Exploring Collatz Dynamics with Human-LLM Collaboration

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. We further prove that, in the modular model, gap lengths and $2$-adic valuations follow geometric distributions, while persistent run lengths are geometric with expected burst length $E[B]=2$; together these predict strict orbit contraction. These results suggest a conditional framework in which convergence would follow from suitable orbitwise hypotheses on burst and gap lengths, which in turn are suggested by an orbit equidistribution conjecture. However, the key hypotheses remain open, and the framework is exploratory rather than a complete reduction. The paper also documents the human-LLM collaboration through which these observations were developed.