Empirical Evidence for Marginal Stability in the Collatz Dynamics: A Structural Analysis via the ΩR-Metric

This paper presents a novel approach to the Collatz Conjecture by introducing a complexitymetric ΩR , derived from additive prime power decompositions, which reveals marginalstability in the dynamics. Empirical analysis, particularly on the historically challengingnumber 703, reveals that the dynamics of this metric exhibit marginal stability, with acalculated Lyapunov exponent of approximately zero. This suggests that the Collatz processis not a chaotic system, but rather a perfectly self-correcting one. We outline a path to aformal proof by leveraging the concepts of ΩR-stability, induction on additive structures, andmodern dynamical systems theory. This work provides a quantitative framework that mayfinally lead to a solution for one of mathematics’ most enduring problems.