Quantitative Collatz Descent to Stretched-Logarithmic Scale in Natural Density (Almost all)

Abstract

This record contains the version submitted to Forum of Mathematics, Sigma on 7 August 2026, together with the corresponding LaTeX source package. The manuscript proves a quantitative stretched-logarithmic descent theorem for the Collatz map in ordinary natural density. For every δ below the explicit endpoint 0.251245530155874..., almost every initial value has an iterate at most exp((log n)^(1−δ)), with a stretched-exponential exceptional-count estimate and a logarithmic witnessing-time bound. A smaller parameter range additionally supports a shrinking-error two-sided orbit envelope and a uniform quantitative first-passage profile. These are almost-all results. They do not prove the Collatz conjecture or exclude exceptional cycles or divergent trajectories. The mathematical results are unchanged from the preceding formalization-oriented edition; this version revises the presentation and literature comparison for submission to Forum of Mathematics, Sigma. The associated Lean 4 formalization is archived separately at https://doi.org/10.5281/zenodo.21797535. Added resolution to minor review comments from peer review.