Lemma: Normal form of a reconstruction

For every $T\in {\mathcal{D}}^{\prime }(X)$ and $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$,

The same identity holds for ${\mathsf{Rec}}^{\circ }$ with $T^{\circ }$, ${{\mathsf{E}}^{\circ }}_{\gamma }$, and ${\beta }_{\gamma }(a){|}_{X^{\circ }}$.

Proof. For $\varphi \in \mathcal{D}(U)$,

by the Leibniz rule. Since $(-1{)}^{|\alpha |}=(-1{)}^{|\beta |}(-1{)}^{|\alpha -\beta |}$, each term is $(-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\alpha }{\beta })⟨{\mathsf{E}}_{\alpha -\beta }(({{\partial }_q}^{\beta }{a}_{\alpha })(\cdot ,0)T),\varphi ⟩$. Substituting $\gamma =\alpha -\beta$ and summing over $\alpha$ gives (eq:normalform) with the coefficient (eq:beta). The punctured case is the same computation on $U^{\circ }$.