A Lyapunov Framework for the Accelerated Collatz Map: Conditional Convergence and the Valuation Lower Bound Conjecture

This work develops a Lyapunov-based framework for the accelerated Collatz (Syracuse) map on odd integers, using the logarithmic potential V(m) = \ln(m). We show that under uniform residue-class distributions modulo powers of two, the expected single-step drift of this potential is exactly negative, with limiting value \ln(3) - 2\ln(2). The central contribution is a Conditional Convergence Theorem: if the Cesàro mean of the 2-adic valuations \nu_2(3m_i+1) along an orbit remains strictly above the critical threshold \ln(3)/\ln(2), then the orbit must terminate at 1 in finite time. This result reduces the Collatz conjecture to a single, precisely stated arithmetic condition—the Valuation Lower Bound Conjecture—concerning orbit-wise valuation averages. Extensive computational experiments reveal no sustained violations of this bound across large samples of orbits, while existing “almost-all” results (Terras, Tao) are shown to be insufficient to resolve the required pointwise condition. The analysis isolates the exact remaining obstruction as a deterministic arithmetic mixing problem and outlines several concrete avenues by which it might be resolved. Although the Collatz conjecture itself remains unproven, this work provides a minimal, rigorous reformulation of the problem that sharply delineates what is known from what remains open. Related work: The Collatz map behaves like a deterministic, finite-memory, mixing system with information loss. De Jesus, Elias. (2026). Memory Erasure in Non-Invertible Dynamics: A Diagnostic Framework with Benchmark Validation. Zenodo. https://doi.org/10.5281/zenodo.18155782