A Formal Refutation of Non-Trivial Integer Cycle in Collatz Dynamics

The Collatz conjecture posits that any positive integer x, when repeatedly subjected to two operations:— 3x+1 when x is odd and x/2 when x is even—ultimately reaches 1 without exception. This paper presents a formal refutation of non-trivial cycles for the accelerated Collatz (Syracuse) operator. We establish an odd-to-odd iterative Diophantine equation and identify the trivial 1-cycle as an ‘equilibrium state’ where all division exponents a_i = 2. Using an exhaustive topological partition, we prove that for any non-trivial integer cycle starting at x > 1, there must exist a permutation of division exponents that deviates from this equilibrium. We demonstrate through Diophantine magnitude bounds and 3-adic valuation analysis that any such deviation triggers a terminal mathematical contradiction. The entire proof—comprising the exhaustive perturbation space and the resulting magnitude collapses—has been formally verified in Lean 4 using only foundational Mathlib axioms.#Collatz #conjecture #Number #theory #Numbertheory #Mathematics #Unsolved #non-trivial #integercycle #3x+1