= Simplicity of SLE from boundary-avoiding restarts
Suppose a chordal <SLE> trace stays in the open <complex upper half-plane> at every positive time almost surely. For every deterministic rational time $s$, the <domain Markov property of a chordal Loewner chain> makes its mapped and centered future a fresh SLE, so its positive-time future stays in the open half-plane as well. The inverse <mapping-out function> then places that future outside the entire old hull. Intersecting these probability-one events over rational $s$ rules out every self-intersection: if two times $u<v$ shared a trace point, choose rational $s$ strictly between them. This proves simplicity using the allowed boundary-avoidance assertion for $0<\kappa\leq4$.
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