Lemma: Uniform coefficient regularity

For each finite $N$, $\Gamma$ has a one-sided $C^N$ extension to $d=0$, and a local $C^N$ extension across it, with

Adding $l$ spectator derivatives requires at most $2N+l$ weight derivatives.

Proof. Appendix app:circle proves that the $j$-th derivative of a circle average in its squared radius is controlled by $2j$ derivatives of its weight. In (eq:fiber), that squared radius is $(S^2)\theta (1-\theta )$; the longitudinal point and frame are smooth on the non-tip domain. A total propagation derivative of order $N$ therefore uses at most $2N$ weight derivatives. The estimate is uniform at $\theta =0,1$.

For a two-sided extension, choose $a_1,\dots ,a_{N+1}$ such that ${{\sum }_{j=1}}^{N+1}{a}_j(-j{)}^k=1$ for $0\le k\le N$. For negative small $d$, use

The normal and mixed jets match through order $N$, with the same type of bound.