Audited Structural Results for the Two-Player 3n ± 1 Game

Abstract

This is a revised and audited version of the work originally published on 8 August 2026. The original version claimed a proof that the two-player 3n ± 1 game has no DRAW positions. Subsequent adversarial auditing revealed gaps in the global proof architecture, and that claim is therefore withdrawn. This manuscript presents only structural and arithmetic results that admit self-contained proofs and have survived adversarial re-checking. These include the exact binary conjugation of the game, constant-tail recurrences, canonical coefficient-source factorization, an exact description of all predecessors of the contracting branch, a proof that every three-move word containing two contracting moves is strictly decreasing, a four-WIN obstruction, exceptional-sibling separation, the six-bit attachment identity, and a strict 2-adic countdown for consecutive exact length-seven returns. The main no-DRAW conjecture remains open. This version does not claim a proof of optimal-play termination for all starting positions. Project repository:https://github.com/Grisha-Pochuev/3n-plus-minus-1-game Telegram:https://t.me/let_people_dance Contact:n_854@mail.ru