On the trapping of ray families by mirrors

What the math gives to ML

The paper provides a constructive trapping mechanism based on a normally hyperbolic invariant manifold (NHIM): a lower-dimensional central waist is invariant, tangent dynamics may remain neutral, while transverse dynamics are hyperbolic and contract trajectories toward it. The stable-manifold theorem predicts a family of states that asymptotically approaches the manifold, while recurrence prevents an open trapped set in the full phase space. This transfers naturally to recurrent neural networks and learned world models by separating latent coordinates into persistent tangent variables and contracting transverse variables. The key falsifiable benefit is controlled long-horizon behavior: transverse perturbations should decay geometrically at a rate predicted by the learned Jacobian without forcing all useful tangent dynamics to contract.