Exploring Collatz Dynamics with Human-LLM Collaboration

Abstract

We develop a quantitative framework for the Collatz conjecture through a human-LLM collaboration, combining exact arithmetic structure, cycle-level probabilistic laws, and a conditional convergence reduction. The central quantitative result is the Per-Orbit Gain Rate theorem, which proves R <= 0.0893 < epsilon = 2 - log_2 3 ~= 0.415, leaving a safety margin of at least 4.65x. A robustness corollary shows that exact equidistribution is unnecessary: it suffices that sum_K delta_K < 0.557. This promotes the Weak Mixing Hypothesis (WMH) to the primary open condition. On the arithmetic side, we refine modular crossing methods and prove that by depth 13 about 91 percent of odd residue classes are already forced to descend below their start. On the odd skeleton, we prove the exact run-length identity L(n) = v_2(n+1) - 1, derive an exact one-cycle crossing criterion, and compute the exact one-cycle crossing density P_1cyc = 0.713725498.... A major breakthrough is that the odd-skeleton valuation process satisfies an exact finite-block law: every prescribed valuation block occurs on a single odd residue class with the expected density. Hence the valuation process is exactly i.i.d. geometric in the natural-density ensemble, and the induced run-compensate cycle types are exactly i.i.d. This yields an exact cycle-level large-deviation theory and an unconditional almost-all crossing theorem in cycle language. We also prove substantial classwise deterministic crossing: about 41.9 percent of odd starts lie in one-cycle residue classes where every representative crosses below its start, and about 50.4 percent lie in two-cycle residue classes with the same universal crossing property. The framework does not yet prove Collatz. The remaining gap is now sharply isolated as a pointwise problem: proving that every deterministic orbit realizes enough of the exact negative cycle drift to cross below its start.