Local finiteness of the number of reflections in semi-dispersing Minkowski billiards

What the math gives to ML

The paper provides a non-Zeno mechanism for hybrid trajectories: in a smooth strictly convex Minkowski geometry, reflections from convex walls cannot accumulate along a trajectory of finite length. The transferable asset is a variational reflection rule together with a monotone travel-time functional whose endpoint bounds imply a path-length inequality near a common wall intersection. This can be used to build adaptive-depth neural networks or mixture-of-experts routers whose latent state moves through convex routing regions while guaranteeing that infinitely many mode switches cannot occur in finite computational time. The most direct implementation is a monitored hybrid residual network that damps routing updates when the local path-length certificate is violated.