Definition: Contact collapse

For $C\in {\mathcal{C}}_{\mathrm{coh}}$, with the physical reduced coefficient ${\bar{a}}_b$ of Definition def:Gmaps,

There is no second ${\chi }_{\mathrm{I}}$ outside ${\bar{a}}_b$: the single switch factor already sits inside ${\bar{a}}_b={\chi }_{\mathrm{I}}{\hat{\bar{a}}}_b$. The second equality holds because $\int dwF(y,w)({\partial }^{\gamma }\delta )(y-w)=({{\partial }_w}^{\gamma }F)(y,y)$ and $\mathcal{G}(C)=\sum C\bar{a}{\partial }^{\gamma }\delta$. ${{\partial }_w}^{\gamma }$ distributes by Leibniz over the factors of ${\bar{a}}_b={\chi }_{\mathrm{I}}(w){\hat{\bar{a}}}_b(y,w;h)$ (the $h$-jet and ${\chi }_{\mathrm{I}}$); the ${\sigma }_{\epsilon }$-jet sits at $y$ and is not hit (XVI-L, prop:Ljet(2)). $\mathrm{Collapse}:C\mapsto E_C$ maps into local tensors that are linear in the $h$-jet, with coefficients that are monomials in the external jets. Four computational readings are used, together with the Stage 2 fixed-data region ledger:

• formal: ${\sigma }_{\epsilon }$- and ${\chi }_{\mathrm{I}}$-jets are independent symbols;

• {$\chi$-plateau}: ${\chi }_{\mathrm{I}}\equiv 1$ locally and ${\sigma }_{\epsilon }$ formal;

• preparation plateau: ${\sigma }_{\epsilon }\equiv 1$ locally and ${\chi }_{\mathrm{I}}$ formal (variable switch; only $J_3$ survives);

• double plateau: ${\chi }_{\mathrm{I}}\equiv 1$ and ${\sigma }_{\epsilon }\equiv 1$ locally, so only $J_3$ terms survive and the coefficients are constant.