 Avoidance probabilities determine a simple chordal curve law

For random proper simple curves from $0$ to infinity in the complex upper half-plane, the probabilities of avoiding every compact H-hull whose closure excludes zero determine the unparameterized trace law. Every point off a simple chordal curve can be connected to a real boundary interval away from zero within one side of its complement. A countable collection of thin rational polygonal tubes attached to that interval therefore recovers the complement from its avoidance indicators. Joint avoidance of finitely many tubes reduces to avoidance of their filled union. If the union blocks every path from zero to infinity the joint event is empty; otherwise filling pockets does not change the avoidance event. The inclusion-exclusion principle gives all finite indicator distributions. These determine the random complement and hence the trace. Simplicity and properness are hypotheses; the assertion is not a general determination theorem for arbitrary random closed sets.