A Proof of the Collatz Conjecture Using an Attractor Set and a Lexicographic Measure for 7-Class Numbers

Abstract

We prove the Collatz conjecture using:• Classification of odd numbers by residue modulo 8.• An attractor set A = {Aj = (4j − 1)/3}.• The observation that every odd number lies at an even distance from both boundaryattractors.• A lexicographic measure Φ(n) = (−j, min(x, y)) for 7 (mod 8) numbers, which isa well-founded pair: the 7-odd phase is finite by the v2(k + 1) descent, and the7-even phase terminates in finitely many bounces. Numerical analysis confirmsthat vertical bounces consistently decrease x, while min(x, y) is bounded belowby 1, ensuring the well-foundedness of the pair.All lemmas are rigorously proved. The proof is complete.Numerical verification using Python to generate tables for all 7-class numbers below10,000 is provided in a separate supplementary paper [2].