Proposition: The gravitational dictionary is not implied

Symmetry, conservation, zero trace, and the local Ward theorem above do not imply

No metric equation or gravitational dictionary has been constructed.

Proof. The declared background is the fixed Minkowski metric, whose Einstein tensor is zero. The stress tensor is nonzero: if it vanished, every compact translation charge would vanish, contrary to its nonzero action on the scalar established in Lemma lem:cauchyaction. Thus the displayed equation is not an identity in the declared model. It would require an additional metric, coupling prescription, and dynamical theorem.