Semi-analytic construction of global transfers between quasi-periodic orbits in the spatial R3BP

The paper’s Theorem 1 is proved via a robust IFS argument that breaks all essential f-invariant circles using a scattering map with action oscillation exceeding ρ2>ρ1, and then applies a standard shadowing lemma to obtain a true orbit from U1 to U2; the key steps are stated precisely and referenced (Theorem 1 hypotheses and conclusion; Theorem 2 for the IFS crossing; Theorem 3 for shadowing) . By contrast, the model’s proof hinges on unproved “angle steering” at each inner iterate: it assumes one can choose n so that the inner map places the angle in a prescribed open set V_I on every step, ensuring ΔI≥ρ2−ε. That reachability is not implied by (I.i)–(I.ii); invoking shadowing does not fix this scheduling gap. Hence the model’s proof is incomplete under the stated hypotheses, while the paper’s argument is sound.