Theorem: The second-order law is a response identity

For $\xi \in {\mathfrak{g}}_p$, with $r,\varphi$ held fixed,

For a smooth free $r$, the second term is zero. For the free quantum field it is an equation-ideal term after admissible compact smearing.

Proof. The background dependence of the source is linear:

The second equality holds because the operator coefficients are supported in $\mathrm{supp}\xi \subset U$, where $\sigma =1$. By (eq:key),

The field ${{\mathcal{L}}_{\xi }}^{c}r$ has compact support. Apply Lemma lem:leftinverse.

For a compact smooth test $A$,

The inner test is smooth and supported in $\mathrm{supp}\xi$. The bilinear version required by the observable is proved in Section sec:ideal.