Oscillatory motion to collision and infinity in the Earth-Moon restricted three body problem

What the math gives to ML

The paper provides a constructive topological mechanism for proving that trajectories follow prescribed itineraries through regions of phase space: h-set covering relations combined with cone invariance and hyperbolic evolution. This can transfer to neural optimization by treating a finite sequence of parameter-space or optimizer-state boxes as certified training regions and verifying that the optimizer map covers the next box while contracting transverse directions. The resulting method predicts a sharp feasibility boundary in learning rate, momentum, or weight scale where covering relations cease to hold. A practical first use is to certify that an optimizer enters and remains in a low-loss basin through a short chain of training-state regions.