 Solution

Let ${\psi }_A=g_A-g_A(0)$, and take another admissible compact H-hull $B$ in the image complex upper half-plane. Form the hull $C=A\cup {\psi }_A^{-1}(B)$, including its relative closure and filling bounded complementary components if needed. Its mapping-out function of a compact H-hull isIndeed, this composition maps the correct remaining domain onto $\mathbb{H}$, and its constant term at infinity is zero. Since ${\psi }_A(0)=0$, the chain rule givesBoth derivatives are positive by local Schwarz reflection principle and boundary orientation. Therefore the assumed avoidance formula givesThe conditioning event has positive probability because $g_A^{\prime }(0)>0$.

It remains to justify that these equalities determine the law. For a proper simple curve from $0$ to infinity, its complement has two sides, each connected to an interval of the real boundary. Any point off the curve lies in a thin polygonal tube attached to that boundary which avoids both the curve and zero; its closure, with an end cap, is an admissible compact H-hull. A countable collection of tubes with rational geometry therefore reconstructs the complement as the union of the tubes avoided by the curve.

The joint distribution of their avoidance indicators is also determined. Avoiding finitely many tubes is equivalent to avoiding the filled union when that union is admissible. If the union separates $0$ from infinity, avoidance is impossible and the joint probability is zero. Otherwise a simple curve cannot enter the bounded complementary pockets without crossing the union, so filling does not change the avoidance event. Finite joint avoidance probabilities consequently come from the same single-hull formulas; inclusion-exclusion principle supplies every finite indicator pattern. These patterns determine the random complement and hence the unparameterized simple trace.

Thus avoidance probabilities determine a simple chordal curve law, and the conditional image law above equals the original law. This proves the derivative avoidance criterion for chordal restriction, giving the chordal restriction property. Simplicity is essential to this reconstruction argument; it should not be replaced by a claim that arbitrary random closed sets are determined by these hull-avoidance tests.