First integrals of dense hard-ball gases

What the math gives to ML

The paper proves a rigidity mechanism for billiard and hard-ball dynamics: free flight forces polynomial first integrals to be position-independent, while elastic collisions restrict momentum dependence through reflection symmetries. For a hard-ball gas with connected collision graph, every polynomial-in-momentum first integral is a polynomial of total kinetic energy, so no additional polynomial conserved quantities survive. A transferable neural construction is an energy-preserving graph latent dynamics model in which propagation is unconstrained between events but pairwise interactions are orthogonal reflections or rotations. The sharp prediction is that connected interaction graphs retain only functions of total latent kinetic energy as low-degree polynomial invariants, while disconnected graphs retain one energy invariant per connected component.