The dynamical jet law

The tensor transformation law of the quantum fluctuation under background diffeomorphisms, which X-E's closing remark of E4 assigned to a second-order quantum metric “transforming by ${\mathcal{L}}_{\xi }\hat{h}$”, is a consequence of ${\mathcal{Q}}^{(1)}[K\xi ]=[{\mathcal{L}}_{\xi },P_{\mathrm{gr}}]$ and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{P}_{\mathrm{gr}}=1$ on compact supports. No field is postulated.