Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

Abstract

We prove that, in ordinary natural density, almost every positive integer admits a shortcut Collatz iterate of polylogarithmic size within logarithmically many steps, while every iterate up to the same witness remains at most n^(1+beta) for every fixed beta > 0. The result supplies an explicit first-passage method threshold for the polylogarithmic exponent, quantitative exceptional-set rates, a logarithmic witnessing clock, method-threshold log-log refinements, and stretched-logarithmic companion results. The proof counts parity words exactly on dyadic shells. Large-scale prefix bounds control orbit height, a terminal odd-step tail controls short blocks that time out, and decreasing thresholds convert every later failure into a direct first passage of the original orbit. This organizes long multi-landing passages in natural density without a linear time-union loss: only O(sqrt(M log M)) cumulative passage times are possible. A separate Lean 4 software record kernel-checks the canonical timeout route to the principal exported theorem, including the exceptional ratio, logarithmic clock, and orbit ceiling. The formalization is supplementary; the manuscript proof is self-contained. All results are almost-all statements. The manuscript does not prove the pointwise Collatz conjecture, exclude exceptional cycles or divergent trajectories, or control the orbit after the selected witness.