Regions without invariant tori of given class for the planar circular restricted three-body problem

The paper states two converse-KAM nonexistence criteria on a fixed energy 3-manifold of the PCR3BP: the general formulation (if ω(η,ξ) changes sign and λ(η)<0, then no invariant torus transverse to ξ passes through s0) and the symmetric formulation (with time-reversal symmetry and appropriate symmetry choices of ξ and η0, a sign change of ω(η,ξ) alone suffices) . The candidate solution reproduces these criteria and gives a geometric proof via a normal covector α along a hypothetical transverse invariant torus, showing sign-invariance of t ↦ α(η(t)) and deriving contradictions under the paper’s trigger conditions. This matches the paper’s logic (η must stay on the same side of a transverse torus; detect linear dependence with ω and the ‘opposite’ direction with λ) and its reversible refinement. Minor presentational differences aside, the arguments are equivalent in substance, and both correctly note the symmetric test still works when ξ ∥ V (even though such tori wouldn’t be transverse) .