The correct XI-G transfer statement

A source tensor and an observable kernel are different kinds of object. In particular, changing $J_3$ to $J_D$ does not by itself determine a change of free-shell class.

Let $\mathcal{K}$ be the space of compact distributional kernels on $M^3$ with two symmetric-tensor slots and one scalar slot, symmetric under interchange of the tensor slots. Set $P_m=\square +m^2$, with $m\ge 0$, and define

Compact support makes these pairings with smooth solutions meaningful.

For $w\in \mathcal{K}$, write

Let ${\mathcal{K}}_{\mathrm{mc}}\subset \mathcal{K}$ denote the kernels representing the predecessor's microcausal polynomial target. The reported XI-G conclusion, for its frozen kernel $w_0$ and its stated hypotheses, is

This is the kernel formulation of its stated non-admission into ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ XIG.