Normalized construction of a smaller residue-20 ancestor

Proved

CollatzWork.residueAncestor_normalized

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 $k,u,r\in\mathbb N$ satisfy $k\ge10$ , $u>0$ , $u\not\equiv0\pmod3$ , and $3^{k+3}u=4r+1$ . Then

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

This is the factorized ancestor construction before replacing k+3 by a single valuation parameter.

Preamble

import Std
import Init.Grind.Ordered.Module
import Definitions.Def_CollatzWork_ConvergenceStatement
import Definitions.Def_CollatzWork_InverseWordBoundaryStatement
import Definitions.Def_CollatzWork_RefinedMersenneChild
import Theorems.Thm_CollatzWork_residueAncestor_prefix_one
import Theorems.Thm_CollatzWork_residueAncestor_prefix_two
import Theorems.Thm_CollatzWork_residueAncestor_refinedTail

Formal statement

theorem CollatzWork.residueAncestor_normalized (k u r : Nat) (hk : 10 ≤ k)
    (hu : 0 < u) (hunit : u % 3 ≠ 0)
    (hguard : 3 ^ (k + 3) * u = 4 * r + 1) :
    ∃ m b : Nat, 0 < m ∧ m < r ∧ m % 27 = 20 ∧ shortcutIter b m = r := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/ResidueAncestor.lean#L90-L124

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