 Avoidance probabilities determine a chordal filling law

The probabilities that a random chordal filling avoids each admissible compact H-hull away from zero determine its law as an unparameterized closed set. Each point of its complement can be joined to the relevant real boundary arc through the complement and lies in a thin rational polygonal tube attached to that arc. These countably many hull tests recover the complement. Their joint avoidance events are tested by filled finite unions; if a union separates zero from infinity, the joint avoidance event is empty. Otherwise connectedness makes avoidance of the union equivalent to avoidance of its filled hull. Thus individual hull probabilities give every finite joint indicator distribution and determine the random complement.