Absence of hidden analytic conserved quantities in harmonically confined rods

What the math gives to ML

The paper gives a constructive symmetry characterization of analytic conserved quantities in harmonically trapped hard-rod dynamics. Free harmonic motion imposes invariance under phase rotations, while finite-size elastic collisions impose permutation invariance on momenta; together these conditions imply that every analytic invariant is a function of total energy and center-of-mass energy. The transferable neural-network mechanism is a symmetry-constrained invariant module or critic for collision-rich world models, with a falsifiable test that learned conserved scalars contain no independent information beyond these two energies.