{"id":"hexagon/2610.00249","pdf":"https://hexagonmath.org/pdf/2610.00249v2","url":"https://hexagonmath.org/abs/2610.00249v2","tags":["msc:11B83","msc:37P99","msc:60G50","Collatz conjecture","arithmetic dynamics"],"score":0,"title":"Explicit positive lower density for the Collatz Conjecture","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Kenny Lau"],"reviews":0,"subject":"math.NT","summary":"Lech Mazur proved that a positive proportion of integers $n$ reach $1$ under the Collatz map in at most $10.46\\log n$ steps. We give a quantitative refinement of his construction. For every $N$, at least $\\lfloor N/2^{2^{2^{163}}} \\rfloor$ odd positive integers $n<N$ reach $1$ within $10.46\\log n$ ordinary steps and $3.4881\\log n$ accelerated steps. We also obtain results for paths that include any prescribed integer $t$ not divisible by $3$. AxiomProver was used to obtain and formalize this result in Lean.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["AxiomProver"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry.html?id=PALOMAR-2026-10-15-000013&version=1"],"human_understanding":"some"}