GLOBAL HYPERSURFACES OF SECTION IN THE SPATIAL RESTRICTED THREE-BODY PROBLEM

The paper proves that for the spatial CRTBP one has an S1-family of global hypersurfaces of section; in particular, for c < H(L1) the page {ξ3 = 0} is a global hypersurface of section, and for c ∈ (H(L1), H(L2)) (μ ∈ (0,1)) a global hypersurface exists as well. This is stated explicitly in Theorem A and summarized in Section 10.1, which also notes that Equation (6.19) implies ξ3 = 0 is a global hypersurface of section . The candidate solution reaches the same conclusions using the decoupled vertical subsystem q̇3 = p3, ṗ3 = −(μ/|q−m|3 + (1−μ)/|q−e|3) q3 (identical to the paper’s Eq. (3.2)-based computation) and a strictly decreasing vertical angle argument . However, the candidate glosses over two technical points that the paper treats carefully: (i) compactness and transversality issues near collisions, where the “physical” angle must be interpolated with a geodesic angle (Eq. (6.19), Lemma 6.4) to extend across the collision locus, and (ii) compactness of pages before regularization. The paper supplies these via first- and second-order estimates and an explicit interpolation near collisions, ensuring that pages are global hypersurfaces on the compact regularized level sets . Thus, both reach the same correct result; the paper’s proof is complete and rigorous, while the model’s outline is correct in spirit but omits the collision interpolation and briefly misstates compactness pre-regularization.