts is performed: the printed representatives of ${\ell }_3$ and of ${\mathcal{J}}_{\mathrm{D}}$ are kept.