Lemma: The defect is a tensor in first derivatives of the graviton

With ${{\Gamma }^{(1)}}_{\lambda \mu \nu }[h]=\frac{1}{2}({\partial }_{\mu }{h}_{\lambda \nu }+{\partial }_{\nu }{h}_{\lambda \mu }-{\partial }_{\lambda }{h}_{\mu \nu })$,

Proof. For any smooth symmetric $h$ and vector field $\xi$,

by expanding $K(h\cdot \xi )={\partial }_{\mu }(h_{\nu \lambda }{\xi }^{\lambda })+{\partial }_{\nu }(h_{\mu \lambda }{\xi }^{\lambda })$ and comparing with ${\xi }^{\lambda }{\partial }_{\lambda }{h}_{\mu \nu }+h_{\lambda \nu }{\partial }_{\mu }{\xi }^{\lambda }+h_{\mu \lambda }{\partial }_{\nu }{\xi }^{\lambda }$. Since $\mathrm{supp}(h\cdot \xi )\subset \mathrm{supp}\xi \subset U$, X-D's representative gives $d_1[K(h\cdot \xi )]=P_m((h\cdot \xi {)}^{\mu }{\partial }_{\mu }{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)$, linearly in $h$. Substituting $\hat{h}$, the kernel of ${\hat{A}}_1[K(\hat{h}\cdot \xi )]$ lies in $(P_m\otimes 1){{\mathcal{D}}^{\prime }}_c$, and Lemma lem:ideal(i) applies.