{"id":"hexagon/2610.00523","pdf":"https://hexagonmath.org/pdf/2610.00523v1","url":"https://hexagonmath.org/abs/2610.00523v1","tags":["math.DS","msc:11B83","msc:37P99","Collatz problem","Fourier decay","formal verification","hitting times","inverse trajectories","lower natural density"],"score":0,"title":"Positive density of Collatz convergence at every rate above 3/log(4/3)","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.NT","summary":"We prove that a positive proportion of positive integers reach 1 within c log x ordinary Collatz steps for every fixed c > 3/log(4/3), with one positive lower natural-density bound independent of c. For each fixed positive target not divisible by three, the same range of times holds with a target-dependent bound, irrespective of its subsequent orbit. Targets divisible by three have predecessor basins of natural density zero. Diagonalization gives one positive-density set with a vanishing excess above the limiting time coefficient. The proof extends weighted inverse trajectories from a fixed…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-09-23-000002&version=1"],"human_understanding":"all"}