Remark: What the image theorem does not authorize

The kernel in (eq:Newtoncontact) is a distribution across $\mathbf{x}=\mathbf{z}$. An additional coincident-point restriction is not supplied. The zeroth derivative already has a noncompact dipole tail. Higher derivatives can include singular contact pieces as well as their off-diagonal tails.

The finite bound $|\gamma |\le m$ does not make the coefficient space finite dimensional: the smooth tangential functions $j_{\gamma }$ remain freely variable. Nor is the Newton map necessarily injective on sources; divergence-free source directions can be invisible.