The Negative Energy N-Body Problem Has Finite Diameter

The paper proves that the JM metric space ME for the Newtonian N‑body problem has finite diameter at negative energy by showing a uniform bound on the distance to the collapsed Hill boundary via an “escape game,” a Lipschitz equivalence between 1/U and dist(q,Δ) derived from a precise distance-to-collision formula, and a positive global escape rate from a union of linear subspaces; integrating √(U−|E|) along unit-speed escapers then yields a finite, q‑independent bound for dE(q,∂ME), hence finite diameter . The candidate solution independently proves the same conclusion by constructing a homothetic expansion path about the center of mass, bounding its JM length via an explicit one-dimensional integral and a sharp scale-free inequality U(q) ≥ G S^{3/2}/√(M I_cm(q)). This furnishes an explicit uniform upper bound on dE(q,∂ME), and thus on diam(ME), with a constant depending only on masses, G, and E. The two arguments differ substantially in method but agree on the main result. The paper appears mathematically sound (there is a minor typographical/slip in the displayed change-of-variables factor around the integral of √(1/(kt)−1), where a factor 1/k should appear rather than k), while the model’s path-based proof is also correct and self-contained.