Exact iterate for an arbitrary finite OOE burst

Proved

CollatzWork.rootDescentBurst

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 $T^0(n)=n$ .

For $k,u\in\mathbb N$ with $u>0$ ,

T^{3k}(8^k u-5)=9^k u-5.

The burst length is unbounded, but the statement describes only a finite prefix and does not claim descent.

Subtraction is truncated on natural numbers; this matters when k=0 and u is less than 5.

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_shortcutIter_OOE

Formal statement

theorem CollatzWork.rootDescentBurst : RootDescentBurstStatement := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/RootDescent.lean#L38-L52

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