Symplectic Tiling Billiards on Complete Affine Tori

What the math gives to ML

The paper provides a concrete projective-dynamics mechanism for complete affine torus tilings: affine holonomy elements with sufficiently large positive translational invariant drive almost every initial direction toward one of two distinguished projective directions, with the basin selected by the sign of a linear functional. The transferable asset is not the billiard geometry itself, but a parameterized family of structured affine recurrent maps with analytically predictable directional attraction. A neural recurrent architecture can use this as a long-horizon stabilization module or as a regularizer on products of hidden-state Jacobians. The mechanism is most plausible for low-dimensional latent state-space models, where projective directions and cumulative affine translation can be computed cheaply and the predicted angular-collapse behavior can be directly falsified.