We prove that the only cycle in the Collatz sequence (the 3n+1 problem) is the trivial cycle 1 → 4 → 2 → 1. The proof combines algebraic reduction to a Diophantine equation, numerical verification up to a computable boundary, and asymptotic analysis beyond that boundary. This result establishes that nontrivial cycles in the Collatz conjecture are impossible.
Vitaly Reznik (Sat,) studied this question.