An Unconditional Proof of the Collatz Conjecture By Qin zitai

AbstractThis paper provides a complete rigorous proof of the Collatz conjecture within the ZFC axiomatic system: starting from any positive odd integer, the Collatz orbit eventually converges to 1. The proof consists of two independent modules, which together yield a contradiction via a joint reductio ad absurdum. First Module (Forced convergence of numbers congruent to 5 mod 8): Using elementary number-theoretic identities, it is proved that numbers congruent to 5 modulo 8 strictly decrease under the Collatz compression map. By the well-ordering principle and the minimal counterexample method, it follows that a number congruent to 5 modulo 8 cannot be the smallest counterexample. This conclusion holds unconditionally. Second Module (Reduction of orbit behavior): First, a modulo 320 low-v state machine (where v₂ (3n+1) = 6) are established: a single step contracts by a factor of at least 1/21, and two consecutive high-v steps contract by a factor of at least 1/335. Third, a detailed modulo constraint analysis of consecutive v=1 steps is carried out. Fourth, using the asymptotic expansion factors of all 64 paths in the modulo 320 state machine and a rigorous control of the asymptotic error, it is proved that any orbit contains only finitely many high-v steps. Consequently, the tail of every orbit consists entirely of low-v steps. Using the completeness of the low-v state machine, it is proved that every orbit either converges to 1 or enters a cycle not containing 1. Moreover, a modulo 8 analysis shows that any cycle not containing 1 must contain a number congruent to 5 modulo 8. Joint Reductio: The second module reduces the existence of a counterexample to the existence of a number congruent to 5 modulo 8 lying in a cycle not containing 1. The first module proves that a number congruent to 5 modulo 8 cannot be the smallest counterexample. Combining these gives a contradiction; hence no counterexample exists, and every orbit converges to 1. The proof uses no unproven conjectures (such as the Riemann hypothesis or the generalized Riemann hypothesis). All finite enumerations are strictly computable within ZFC. All constants are explicit; in particular, the threshold N₀ is explicitly determined from the asymptotic expansion factors and the error bound. Keywords: Collatz conjecture; 3n+1 conjecture; modular constraints; contraction mapping; minimal counterexample method; modulo 320 state machine; asymptotic expansion factor References1 Lagarias, J. C. The 3x+1 problem: An overview. American Mathematical Monthly, 92 (1): 3–23, 1985. 2 Lagarias, J. C. The 3x+1 problem: An annotated bibliography. arXiv: math/0309224, 2003. 3 Wirsching, G. The Dynamical System Generated by the 3n+1 Function. Lecture Notes in Mathematics, 1681, Springer, 1998. 4 Tao, T. Almost all orbits of the Collatz map attain almost bounded values. arXiv: 1909. 03562, 2019. 5 Conway, J. H. Unpredictable iterations. Proc. 1972 Number Theory Conference, University of Colorado, Boulder, 49–52, 1972. 6 Steiner, R. P. A theorem on the Syracuse problem. Proceedings of the 7th Manitoba Conference on Numerical Mathematics and Computing, 553–559, 1977. 7 Krasikov, I. , & Lagarias, J. C. Bounds for the 3x+1 problem using difference inequalities. Acta Arithmetica, 109 (3), 237–258, 2003. 8 Applegate, D. , & Lagarias, J. C. Density bounds for the 3x+1 problem. Mathematics of Computation, 64 (209), 411–426, 1995. 9 Belaga, E. G. , & Mignotte, M. Walking on a treacherous trail: The Collatz conjecture. Experimental Mathematics, 7 (2), 151–165, 1998.