 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 $0<\kappa \le 4$.