Proposition: Exact varied-action identity with all switch terms

For every configuration and every compactly supported $F$ as above,

Through second order,

where ${\mathfrak{L}}^{(1)}[h,\varphi ]=\frac{1}{2}({a_1}^{\mu \nu }[h]{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{b}_1[h]{\varphi }^2)$ and, for a $\kappa$-dependent $\varphi$, the displayed products are expanded coefficientwise.

Proof. By (eq:Lg) and the naturality (eq:nat1) of ${\mathfrak{L}}_q$,

Lemma lem:Z turns the second term into $g{\partial }_{\mu }{Z_F}^{\mu }={\partial }_{\mu }(g{Z}_F^{\mu })-({\partial }_{\mu }g){Z_F}^{\mu }$. The divergence of the compactly supported $g{Z}_F$ integrates to zero. The expansions are (eq:scalarpull) and Lemma lem:Z with ${\mathfrak{L}}_q={\mathfrak{L}}_{\eta }+\kappa {\mathfrak{L}}^{(1)}+O({\kappa }^2)$.