Lemma: Uniform angular Taylor expansion

For $M\in {\mathbb{N}}_0$, on the compact parameter ranges above,

uniformly for smooth $F$ with the displayed derivative bound on a fixed neighborhood of the Taylor segments.

Proof. Taylor-expand in $X$ through order $2M+1$. Odd angular moments vanish. At degree $2c$, the averaged term is

This follows by integration of the even monomials on the unit circle. Expand its coefficient in $(u,v)$ through degree $M-c$. Collecting $a+b+c=n$ gives (eq:appLn).

The transverse remainder is $O(R^{2M+2})$. Each longitudinal remainder is $O({\delta }^{M-c+1})$ multiplied by ${\delta }^c$. Thus every error is $O({\delta }^{M+1})$. The indicated $2M+2$ derivative bound controls all these terms.