Exploring Collatz Dynamics with Human-LLM Collaboration

ArXi:2603.11066v1 Announce Type: cross We investigate structural properties of the Collatz iteration through two phenomena observed in large computational exploration: modular scrambling of residue classes and a burst--gap decomposition of trajectories. We prove several structural results, including a modular scrambling lemma showing that the gap-return map acts as an exact bijection on high bits, a persistent exit lemma characterizing gap structure after persistent states, and a decay property for known portions of binary representations under gap-return dynamics.