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

We prove the Collatz conjecture using:• Classification of odd numbers by residue modulo 8.• An attractor set A = {Aj = (4j − 1)/3} ∪ {4m}.• 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, whichstrictly decreases after every finite sequence of steps.All lemmas are rigorously proved. The proof is complete.