The Restricted Three-Body Problem as a Perturbed Duffing Equation

The paper’s Main Theorem claims transversality for all but finitely many mass ratios uniformly in large negative Jacobi constant, but the body of the paper proves transversality only on a small-mass-ratio interval (Proposition 4.1 assumes 0 < ρ ≪ ε) and does not supply the missing step that upgrades an open interval in ρ to “all but finitely many” mass ratios. The candidate model solution, while aligning with the Duffing reduction and Melnikov setup, relies on a uniform-in-ρ, power-law small remainder D = M0 + O(ε^α) in the ε → 0 regime; however, the paper’s derivation shows the relevant Fourier/Melnikov coefficients are exponentially small in 1/ε^3 (Appendix C), so a power-law remainder can easily dominate the leading Melnikov term and the persistence argument is not justified. Hence both the paper (overstated conclusion) and the model (insufficient control in the exponentially small regime) are incomplete.