Variational Construction of Homoclinic and Heteroclinic Orbits in the Planar Sitnikov Problem

The paper’s normalized-functional approach (Theorem 2 with the Ω/PN/SN setup and the J functional) is conceptually sound and largely self-contained, but one tail-identification step—showing that the minimizing orbit’s right tail converges to the chosen extremal periodic solution γ+ rather than merely to some periodic minimizer in N(b+)—is asserted rather than fully justified (see the proof of Theorem 2 and the preceding normalization/ordering results). The candidate solution is closer in spirit to the paper but leaves key variational-compactness and closedness issues unaddressed, and it appeals to the very 2025 preprint under audit for crucial details, so it is also incomplete. See the paper’s statement of Theorem 2 and surrounding setup for context , and the proof outline using Γ±∞ and J in Section 3 .