Definition on arbitrary test functions

Let $F_{IJ}(y_1,y_2,x)$ be an arbitrary compactly supported smooth test field of the appropriate component type on $M^3$. Define

The right-hand side pairs a distribution with a smooth function. It is well defined because $B$ is compactly supported. Any auxiliary cutoff equal to one near $\mathrm{supp}B$ gives the same value.

For clarity, the corresponding distribution before integrating out $y$ can be defined on a general test $G(y,y_1,y_2,x)$ on $M^4$:

Thus derivatives on the source deltas are transferred to the test before evaluation on the diagonal. They are not silently applied to a singular root kernel.

With

the distribution $\mathcal{K}$ is a finite sum of derivatives in $y_1,y_2$ of the pushforwards ${\iota }_{\ast }(cB)$. Its support is compact, so ${\pi }_y$ is proper on that support. The notation

is understood through (eq:lifted-test). It is this particular diagonal construction, not a claim that arbitrary pairs of distributions may be multiplied.