Stability of geodesic-ray data, horofunctions, and rectifiability of fixed-shape slices in the Newtonian $N$-body problem

For the Newtonian $N$-body problem at nonnegative energy, we study solution sets selected by the Jacobi–Maupertuis variational principle and by the associated stationary Hamilton–Jacobi equation. We prove a compactness/stability theorem for classical initial data generating geodesic rays: limits in the ambient phase space remain collision-free, generate geodesic rays, and carry locally relatively compact normalized Busemann functions. The limiting horofunction of normalized Busemann functions yields a viscosity solution of the limiting stationary equation. For a fixed collision-free hyperbolic limit shape $a$, we also prove closedness of the corresponding slice of geodesic-ray data. Finally, after passing to the reduced configuration space $X$, we show that this fixed-shape slice is countably $d(N-1)$-rectifiable in phase space and has Hausdorff dimension exactly $d(N-1)$. Thus the paper combines phase-space compactness of calibrated minimizing motions with a geometric-measure description of a fixed-shape Hamilton–Jacobi calibrated slice.