Wide existence of superhyperbolic motions in the Newtonnian N -body problem

The paper claims to prove NSH is closed (hence SH open) using weak KAM/variational tools, but the main argument contains critical gaps: it (i) implicitly assumes the availability of non‑superhyperbolic trajectories with prescribed (x0,v0) while Theorem 0.5 only ensures existence for prescribed x0,a,h, not v0; and (ii) uses an equality identifying φ_h with the action along γ_n segments without establishing that the chosen γ_n are free‑time minimizers (the subsequent attempt to prove calibration depends on this equality), making the proof circular. The model’s solution uses a limsup-based lower semicontinuity argument but incorrectly identifies {L=+∞} with a union of superlevel sets; it is actually a countable intersection of open sets, so the openness conclusion does not follow. Therefore, both arguments, as written, are incomplete.