Formal immersions of the circle in the plane are classified by the winding number

$T{S}^1$ is trivial with the global frame $({\mathrm{\partial }}_{\theta })$: the field is smooth and nowhere zero at every point of the circle, so it is a global frame by [F3]. Hence $E=V(T{S}^1,{\epsilon }^2)\cong S^1\times S^1$ by the triviality clause of [F1], and its fibres are the isometric injections $T_{\theta }{S}^1\to {\mathbb{R}}^2$, identified with $S^1$ by [F4].

Under the trivialisation of step 1.1, a smooth section of $E$ is exactly a smooth map $S^1\to S^1$, so $\mathrm{\Gamma }\cong C^{\mathrm{\infty }}(S^1,S^1)$; a section $F$ corresponds to $\theta \mapsto$ the coordinate of $F_{\theta }({\mathrm{\partial }}_{\theta })$ in the trivialisation, a nowhere-zero continuous function for a monomorphism. Writing $v(\theta )=F_{\theta }({\mathrm{\partial }}_{\theta })$ for a bundle monomorphism over the identity, normalisation $v\mapsto v\mathrm{/}\mid v\mid$ is a homotopy of nowhere-zero maps, because the straight segment from $v(\theta )$ to $v(\theta )\mathrm{/}\mid v(\theta )\mid$ stays in the open ray through $v(\theta )$ and misses $0$.

Degree classifies the smooth section components. A smooth circle map has a smooth angular lift $\alpha :\mathbb{R}\to \mathbb{R}$ with $\alpha (\theta +2\pi )=\alpha (\theta )+2\pi d$, where $d$ is its degree: the continuous lift in the circle-loop model is smooth on each local inverse branch of the exponential. The linear interpolation $(1-t)\alpha (\theta )+td\theta$ exponentiates to a smooth path of circle maps to the standard degree-$d$ map, continuous in the weak $C^{\mathrm{\infty }}$ topology. Thus equal degrees give a path of smooth sections; conversely any such path is a continuous homotopy and preserves degree by [F5]. Every integer occurs via $\theta \mapsto e^{id\theta }$, so ${\pi }_0(\mathrm{\Gamma })\cong \mathbb{Z}$.

The winding invariant $w(f,F)$ is the degree of the section of $(f,F)$ in the fixed trivialisation of step 1.1, so $w$ is constant on path components and induces the bijection ${\pi }_0(\mathrm{\Gamma })\cong \mathbb{Z}$ of step 3.1. For the derivative $(f,df)$ of an immersion, the corresponding section is the velocity map $v(\theta )=d{f}_{\theta }({\mathrm{\partial }}_{\theta })$, whose normalisation is ${\tau }_f$; by step 2.1 $v$ and ${\tau }_f$ are homotopic through nowhere-zero maps, so they have the same degree, and that degree is $\mathrm{rot}(f)$ by [F2].

Two formal immersions $(f,F)$, a smooth $f$ with a smooth bundle monomorphism $F$ over $f$ in the sense of [F6], lie in the same path component of $\mathrm{FImm}(S^1,{\mathbb{R}}^2)$ exactly when $w$ agrees: by [F1], $\mathrm{FImm}(S^1,{\mathbb{R}}^2)$ is homotopy equivalent to $C^{\mathrm{\infty }}(S^1,{\mathbb{R}}^2)\times \mathrm{\Gamma }$ via polar normalization, and $C^{\mathrm{\infty }}(S^1,{\mathbb{R}}^2)$ is contractible, so path components of the product correspond bijectively to path components of $\mathrm{\Gamma }$, which are classified by the degree by step 3.1; the winding invariant is that degree. This fixes the normalisation: the round unit circle traversed once in the positive direction has $w=1$ and its reverse has $w=-1$, matching the sign convention of [F2].