The source response map

Before composing either gravitational leg, set

with the contracted metric index understood in the source kernel.

This is the untested kernel of $r_h(b)=\mathcal{P}{E}_0^{\mathrm{R}}({\mathcal{J}}_{\mathrm{D}}^{\prime }[G]h)(b)$. It retains the residual variable.

On the separated patch $b\in B$, $a\in A$, the endpoints $b,a$ never coincide. Moving the external derivatives in (eq:Qincidence) outside the integral writes (eq:responsekernel) as a finite sum of derivatives, of total order at most two, of

The differentiated coefficients are included in this identity; they are not omitted remainder terms.