Bounded oscillation of the symplectization coordinate on circles about a puncture

Let $S=H^{-1}(1)\subset\mathbb R^{2n+2}$ be a strictly star-shaped level with a $d\lambda$-compatible complex structure $J$ on $\xi$, and let $u=(a,w):\dot D=D\setminus\{0\}\to\mathbb R\times S\cong\mathbb R^{2n+2}\setminus\{0\}$ be a finite-energy $\tilde J$-holomorphic curve on the punctured unit disc. Assume the gradient bound $|z|\,|\nabla u(z)|\le C$ for $0<|z|<r$, where $|\nabla u|=\|Da\|+\|Dw\|$. Then the coordinate $a$ has oscillation at most $6\pi C$ on every circle about the puncture of radius less than $\min(r,1)$: if $z\neq0$, $|z|<r$, $|z|<1$ and $|z'|=|z|$, then $|a(u(z))-a(u(z'))|\le 6\pi C$.

This is used in the zero-charge case of the asymptotic description of finite-energy punctured discs (Hofer; Wendl, Theorem 9.8): it lets one choose circles about the puncture on which $a$ lies entirely on one side of a given level.

Formalization Note $\mathbb R\times S$ is identified with $\mathbb R^{2n+2}\setminus\{0\}$ by $(a,x)\mapsto e^{2a}x$; $a$ is aCoord H and $|\nabla u|$ is gradNorm H u. The bound $6\pi C$ is a convenient explicit constant.