Common kernels after a smooth residual test

For an arbitrary $G\in {C_c}^{\infty }(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$, pair the residual metric slot of $K_{\mathrm{ret}}$ against $G$. Denote the resulting attachment kernel by $L_G$:

Testing the residual slot is a distributional pairing. It neither changes the observable nor sets $z=a$ in branches where that equality is absent.