Hyperbolic and Bi-Hyperbolic Solutions in the Planar Restricted (N+1)-Body Problem

The paper explicitly proves both halves of the claim. Theorem 1.1 constructs, for any initial point/time, hyperbolic half-orbits with prescribed asymptotic direction and speed √(2h); see the statement and proof outline including the blow-up argument to handle arbitrary period T (Theorem 1.1 and its proof) . More importantly, Theorem 1.2 proves that for every pair (θ−, θ+) and every h>0 there exists a bi-hyperbolic solution with those prescribed asymptotics, with at most one collision; this is achieved by a variational construction using a “tied” topological constraint and free-time minimizers (Definitions/Propositions in Section 4), culminating in the stated theorem and its proof . The candidate’s assertion that the bi-hyperbolic result is likely open as of 2022-11-02 is therefore incorrect.