Guarded burst followed by halving descends below its start

Proved

CollatzWork.rootDescent

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 $k,u,m\in\mathbb N$ be positive and assume $2^k m+5=9^k u$ . Then

T^{4k}(8^k u-5)=m<8^k u-5.

The displayed exit equation is essential and is not asserted for every positive start.

Preamble

import Std
import Init.Grind.Ordered.Module
import Definitions.Def_CollatzWork_ConvergenceStatement
import Definitions.Def_CollatzWork_InverseWordBoundaryStatement
import Definitions.Def_CollatzWork_RefinedMersenneChild
import Definitions.Def_CollatzWork_RootDescentStatement
import Theorems.Thm_CollatzWork_rootDescentBurst

Formal statement

theorem CollatzWork.rootDescent : RootDescentStatement := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/RootDescent.lean#L94-L119

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