Ejection-collision orbits in two degrees of freedom problems in celestial mechanics

The paper proves Theorem 1 (existence, for every m ≥ 1, of ECOs of pure type (a,…,a) and (b,…,b)) by constructing suitable arcs on the Σ-sections and iterating the Poincaré map backward, using an ordering of intersection sequences on the collision manifold C to force intersections with W^u(E+) and W^s(E−) (see Definition of Σ, equation (7)–(8), and the proof of Theorem 1) . The model captures many correct ingredients (the McGehee regularization, the first integral (8), transversality at θ=θa,b, and the use of W^u(E+), W^s(E−)), but its Phase-2 argument contains a decisive error: it asserts that all forward orbits in W^u(E+) are ECOs converging to E−. In the paper, W^u(E+) meets C along escaping orbits Γu± with v → +∞ (forward time), and the gradient-like flow on C implies forward attraction to E+ (not E−) on C; only special orbits in W^u(E+) intersect W^s(E−) to form ECOs . The model also relies on an unproven contraction property for single-side return maps Pa, Pb near Ka, Kb, whereas the paper avoids this by a robust preimage-and-ordering construction on Σ and C . Hence, the paper’s proof is correct; the model’s proof is flawed in key logic.