A smaller residue-20 ancestor under a factorization guard

Proved

CollatzWork.residueAncestor

by Sodelin · Sep 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

collatz-work-import

Let $T:\mathbb N\to\mathbb N$ be the shortcut Collatz map: $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$ . Write $T^k$ for its $k$ -fold iterate, with $T^0(n)=n$ .

Let $v,u,r\in\mathbb N$ satisfy $v\ge13$ , $u>0$ , $u\not\equiv0\pmod3$ , and $3^v u=4r+1$ . Then

\exists m,b\in\mathbb N,\quad 0<m<r,\quad m\equiv20\pmod{27},\quad T^b(m)=r.

This gives a smaller positive ancestor for roots satisfying the explicit valuation guard. It asserts no coverage of roots outside that guard.

Preamble

import Std
import Init.Grind.Ordered.Module
import Definitions.Def_CollatzWork_ResidueAncestorStatement
import Theorems.Thm_CollatzWork_residueAncestor_normalized

Formal statement

theorem CollatzWork.residueAncestor : ResidueAncestorStatement := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/ResidueAncestor.lean#L126-L130

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