Paper on : Natural-density for Collatz: T_min(n) ≤ C (log n)^A in O(log n) steps (almost all n)
What this paper proves (almost every , natural density — not the full Collatz conjecture):
T_min(n) ≤ C (log n)^A
within O(log n) shortcut steps
for every fixed A above an explicit critical exponent A_FP ≈ 9.99plus a bound on the orbit up to that same time.
Paper: https://doi.org/10.2139/ssrn.7290240
Proof walkthrough: https://shaikidris.github.io/
Lean4 formalization: https://github.com/shaikidris/FirstPassageLinearTransport
Not the same as the recent “bridges” from Tao’s logarithmic-density theorem to natural density. Those aim at every target that goes to infinity. This is a standalone natural-density argument with a fixed polylog target, a logarithmic clock, and quantitative rates.
Prior almost-all natural-density scale: Korec got powers n^θ (θ > 0.79); Inselmann later got every n^ε power . Here the landing is polylogarithmic in .
Approach: Count parity words exactly on eachdyadic shell [2^M, 2^{M+1}). Large-scale prefix bounds control the orbit; a terminal odd-step “timeout” handles small blocks that don’t cross their next threshold in time. Decreasing thresholds turn later failures into direct first passages from the original shell. That organizes long multi-landing passages in natural density without a linear time-union loss — only O(sqrt(M log M)) cumulative passage times
All of this is weaker almost-all, not “for every n.”
Happy to answer questions / take corrections.