{"id":"palomar/PALOMAR-2026-09-23-000002","url":"https://github.com/shaikidris/CollatzConvergencePositiveDensity/tree/f0fda887ee60dc8ac4ef0d864935148b03734726","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-23-000002-v1.json","repo":"https://github.com/shaikidris/CollatzConvergencePositiveDensity","tags":["math.NT","msc:11B83","msc:37P99","number-theory","collatz","natural-density","lean"],"score":0,"title":"Positive density of Collatz convergence at every rate above 3/log(4/3)","trust":"high","domain":"math.NT","source":"palomar","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.NT","summary":"A Lean 4 formalization of eighteen selected Phase 1 statements, led by a single positive lower natural-density constant for the set of positive integers reaching 1 within c log x ordinary Collatz steps, valid for every fixed c > 3/log(4/3). It also covers fixed-target hitting, the exact target criterion, individual-root consequences, vanishing-clock-loss variants, depth-eleven density bounds, exact finite fan allocation, real-moment comparison, target profiles, and bounded actual reference entropy with explicit tail estimates and zero normalized growth. Density witnesses and levels are…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"formalizes":[{"title":"Positive density of Collatz convergence at every rate above 3/log(4/3)","identifier":"doi:10.5281/zenodo.22871622","relationship":"formalizes"}]}