Raising a graph loop over one puncture adds a small loop around it to the Gauss linking integral

Setting. $E:\mathbb R^2\to\mathbb R^4$ is $C^2$ near the closed unit disk $\overline{\mathbb D}$ with $|E|=1$ on $\overline{\mathbb D}$; $\gamma:\mathbb R\to S^3$ is a $C^2$ loop of period $1$ meeting $E(\overline{\mathbb D})$ only over a finite set $Z$; $N\in S^3$ misses $\gamma$ and $E(\overline{\mathbb D})$. For a centre $c\in\mathbb D$ use strip coordinates $\Psi(\mu,s)=c+\mu(\partial(s)-c)$, $\partial(s)=(\cos2\pi s,\sin2\pi s)$, and for a $1$-periodic $h$ with values in $[0,1]$ the graph loop $\ell_h(s)=\Psi(h(s),s)$.

Statement. Let $z=\Psi(a_0,s_0)$ and $0<\delta<1/4$. Let $g,b$ be $C^2$ and $1$-periodic with $g\ge0$, $b\ge0$, $g+b\le1$, $b=0$ outside $s_0+(-\delta,\delta)+\mathbb Z$, and $g(s_0)<a_0<g(s_0)+b(s_0)$. Assume the only point of $Z$ in $\Psi([0,1]\times[s_0-2\delta,s_0+2\delta])$ is (possibly) $z$, and that $\ell_g,\ell_{g+b}$ miss $Z$. Then for all small $\rho>0$

Suggested proof (homotopies and a substitution only).

1. Lens loop. With $s(\sigma)=s_0-\delta'\cos2\pi\sigma$ ($\delta<\delta'<2\delta$, so $b=0$ near $s_0\pm\delta'$) let $\Lambda(\sigma)=(g(s(\sigma))+\chi(\sigma)\,b(s(\sigma)),\,s(\sigma))$ where $\chi$ is the indicator of $[0,\tfrac12]+\mathbb Z$; $\Lambda$ is $C^2$ and $1$-periodic. The outer Gauss integrand depends only on the point and the velocity of the loop and is linear in the velocity, so substituting $s=s(\sigma)$ on each half gives $\mathrm{Gauss}(E\circ\Psi\circ\Lambda)=\int_{s_0-\delta'}^{s_0+\delta'}(f_{g+b}-f_g)\,ds=\mathrm{Gauss}(E\circ\ell_{g+b})-\mathrm{Gauss}(E\circ\ell_g)$, because $\ell_{g+b}=\ell_g$ off the interval.
2. Lens to ellipse. The straight-line homotopy (in the first coordinate only) from $\Lambda$ to the ellipse $(m+\eta\sin2\pi\sigma,\,s(\sigma))$, $m\pm\eta=$ the values $g(s_0)+b(s_0)$ and $g(s_0)$, stays in $[0,1]\times[s_0-\delta',s_0+\delta']$ and keeps the first coordinate $>a_0$ at $\sigma=\tfrac14$ and $<a_0$ at $\sigma=\tfrac34$, the only times with $s=s_0$; so it misses $z$ and, by hypothesis, all of $Z$. Shrink the ellipse to one centred at $(a_0,s_0)$ in the same way.
3. Ellipse to round circle. $\det D\Psi(a_0,s_0)=2\pi a_0(1-\langle c,\partial(s_0)\rangle)>0$; for a small ellipse $\Psi((a_0,s_0)+r w(\sigma))$ is homotopic to $z+r\,D\Psi\, w(\sigma)$ (Taylor, the error is $O(r^2)$ against a lower bound $\ge cr$), then within $\mathrm{GL}^+(2)$ (write the linear loop as $\alpha e^{2\pi i\sigma}+\beta e^{-2\pi i\sigma}$ with $|\alpha|>|\beta|$, kill $\beta$, rotate $\alpha$ to $|\alpha|$) to $z+\rho\,\partial$, all inside a small punctured disk around $z$. Homotopy invariance of the Gauss integral (gaussLinkingIntegral_planar_homotopy) finishes the proof.