Proposition: Conservation and trace do not source the host

A conserved traceless tensor need not satisfy Einstein's equation for the prescribed host.

Proof. For a nonzero constant ${\rho }_0$, let

The forms are parallel, so $\tau$ is conserved; its $G$-trace vanishes. The regular host is locally flat and has zero Einstein tensor. Thus $\mathrm{Ein}(G)={\kappa }_{\mathrm{grav}}\tau$ fails for nonzero ${\kappa }_{\mathrm{grav}}$ at zero cosmological constant. No distributional source equation at the defect is constructed.