Exploring Collatz Dynamics with Human-LLM Collaboration

arXiv:2603.11066v3 Announce Type: replace-cross Abstract: We develop a structural and quantitative framework for analyzing the Collatz map through modular dynamics, valuation statistics, and combinatorial decomposition of trajectories into bursts and gaps. We establish several exact and asymptotic results, including an affine scrambling structure for odd-to-odd dynamics, structural decay of residue information, and a quantitative bound on the per-orbit contribution of expanding primitive families via a phantom gain analysis. In particular, we prove that the average phantom gain remains strictly below the contraction threshold under uniform distribution, with a robust extension under bounded total-variation discrepancy. Building on these components, we reduce the convergence of Collatz orbits to an explicit orbitwise regularity condition: agreement between time averages and ensemble expectations for truncated observables, together with a tail-vanishing condition. Under this condition, formulated in terms of weak mixing or controlled discrepancy, the orbit converges. Accordingly, the present work should be interpreted as a structural and conditional reduction of the Collatz conjecture, rather than a complete proof. It isolates the remaining obstruction as a single orbitwise upgrade from ensemble behavior to pointwise control, while establishing several independent exact results that may be of separate interest.