= Boundary approach forces a small SLE Bessel gap
{title2=$\operatorname{dist}(x,\gamma)\lesssim r\ \Rightarrow\ \inf_t(g_t(x/2)-\xi_t)\lesssim r$}
For $0<\kappa<4$ and $x>0$, stop the simple chordal <SLE> trace on first entering the radius-$r$ half-disc about $x$, with $r<x/4$. A crosscut from its tip to $x$ separates the positive-side boundary arc containing $x/2$ from infinity. <Harmonic measure> comparison with the half-disc bounds the length of that arc's mapped interval, $g_t(x/2)-\xi_t$, by a universal constant times $r$. This centered boundary image is a positive <Bessel process> of dimension $1+4/\kappa>2$ and has a strictly positive all-time minimum. Hence the full trace stays a positive distance from each fixed nonzero real point. Reflection gives the negative-side version.
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