A loop crossing a disk transversally crosses it at only finitely many points

Let $E:\mathbb{R}^2\to\mathbb{R}^4$ be $C^1$ on an open neighbourhood of the closed unit disk $\overline{\mathbb D}$ and let $\gamma:\mathbb{R}\to\mathbb{R}^4$ be a $C^1$ loop of period $1$. Suppose that at every crossing $E(v)=\gamma(t)$ with $v\in\overline{\mathbb D}$,

(rows of a $4\times4$ matrix). Then the set of crossing points $\{v\in\overline{\mathbb D}:\exists t,\ E(v)=\gamma(t)\}$ is finite.

Proof. At a crossing the three vectors $\gamma'(t),\partial_1E(v),\partial_2E(v)$ are linearly independent, so the map $(t,v)\mapsto\gamma(t)-E(v)$ from $\mathbb{R}^3$ to $\mathbb{R}^4$ has injective differential there and is injective near $(t,v)$ (inverse function theorem / local left inverse). Hence crossings are isolated in $[0,1]\times\overline{\mathbb D}$, a compact set, so there are finitely many pairs $(t,v)$ with $t\in[0,1]$, and by periodicity every crossing point $v$ arises from such a pair.