2-Adic Finite-Certificate Descent Closure for the 3x + 1 Collatz Problem

The Collatz problem asks whether every positive integer reaches the classical cycle 1 ? 4 ? 2 ? 1; for the shortcut map used here, this is the cycle 1 ? 2 ? 1. This paper develops a ranked 2-adic residue automaton framework for a certificate-based closure of the problem. Each parity word is converted into an exact affine iterate, a unique residue cylinder, and a descent threshold. Non-descending cylinders are then organized by an author-defined carry-pressure rank: either a cylinder descends above a finite height, or it transitions within bounded time to a strictly lower-ranked cylinder. The central theorem proves that a finite total certificate of this kind, together with a checked base interval, forces global convergence by well-founded induction. The manuscript therefore isolates the whole infinite problem into a finite residue-rank certificate whose accepted replay leaves no density-one exception, probabilistic residue, or unranked divergent orbit.