Proposition: Reduction of G

In the formal universal-jet reading of Definition def:Gmaps (with the presentation of Theorem thm:M2): Then ${\mathcal{G}}_{\mathrm{formal}}(C)=0$ if and only if $C_N{|}_{U_N}=0$ for every $N$. Hence

Proof. Coherence gives $\mathcal{G}(C)={\sum }_{\gamma }{F}_{\gamma }(y,w){\partial }^{\gamma }\delta (y-w)$ with $F_{\gamma }={\sum }_{N,b}{C}_{N,b,\gamma }{\bar{a}}_b={\chi }_{\mathrm{I}}(w){\sum }_{N,b}{C}_{N,b,\gamma }{\hat{\bar{a}}}_b$. Taylor expansion of $F_{{\gamma }^{\prime }}(w+z,w)$ in $z$ gives the unique canonical form ${\sum }_{\gamma }{g}_{\gamma }(w){\partial }^{\gamma }\delta (y-w)$, with $g_{\gamma }={\sum }_{{\gamma }^{\prime }\ge \gamma }(-1{)}^{|{\gamma }^{\prime }-\gamma |}(\genfrac{}{}{0ex}{}{{\gamma }^{\prime }}{{\gamma }^{\prime }-\gamma }){{\partial }_y}^{{\gamma }^{\prime }-\gamma }{F}_{{\gamma }^{\prime }}{|}_{y=w}$. Proceed by descending induction on $|\gamma |$: the top coefficient is $F_{\gamma }(w,w)$. In the formal reading the $\sigma$-jet symbols (at $y$) and the $h$-jet and $\chi$ symbols (at $w$) are independent, and evaluation on the diagonal does not identify any two of them. So $F_{\gamma }(w,w)=0$ for all data if and only if ${\sum }_b{C}_{N,b,\gamma }{\hat{\bar{a}}}_b\equiv 0$ as a polynomial (the factor ${\chi }_{\mathrm{I}}(w)$ in ${\bar{a}}_b$ is independent), and then all its $y$-derivatives vanish too. Thus $\mathcal{G}(C)=0$ if and only if ${\sum }_b{C}_{N,b,\gamma }{\hat{\bar{a}}}_b\equiv 0$ for every $\gamma$. By the reduced expansion this means ${\sum }_b{C}_{N,b,\gamma }{c}_{m,b}=0$ for all $m$, i.e.\ $C_N(\cdot {)}_{\gamma }$ annihilates $U_N$. Monomials of different $N$ are disjoint (preparation order $2-s$, $h$-order $2-N+s$, checked in contact_space.py), so the conditions split by $N$.