Existence of partially hyperbolic motions in the Newtonian N–body problem

The uploaded paper (Burgos, 2020) proves the existence of partially hyperbolic geodesic rays by a clean horofunction/viscosity-solution argument: approximate a collisional final configuration b ∉ Ω by collisionless a_n with ||a_n||^2/2 = h, build hyperbolic free-time minimizers γ_n via Maderna–Venturelli, pass to a limiting motion ζ by ODE continuity of solutions (fixed x, convergent initial velocities), then use horofunctions to show ζ is a global h–calibrating curve and hence a free-time minimizer; Marchal–Saari then gives ζ(t)=b′t+o(t), and Chazy’s lemma rules out b′∈Ω, so ζ is partially hyperbolic. This proof is explicit in the PDF (Theorem 1.1 and Section 3) and its preliminaries (Sections 2.2–2.3) . By contrast, the model’s solution attempts a limit-of-hyperbolic-rays compactness argument via Tonelli/Marchal to extract a uniformly convergent subsequence on [0,S] for each S, but it does not justify the uniform action/equi-absolute-continuity bounds independent of k that are needed to apply Arzelà–Ascoli on variable-endpoint minimizing segments; this is the key technical gap. It also omits the energy-normalization ||a^k||^2/2=h required by the Maderna–Venturelli existence theorem as invoked in the paper. Consequently, as written, the model’s proof is incomplete, while the paper’s proof is correct.