Christoffel words as extremal structures in Collatz dynamics

What the math gives to ML

The paper provides a constructive extremal principle for binary schedules: among length-N words with exactly r active positions, Christoffel words distribute the active symbols as evenly as possible and uniquely maximize a rotation-invariant positional functional. This is transferable to neural-network systems that periodically activate an expensive operation, such as SAM, second-order preconditioning, checkpointing, auxiliary adapters, or MoE experts, while maintaining a prescribed activation rate. The most testable use is to replace bursty or random activation schedules by a cyclic Christoffel schedule and measure whether the predicted bounded discrepancy reduces optimization-event clustering and gradient variance.