Theorem: The cubic response obeys the complete second-order law

For every smooth $\gamma$ and $\xi \in {\mathfrak{g}}_U$,

If $A\gamma =0$ on $U$, the last term vanishes and

Proof. By the proof of Proposition prop:Etot, $J_3[\gamma +K\xi ]-J_3[\gamma ]={{\mathcal{L}}^{\mathrm{dens}}}_{\xi }(A\gamma )-A{s}_{\xi }$. Apply $-{G_{\mathrm{gr}}}^{\mathrm{R}}=\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ and Lemma lem:GA ($s_{\xi }$ is compactly supported in $U$): $-{G_{\mathrm{gr}}}^{\mathrm{R}}(-A{s}_{\xi })=s_{\xi }-2\mathcal{R}{{\mathsf{E}}_0}^{\mathrm{R}}D{s}_{\xi }=s_{\xi }+K{\zeta }_{\mathrm{gf}}[s_{\xi }]$. If $A\gamma =0$ on $U\supset \mathrm{supp}\xi$, then ${{\mathcal{L}}^{\mathrm{dens}}}_{\xi }(A\gamma )=0$. The divergence statement is Proposition prop:gfprops(c).