Strict Separation of Antiperiodic Degeneracy Curves for Elliptic Euler and Lagrangian Solutions

Abstract

We prove strict separation of antiperiodic degeneracy curves associated with elliptic Euler and Lagrangian solutions of the planar three-body problem. For every eccentricity $0<e<1$, each Euler pair issuing from a common circular parameter is strictly ordered in the extended family $β>0$, and the two Lagrangian curves remain disjoint. Every antiperiodic degeneracy point in these families with $0<e<1$ has a one-dimensional full kernel. Rotation and reflection decomposition yield, for each model, two half-integer Fourier block operators with identical tails and different first blocks. In the Euler case, a positive definite first-block perturbation gives a definite connecting crossing form; a spectral-flow rectangle and sectorwise index counts then force strict separation. In the Lagrangian case, exact tail elimination and an explicit estimate for a two-dimensional Schur complement exclude simultaneous degeneracy in the two reflection sectors.