Critical homoclinics in a restricted four body problem numerical continuation and center manifold computations

The paper frames its four statements explicitly as conjectures supported by numerics and normal-form calculations, not as theorems, so it is intentionally incomplete regarding rigorous proofs (see “Conclusions,” Conjectures 1–4, and the surrounding discussion) . The candidate model correctly reproduces the characteristic polynomial and the identification of the critical curve D via b=ΩxxΩyy−Ωxy^2=0 (matching Eq. (5)) but makes a key factual error in asserting that L0 is never a saddle‑focus. The paper’s own configuration summary states that L0 has saddle‑focus stability for some mass values, and its review of the triple Copenhagen computations treats homoclinics at a saddle‑focus equilibrium at L0 (e.g., Section 1.3 and Fig. 2 caption) . Consequently, the model’s attempted refutation of Conjecture 1 (on Belyakov–Devaney transitions) is not supported. Overall: the paper remains conjectural; the model’s core counterclaim about L0’s stability is incorrect; hence both are incomplete for a final resolution.