Proposition: The jet law is dynamical

Let $\xi \in {\mathfrak{g}}_U$. With $\hat{\varphi }$ and $\hat{h}$ held fixed,

and for every compactly supported smooth symmetric $w$,

Consequently the total second-order metric $k=k_{\mathrm{cl}}+\hat{k}$ transforms under the frozen action of ${\mathcal{G}}_U$ by the complete two-jet law of X-A,

modulo $\mathfrak{J}$ after smearing, although $\hat{h}$ itself is not transformed.

Proof. $\mathsf{S}$ depends on $h_{\mathrm{cl}}$ only through the linear term $-{\sigma }_{\epsilon }{\mathcal{Q}}^{(1)}[h_{\mathrm{cl}}]\hat{h}$, so $\mathsf{S}[h_{\mathrm{cl}}+K\xi ]-\mathsf{S}[h_{\mathrm{cl}}]=-{\sigma }_{\epsilon }{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}=-{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}$, because $\mathrm{supp}{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}\subset \mathrm{supp}\xi \subset U\subset \{{\sigma }_{\epsilon }=1\}$ by Lemma lem:fluct and (eq:separation). By (eq:key), $-{\mathcal{Q}}^{(1)}[K\xi ]\hat{h}=P_{\mathrm{gr}}({\mathcal{L}}_{\xi }\hat{h})-{\mathcal{L}}_{\xi }(P_{\mathrm{gr}}\hat{h})$. Both terms are compactly supported in $\mathrm{supp}\xi$. Apply ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}$ and Lemma lem:DRP: ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}{\mathcal{L}}_{\xi }\hat{h}={\mathcal{L}}_{\xi }\hat{h}$. This is (eq:jetlaw). For (eq:jetlaw-ideal), $\int w\cdot {{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}=\int {{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cdot {\mathcal{L}}_{\xi }{P}_{\mathrm{gr}}\hat{h}=-\int {\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w\cdot P_{\mathrm{gr}}\hat{h}=-\int P_{\mathrm{gr}}({\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w)\cdot \hat{h}$, by the last two statements of Lemma lem:DRP and the formal self-adjointness of $P_{\mathrm{gr}}$; ${\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w$ is smooth and supported in $\mathrm{supp}\xi$, so $\hat{h}(P_{\mathrm{gr}}{\mathcal{L}}_{\xi }{{\Delta }_{\mathrm{A}}}^{\mathrm{gr}}w)$ is a generator of $\mathfrak{J}$. The last statement combines (eq:jetlaw) with the frozen law $k_{\mathrm{cl}}\mapsto k_{\mathrm{cl}}+{\mathcal{L}}_{\xi }{h}_{\mathrm{cl}}+K\zeta +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$.