XI-F's representation of the cubic kernel, copied

${\mathcal{C}}_3(f,g)[\gamma ,\varphi ]={\int }_{[0,T]\times {\mathbb{R}}^3}{\beta }_{\mu \nu }(y){J_3}^{\mu \nu }[\gamma ](y){\mathrm{d}}^{4}y$, with $\beta ={{\mathsf{E}}_0}^{\mathrm{A}}(\mathcal{P}\alpha )$ and $\mathrm{supp}\beta \subset {\mathcal{D}}_f$. The degree-$(2,1)$ kernel has the form

The factor $\theta (y^0)$ is forced: ${{\mathsf{E}}_0}^{\mathrm{R}}$ truncates every source at $t=0$, and the Einstein–Hilbert density carries neither $g$ nor ${\sigma }_{\epsilon }$.

Here $\alpha =\alpha [f,g,\varphi ]$ is XI-F's smooth kernel of ${\mathcal{A}}_1$, linear in $\varphi$, and $j_3$ is the constant-coefficient bilinear local kernel of $J_3=2{\mathcal{H}}_2$, so that for all smooth ${\gamma }_1,{\gamma }_2,\varphi$ on $M$,

where $J_3(\cdot ,\cdot )$ is the symmetric bilinear form with $J_3[\gamma ]=J_3(\gamma ,\gamma )$. The kernel $u$ is compactly supported on $M^3$, in $\{y_1=y_2\in {\mathcal{D}}_f\cap \{y^0\ge 0\}\}\times {\mathcal{D}}_f$; no spatial decay of ${\gamma }_a$ is needed in (eq:upair), exactly as in XI-F's witness.

Theorem F1 (XI-F). For every frozen switch and every mass there are admissible observation data for which $\pm \mathrm{d}t$ lies in the wavefront set of this kernel at $\Sigma$. The kernel leaves ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. The failure is not finite rank and not a loop.

The data of that theorem, called the F1 data below, are: $t_g:=\min \{t:(t,\mathbf{x})\in \mathrm{supp}g\}>0$; $f\in {C_c}^{\infty }({M_T}^{\circ })$, $f\ge 0$, $f\not\equiv 0$, $\mathrm{supp}f\subset (0,t_g)\times {\mathbb{R}}^3$, $\cos (mt)>0$ on $\mathrm{supp}f$; $\varphi ={\varphi }_m$ with ${\varphi }_m=\sin (mt)/m$ for $m>0$ and ${\varphi }_m=t$ for $m=0$. XI-F's Step 1 computes, for these data,

the last because ${\beta }^{0i}={{\mathsf{E}}_0}^{\mathrm{A}}(-\frac{1}{2}{\sigma }_{\epsilon }{\partial }_i{b}^0)$ with $b^0$ of compact spatial support per slice. $B$ is smooth on $[0,T]$ and vanishes near $T$.

The open remark (XI-F). On the free shell, a single-slot boundary trace equals a bulk functional with a source flat at $\Sigma$, hence has a smooth representative. The boundary term carried by $u$ is a bilinear form in the Cauchy data of both metric slots, weighted by $\beta {|}_{\Sigma }$. Whether it has a representative satisfying the microcausal exclusion after both slots are restricted to free data is not proved.