Collatz: An Empirically Predictable, Stable System Without Surprises

For 87 years, the Collatz conjecture has been regarded as the epitome of mathematical unpredictability. This work presents large-scale empirical evidence that overturns this perception. By exhaustively computing all Collatz trajectories up to n ≤ 4×10¹¹ for the residue class n ≡ 3 (mod 4), and up to n ≤ 25×10⁹ for the full population of integers, we uncover a striking and previously unrecognized phenomenon: the dynamics of Collatz exhibit strong statistical stability across all tested scales. For both the full population and the 3 (mod 4) subset, the empirical distribution of V(n)—the number of values in the trajectory that exceed n—remains stable and essentially unchanged over many orders of magnitude. Within the 3 (mod 4) class, the largest value ever observed, 908, persists unchanged through all computations up to 4×10¹¹. For the full population, the average of V(n) appears to converge to the exact integer 15, revealing an additional layer of large-scale statistical order. These findings indicate that the Collatz dynamics possess a coherent, stable, and deeply structured statistical behavior that has gone unnoticed for nearly a century.