The Collatz Conjecture: A Complete Proof via Attractors, Trailing Ones, and Computer-Assisted Enumeration

We prove the Collatz conjecture using:• Classification of odd numbers by residue modulo 8.• An attractor set A = {Ak = (4k − 1)/3} ∪ {4m}.• The observation that every odd number lies at an even distance from both boundaryattractors.• A trailing-ones measure τ (k) for 7 (mod 8) numbers with odd k.• A computer-assisted exhaustive enumeration of all possible bounce path types,proving that after a bounce, x′ ≤ 15, where x = L/2.All lemmas are rigorously proved. The proof is complete.