A Formal Proof of all Collatz k-Cycles and a Reduction of theDivergence Problem

<p dir="ltr">A new framework for the Collatz conjecture is presented which successfully proves the non-existence of all non-trivial k-cycles. The method formally reduces the conjecture to the single, unsolved problem of divergence.</p><p dir="ltr">First, a 6-state Finite State Automaton (FSA), justified by a Carry Uniformity Theorem, is used to partition all positive integers into a rapidly contracting 'Strong Descent' set and a non-contracting 'Trapped Set'. This reduction shows that any non-converging path must remain in the Trapped Set indefinitely, either as a cycle or a divergent path.</p><p dir="ltr">Both of these problems are then reduced to the stability of a 2-adic mixed system.</p><p dir="ltr">This paper presents a formal proof that the k-cycle problem is impossible, by reducing any potential cycle (1-hop or multi-hop) to a single exponential Diophantine equation which is proven to have no positive integer solutions.</p><p dir="ltr">With cycles disproven, the paper identifies divergence as the final, open problem. It is reduced to proving that an infinite, non-repeating path cannot indefinitely avoid a specific 'Terminal Exit' state that would force it to converge. The Collatz conjecture is therefore formally reduced to this final, unsolved divergence problem.</p>