C. Jet bounds after contact

Write $s:=|a|+|b|$ (source-slot order), $v:=|p|+|q|$ (vertex order), so $N_{\lambda }=s+v$ for an ordered placement contributing to label $\lambda$, and recall from L.1–L.2:

• ${L^{pq}}_{ij}(w;h)$ is linear in the residual-metric jet of order $2-v$ (XVI-K, Proposition prop:hess(1); Remark after Definition def:La);

• the preparation jet of ${\sigma }_{\epsilon }$ sits in $P_A(y)$, of order $2,1,0$ for $A=C_2,M_2,J_3$ respectively (Theorem thm:budget), and is not differentiated by ${\partial }_w$;

• ${\chi }_{\mathrm{I}}(w)$ sits undifferentiated in $a_{\lambda }$ before contact.