The asymptotic winding at $\infty$ is one more than the winding of $\pi\partial_s u$ along large circles

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level, $J$ a $d\lambda_0$-compatible complex structure on $\xi$, and $\tilde J$ the induced almost complex structure on $\mathbb{R}^4\setminus\{0\}\cong\mathbb{R}\times S$. Let $U\subset\mathbb{C}$ be open with $\{z:|z|>R\}\subset U$, and let $u$ be smooth, nonvanishing and $\tilde J$-holomorphic on $U$. Suppose that the $\xi$-component $\pi\,\partial_s u$, read in Hryniewicz's global frame of $\xi$, has winding number $k$ along every circle $t\mapsto\rho e^{2\pi it}$ with $\rho>R$. Then in the positive cylindrical coordinates $(s,t)\mapsto e^{2\pi(s+it)}$ at $\infty$,

This expresses the asymptotic winding at the positive puncture $\infty$, which is defined in cylindrical coordinates, through the winding of $\pi\partial_s u$ along large circles in the plane coordinate $z$.

Formalization Note $\mathbb{R}^4$ is V 1, $\tilde J$ is Jtilde H J, and $\pi\partial_s u$ in the global frame is piDs H u : ℂ → ℂ. The circles are ccwLoop ρ t $=\rho e^{2\pi it}$. The conclusion is HasWindInfAt H u cylInf 1 (k + 1): for all large $s$ the loop $t\mapsto\pi\partial_s w(s,t)$ of $w=\mathrm{pr}\circ u\circ\mathrm{cylInf}$ has winding number $k+1$. No energy bound or asymptotic hypothesis is assumed.