Input statement: XII-D: the six handed objects

On the sheet of Section sec:sheet the following are constructed in XIID and used here without re-proof.

1. The response

   $$u_{\mathrm{D}}[\gamma ]=-\mathcal{P}{E}_0^{\mathrm{R}}({\mathcal{J}}_{\mathrm{D}}[\gamma ]),{\mathcal{J}}_{\mathrm{D}}={\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon },L{u}_{\mathrm{D}}={\mathcal{J}}_{\mathrm{D}},L=-\square \mathcal{P}.$$

   It is smooth, cutoff independent, and supported in $J^{+}(\mathrm{supp}{\mathcal{J}}_{\mathrm{D}}[\gamma ])\cap M_T$.

2. The completed two-leg insertion

   $${\mathcal{C}}_{3,\mathrm{D}}(f,g)[\gamma ,\varphi ]={\mathcal{A}}_1[f,g](u_{\mathrm{D}}[\gamma ];\varphi ),$$

   with both the coordinate and the switched scalar leg (Section sec:sheet).

3. The first root ${\alpha }_{f,g}[\varphi ]$, its transpose ${\beta }_{f,g}[\varphi ]=-{E_0}^{\mathrm{A}}(\mathcal{P}{\alpha }_{f,g}[\varphi ])$, and the compact collar root $a_{f,g}[\varphi ]=\vartheta {\beta }_{f,g}[\varphi ]$, with the derived identity ${\mathcal{C}}_{3,\mathrm{D}}=\int a_{f,g}[\varphi ]\cdot {\mathcal{J}}_{\mathrm{D}}[\gamma ]$.

4. The admission theorem: the polarized kernel $t_{\mathrm{D}}$ of ${\mathcal{C}}_{3,\mathrm{D}}$ is compact and $\mathrm{WF}(t_{\mathrm{D}})\subset Z_3$, hence ${\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.

5. The class $Q_H{\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{E}}_H$, the regular ordering correction ${\alpha }_d{\mathcal{C}}_{3,\mathrm{D}}={\mathcal{C}}_{3,\mathrm{D}}+\hslash L_d$, and ordering stability of the enlarged named sector.

6. The signed ledger. For $\xi \in {\mathfrak{g}}_U$ and every smooth $\gamma ,\varphi$, with $s_{\xi }[\gamma ]={\mathcal{L}}_{\xi }\gamma +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ and ${\rho }_{\xi }[\gamma ]={E_0}^{\mathrm{R}}D{s}_{\xi }[\gamma ]$,

   $$\begin{array}{rl}{u}_{\mathrm{D}}[\gamma +K\xi ]-u_{\mathrm{D}}[\gamma ]& =-s_{\xi }[\gamma ]+K{\rho }_{\xi }[\gamma ]-\mathcal{P}{E}_0^{\mathrm{R}}{{\mathcal{L}}_{\xi }}^{\mathrm{dens}}(A\gamma ),\\ {\Delta }_{\xi }{\mathcal{C}}_{3,\mathrm{D}}+{\mathcal{A}}_1(s_{\xi };\varphi )& ={\mathcal{A}}_1(K{\rho }_{\xi };\varphi )-{\mathcal{A}}_1(\mathcal{P}{E}_0^{\mathrm{R}}{{\mathcal{L}}_{\xi }}^{\mathrm{dens}}(A\gamma );\varphi ),\end{array}$$

   together with the causal plateau proposition: if $P_m\varphi =0$ and $g=1$ on a neighbourhood of $J^{-}(\mathrm{supp}f)\cap \mathrm{supp}(K{\rho }_{\xi })$, then ${\mathcal{A}}_1(K{\rho }_{\xi };\varphi )=0$.

XII-D refused: an interacting BRST class, the complete second-order relational construction, a loop value, a quantum retarded product, a state, and a sharp limit.