Lemma: Necessary source conservation

If ${\mathcal{J}}_g({\varphi }_0)$ is nonempty, then

For the source-response subclass ${\gamma }_0=0$, this is already the compatibility condition for

Proof. The linearized Bianchi identity ${\partial }_{\mu }{G_{\mathrm{lin}}}^{\mu \nu }=0$ follows directly from (eq:Glin). Also ${\partial }_{\mu }{T_0}^{\mu \nu }=(P_m{\varphi }_0){\partial }^{\nu }{\varphi }_0=0$. Taking the divergence of (eq:source-jet) proves the assertion in that subclass.

For arbitrary ${\gamma }_0$, evaluate the order-one coefficient of the exact Noether identity on a purported jet. Since the metric Euler coefficients vanish, its left side vanishes. Since $P_m{\varphi }_0=0$, the order-zero scalar Euler term vanishes. The order-zero value of ${\mathcal{B}}_{\nu }$ is precisely $({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }[{\varphi }_0]$. Equation (eq:noether) therefore forces (eq:compatibility). This argument permits arbitrary vacuum leading metric data and does not presume a harmonic section.