Definition: Three graph maps and two named variants

Write ${\bar{a}}_b:={\sum }_{\lambda }{R}_{\lambda b}{a}_{\lambda }$ for the reduced coefficient of a basis kernel $b$, using the physical (switched) $a_{\lambda }={\chi }_{\mathrm{I}}{\hat{a}}_{\lambda }$ of Definition def:aGeneral. Equivalently ${\bar{a}}_b={\chi }_{\mathrm{I}}(w){\hat{\bar{a}}}_b$ with the switch-free factor ${\hat{\bar{a}}}_b:={\sum }_{\lambda }{R}_{\lambda b}{\hat{a}}_{\lambda }$ (built from the table $\hat{a}$). For $C\in {\mathcal{C}}_{\mathrm{coh}}$ put

(equivalently $\sum C_{N,b,\gamma }{\chi }_{\mathrm{I}}(w){\hat{\bar{a}}}_b^{\mu \nu }(y,w)({\partial }^{\gamma }\delta )(y-w)$; the preferred form matches $E_C$ below).

1. ${\mathcal{G}}_{\mathrm{formal}}$, the formal universal-jet reading. External data $\sigma$ (preparation), $\chi$ (switch) and the field $h$ are arbitrary smooth functions. ${{\mathcal{T}}_{\mathrm{collision}}}^{\mathrm{formal}}$ is the space of collision tensors ${\sum }_{\gamma }{F}_{\gamma }(y,w){\partial }^{\gamma }\delta (y-w)$ whose coefficients are polynomials in the jet symbols: $F_{\gamma }\in \mathbb{Q}[\sigma \text{-jet}(y)]\otimes \mathbb{Q}[h\text{-jet}(w),\chi (w)]\otimes {\mathbb{Q}}^{10}$ (output components). ${\mathcal{G}}_{\mathrm{formal}}(C)=0$ means vanishing for all smooth data. Since every finite jet at a point is realised by a polynomial function, this is equivalent to vanishing as a polynomial identity in independent jet symbols.

2. ${\mathcal{G}}_{\mathrm{background}}$ (the XVI-L specialization): $h=(w^1{)}^2\eta$, with $\sigma ,\chi$ arbitrary; ${{\mathcal{T}}_{\mathrm{collision}}}^{\mathrm{background}}$ has coefficients in $\mathbb{Q}[\sigma \text{-jet}(y)]\otimes \mathbb{Q}[w^1]\chi (w)\otimes {\mathbb{Q}}^{10}$.

3. ${\mathcal{G}}_{\mathrm{fixed}}$: the actual fixed preparation ${\sigma }_{\epsilon }$ and switch ${\chi }_{\mathrm{I}}$, with $h$ arbitrary; ${{\mathcal{T}}_{\mathrm{collision}}}^{\mathrm{fixed}}$ is the space of $h$-linear maps into ${\mathcal{D}}^{\prime }({\mathbb{R}}^4\times {\mathbb{R}}^4)$.

Named variants of the formal reading are time-only, where $\sigma$ depends on time only and only $\sigma ,{\partial }_0\sigma ,{{\partial }_0}^2\sigma$ survive, and plateau, on $\mathrm{supp}{\chi }_{\mathrm{I}}\cap \{{\sigma }_{\epsilon }=1\}$, where only the $J_3$ sector survives.