Proposition: Stress identities do not imply Einstein dynamics

Symmetry, conservation, and zero trace of a tensor on a prescribed metric do not imply

Proof. In any regular Minkowski chart of the host, choose a nonzero constant null covector $k$ and set $t_{\mu \nu }=k_{\mu }{k}_{\nu }$. It is symmetric, constant and conserved, nonzero and traceless. The local Einstein tensor is zero. The displayed equation fails for $G_{\mathrm{N}}\ne 0$. This local counterexample does not require an assertion about a distributional stress tensor on the singular worldsheet.