Residual algebra and the action of

X-D proved that such layer functionals converge, as $\epsilon \to 0$, to the homogeneous solution with velocity $-a^{0\mu }[h]{|}_{\Sigma }$. That limit is not taken here. It is recorded because it is the place where a sharp limit of a quantum-metric dressing will differ in kind from X-D's smooth scalar tests: it is a trace of $\hat{h}$ on $\Sigma$.