{"id":"palomar/PALOMAR-2026-08-26-000005","url":"https://github.com/shaikidris/FirstPassageLinearTransport-Palomar/tree/39a6ba1940a1523e7491e32889f61afd235bb8df","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-26-000005-v4.json","repo":"https://github.com/shaikidris/FirstPassageLinearTransport-Palomar","tags":["math.NT","msc:11B83","msc:37P99","collatz","number-theory","lean","probability"],"score":0,"title":"Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density","trust":"high","domain":"math.NT","source":"palomar","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.NT","summary":"A Lean 4 formalization of four strengthened almost-all results for shortcut and raw Collatz orbits in ordinary natural density: critical polylogarithmic descent with arbitrary positive divergent target and height multipliers, centered trajectory tubes, and actual first-passage clocks; the fixed-exponent common-landing theorem with the exact upper exceptional-rate endpoint; an explicit reciprocal-order lower bound forced by the joint height ceiling; and a quantitative stretched-logarithmic common-landing theorem with smaller relative-height ceilings. These are almost-all statements and do not…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-06","reactions":{},"formalized":true,"formalizes":[{"title":"Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density","identifier":"doi:10.2139/ssrn.7290240","relationship":"formalizes"}]}