Collatz: An Empirically Predictable, Stable System Without Surprises

For 87 years, the Collatz conjecture has been treated as the epitome of mathematical unpredictability. This work presents large-scale empirical evidence that overturns this narrative. By exhaustively computing every Collatz trajectory for n ≤ 4×10¹¹, we uncover a striking and previously unrecognized phenomenon: the dynamics of Collatz are not chaotic at all, but remarkably stable, governed by persistent statistical regularities that repeat across all tested scales. To focus on the only residue class capable of producing a counterexample, we restrict attention to integers n ≡ 3 (mod 4). For each such n, we count how many values in its Collatz trajectory rise above n. While individual trajectories fluctuate wildly, the statistical distribution of this quantity becomes stable and remains essentially unchanged over twelve orders of magnitude, from small samples up to n = 10¹⁰. At the opposite extreme, the largest value ever observed in any trajectory, 908, remains unchanged across all computations up to n = 4×10¹¹. This paper documents this surprising large-scale order and argues that the Collatz dynamics exhibit a coherent and deeply stable statistical structure that has gone unnoticed for nearly a century.