Normal jets as a consequence, not a second proof

On a fixed patch, write $v=(\sigma ,\mathbf{w})$, and let $u$ denote all tangential variables, including $d$. A fixed local cutoff and the integral Taylor formula give

with smooth compactly supported ${\Phi }_j$ and finite-order seminorm bounds on a slightly larger patch. The residue kills the normal-vanishing terms. Lemma A already controls those terms and the trace term uniformly through $d=0$. This observation does not restrict $A(r)$ to the relation and is not used as an alternative proof of the radial estimate.