Lemma: Exact squared-radius identity

For $j\ge 1$,

In particular,

Mixed derivatives with $l$ moving-base or frame parameters require at most $2j+l$ derivatives of $F$.

The Taylor expansion

has

Proof. The divergence theorem on a disk gives

Iteration and the elementary beta-integral identity give the displayed formula. Its right-hand side extends continuously to $h=0$. The derivative bound follows by integrating $(1-v{)}^{j-1}$.

For a moving base and frame, use $F_{\eta }(u)=w(p(\eta )+E(\eta )u)$ on a fixed bounded $u$-domain before applying the identity. Parameter derivatives cost at most $l$ additional weight derivatives. The remainder estimates follow from the one-variable Taylor formula applied to ${{\partial }_h}^q{\mathcal{M}}_F$. No inverse powers of $\sqrt{h}$ remain uncontrolled.