Consecutive Collision Orbits in the Restricted Three-Body Problem above the First Critical Energy Value

The paper proves Theorem A: (i) for energies slightly above H(L1) there is either a periodic symmetric collision orbit or infinitely many symmetric consecutive collision orbits with each primary, and (ii) for generic energies and all but finitely many mass ratios there are at least two geometrically distinct symmetric consecutive collision orbits with each primary. This is stated and outlined in the introduction and formalized later via Corollaries 4.2 and 4.4 . The technical backbone is: (a) Moser/Birkhoff regularization makes the relevant energy hypersurfaces contact type and turns collision loci into Legendrian circles (Theorems 2.1–2.2 and Proposition 2.4) ; (b) the anti-symplectic symmetry lifts and its fixed locus yields a Legendrian Γ; symmetric consecutive collision orbits are obtained by concatenating a chord from Λ to Γ with its reflected image (Remark 2.3) ; (c) an equivariant contactomorphism identifies the Birkhoff-regularized hypersurface with the standard S1×S2 model sending Λ,Γ to meridians, where the Z2-equivariant Lagrangian Rabinowitz Floer homology is computed and shown to be nontrivial in every degree, implying the existence dichotomy (Proposition 3.3, Proposition 4.1, Corollary 4.2) ; and (d) an analytic continuation argument in the mass ratio, together with the generic absence (in the rotating Kepler limit) of symmetric periodic collision orbits with an odd number of collisions, yields at least two distinct symmetric consecutive collision orbits generically (Corollary 4.4) . The candidate solution mirrors this structure nearly verbatim: regularize, interpret symmetric consecutive collisions as Reeb chords, compute Z2-equivariant LRFH in a standard model with Λ,Γ meridians, deduce the dichotomy, and then use a rotating Kepler/genericity and analytic continuation argument to obtain two orbits. Minor divergences are cosmetic: the model mentions a connected sum U* S^2 # U* S^2 (the Moser picture) before switching to the Birkhoff/S1×S2 model used for the homology computation, whereas the paper cleanly works in the Birkhoff model and proves the required contactomorphism to S1×S2. The model’s stronger aside about “countably many resonant energies” in the rotating Kepler case is not needed for the paper’s odd-collision exclusion, but it does not contradict the paper’s argument. Overall, the logical steps, hypotheses, and conclusions are aligned; the proofs are substantially the same in method and structure.