Open Research Index

Polylogarithmic Descent for Almost All Collatz Orbits in Natural Density

Idris Ali Shaik

Palomar Registrypalomar/PALOMAR-2026-08-26-000005Sep 6, 2026math.NTscore 0

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…

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