Proposition: Decaying leading inverse

Let $\mathcal{X}=N\mathcal{S}({\mathbb{R}}^3)$. For Schwartz ${\Pi }_{\mathrm{tr}},A_i$, the unique descriptor in ${\mathcal{X}}^4$ sending the leading conditions to zero is

Equivalently,

Proof. The required descriptor is $-{\mathcal{M}}^{-1}\chi$. Newton inversion gives existence. The difference of two descriptors is smooth harmonic and decays, hence vanishes.