Three-step odd–odd–even shortcut identity

Proved

CollatzWork.shortcutIter_OOE

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 every $z\in\mathbb N$ ,

T^3(8z+3)=9z+4.

This is the elementary block underlying the guarded burst construction.

Preamble

import Std
import Init.Grind.Ordered.Module
import Definitions.Def_CollatzWork_ConvergenceStatement
import Definitions.Def_CollatzWork_InverseWordBoundaryStatement
import Definitions.Def_CollatzWork_RefinedMersenneChild

Formal statement

theorem CollatzWork.shortcutIter_OOE (z : Nat) :
    shortcutIter 3 (8 * z + 3) = 9 * z + 4 := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/RootDescent.lean#L14-L30

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