Shortcut convergence is equivalent to universal positive descent

Proved

CollatzWork.descentCriterion

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 $T^0(n)=n$ . Write $\mathcal C(n)$ for the assertion that $T^j(n)=1$ for some $j\in\mathbb N$ .

Then

\bigl(\forall n>0,\ \mathcal C(n)\bigr)\iff\bigl(\forall n>1,\ \exists k>0,\ 0<T^k(n)<n\bigr).

The right-hand side is a universal hypothesis equivalent to the conjecture, rather than an established all-start descent theorem.

Preamble

import Std
import Init.Grind.Ordered.Module
import Definitions.Def_CollatzWork_ConvergenceStatement
import Theorems.Thm_CollatzWork_allPositiveConverge_of_smallerCoalescence

Formal statement

theorem CollatzWork.descentCriterion : DescentCriterionStatement := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/Convergence.lean#L97-L102

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