Collatz Dynamics II: Drift–Compression Dynamics and Global Convergence

This paper develops a fully deterministic framework for global convergence in the Collatz map, forming the second installment of the Collatz Dynamics program. The central tool is the drift–compression mechanism, expressed as a Lyapunov-type inequality that enforces strict monotone descent of the potential function along every orbit. Coupled with a structural drift constraint, this mechanism guarantees absolute contraction and rules out the possibility of divergence.

In contrast to prior approaches that rely on probabilistic heuristics, asymptotic growth models, or computational verification, the present argument is entirely algebraic and dynamical. It establishes, in closed form, that every Collatz trajectory undergoes inevitable compression into the trivial cycle.

Together with Collatz Dynamics I: Skeleton Bound and the Elimination of Non-Trivial Cycles, which resolves the cycle subproblem, the present paper completes the structural foundation for a deterministic proof of the Collatz conjecture. This two-part program demonstrates that the conjecture admits a resolution not through approximation or simulation, but through a unified algebraic–dynamical framework.