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

Related work: De Jesus, Elias. (2026). Projection-Induced Coherence in Prime Gap Statistics: Baseline Geometry, Operator Alignment, and Residual Effects. Zenodo. https://doi.org/10.5281/zenodo.18165539 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. Remark (Threshold Interpretation). The safe threshold τ* should be understood as an observable address rather than a fundamental constant. The contraction mechanism (expected drift ln(3/4) < 0) operates for all orbits under uniform sampling. The threshold τ* marks where this mechanism becomes provable via finite-m Lyapunov bounds. The gap τ* − τ ≈ 0.152 reflects the resolution penalty imposed by discrete arithmetic, analogous to aliasing in signal processing or coarse-graining in statistical mechanics. The mechanism persists below τ*; only its detectability via our proof technique is scale-limited.