Lemma: Vertex identity

X-A's first-order switched vertex is the metric variation of the switched density:

so $\delta V_{1,g}/\delta h_{\mu \nu }=-\frac{1}{2}g{T}^{\mu \nu }[\varphi ]$ and $\delta V_{1,g}/\delta \varphi =-{\mathsf{L}}_{1,g}[h]\varphi$.

Proof. $-\frac{1}{2}{h}_{\mu \nu }{T}^{\mu \nu }=-\frac{1}{2}{h}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +\frac{1}{4}H(\partial \varphi \cdot \partial \varphi -m^2{\varphi }^2)=\frac{1}{2}{a}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -\frac{1}{2}{m}^{2}b{\varphi }^2$ by (eq:density). The $\varphi$-derivative is X-C's Hessian ledger $V^{\prime \prime }=-\mathsf{L}$, extended here to the mixed block.