Input statement: XII-E and XII-F: the coefficient and its witness

The scalar-linear coefficient is ${{\Phi }^{(2)}}_{\mathrm{D}}={{\mathcal{A}}_2}^{\mathrm{sc}}[f,g](\gamma ,0;\varphi )-{\mathcal{C}}_{3,\mathrm{D}}$, with the slot $r_{\mathrm{D}}=-u_{\mathrm{D}}=\mathcal{P}{E}_0^{\mathrm{R}}{\mathcal{J}}_{\mathrm{D}}$; the full displayed coefficient is $\mathcal{O}={{\Phi }^{(2)}}_{\mathrm{D}}+{\mathcal{A}}_1[f,g](r_g[\varphi ];\varphi )$, $r_g=-\frac{1}{2}\mathcal{P}{E}_0^{\mathrm{R}}(g{T}_0[\varphi ])$. For $\xi$ compact in the plateau and $P_m\varphi =0$,

with $s_{\xi }={\mathcal{L}}_{\xi }\gamma +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$, ${\rho }_{\xi }={E_0}^{\mathrm{R}}D{s}_{\xi }$. The companion vanishes under the causal plateau condition (CP). On the compact free-shell tuple $m=0$, $\psi =t^2+(x^1{)}^2$, $\gamma =\psi \eta$, $\varphi =\psi$, $b=D\gamma =(-2t,2x^1,0,0)$, with ${\xi }_{\ast }=a(t)\zeta (\mathbf{x}){\partial }_t$ and (CP) holding,

A rule with ${Y^{\gamma }}_{\xi }={Y^{\varphi }}_{\xi }=0$ changes the lift by zero. No cancelling map was supplied XIIE,XIIF.