The parity-compatible refined child reaches the same endpoint

Proved

CollatzWork.refinedChild_iter

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 natural parameters $L,\varepsilon,z$ , put $A=4z+2\varepsilon+1$ , $P=2^LA-1$ , and $Q=3\cdot2^{L-2}A-1$ . When $L<2$ , the exponent $L-2$ uses truncated natural subtraction.

Assume $L\ge2$ , $\varepsilon\le1$ , and $\varepsilon\equiv L\pmod2$ . Then

T^L(Q)=\left\lfloor\frac{3^LA-1}{4}\right\rfloor.

This gives the child side of the guarded coalescence construction.

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_oddRun
import Theorems.Thm_CollatzWork_compatibleProduct_mod_four

Formal statement

theorem CollatzWork.refinedChild_iter (L epsilon z : Nat) (hL : 2 ≤ L)
    (hepsilon : epsilon ≤ 1)
    (hparity : epsilon % 2 = L % 2) :
    shortcutIter L (refinedChild L epsilon z) =
      (3 ^ L * refinedA epsilon z - 1) / 4 := by sorry

Source

https://github.com/Sodelin/Collatz-Conjecture-Work/blob/026aa4ad4be6453a005ab950b160a9f2204c5271/lean/CollatzWork/RefinedMersenneChild.lean#L173-L227

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