E2: the sheet, the two legs, and the scalar completion

Use $M={\mathbb{R}}^4$, $\eta =\mathrm{diag}(1,-1,-1,-1)$, $\square ={{\partial }_t}^2-\Delta$, $P_m=\square +m^2$, $m\ge 0$, $M_T=[0,T]\times {\mathbb{R}}^3$, ${M_T}^{\circ }=(0,T)\times {\mathbb{R}}^3$, $0<\epsilon <T$, the fixed ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ of XII-D, the switch $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ with $P_g=\mathrm{int}\{g=1\}$, the plateau

and the observation $f\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$, $K=\mathrm{supp}f$. Fields are ambient smooth configurations; no spatial decay is required. Parentheses include $\frac{1}{2}$. Put

The operators ${E_0}^{\mathrm{R}},{E_0}^{\mathrm{A}},{E_m}^{\mathrm{R}},{E_m}^{\mathrm{A}}$ are the zero-initial-data and zero-final-data slab inverses of $\square$ and $P_m$, defined on smooth sources by finite propagation, and agreeing with the Minkowski retarded and advanced operators on past or future compact sources BGP,XIB.