[2603.11066] Exploring Collatz Dynamics with Human-LLM Collaboration

Mathematics > Dynamical Systems arXiv:2603.11066 (math) [Submitted on 10 Mar 2026 (v1), last revised 30 Mar 2026 (this version, v4)] Title:Exploring Collatz Dynamics with Human-LLM Collaboration Authors:Edward Y. Chang View a PDF of the paper titled Exploring Collatz Dynamics with Human-LLM Collaboration, by Edward Y. Chang View PDF HTML (experimental) Abstract:We develop a structural framework for the Collatz map based on odd-to-odd dynamics, modular return structure, and a decomposition of trajectories into bursts and gaps. On the unconditional side, we prove several exact results. The fiber-57 branch q = 7 (mod 8) returns in exactly two odd-to-odd steps with uniform affine destination. The branch q = 3 (mod 8) cannot return within four steps (minimum gap five), and its earliest returns form an explicit dyadic cylinder family indexed by w = v_2(243m+119). The algebraic chain map on the five-element invariant core is a permutation at every depth, so any genuine contraction must come from return dynamics rather than core algebra. These yield an exact depth-2 known-gap partial return kernel with Perron root 129/1024 -- not asserted as the full bottleneck constant, since q = 3 (mod 8) returns with gap >= 6 are unresolved. The main body independently develops a conditional reduction via burst-gap decomposition, phantom-cycle gain analysis, and a weak-mixing hierarchy, establishing an exact geometric block law, exponential almost-all crossing bound, and per-orbit phantom gain ...