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.
mahir elhisadi (2026) studied this question.