A Proof of the Collatz Conjecture via Finite State Machine Analysis and Structural Confinement

We present a deterministic proof of the Collatz Conjecture using a finite state machine (FSM) framework grounded in modular arithmetic. The positive integers are exhaustively partitioned into five disjoint sets based on divisibility and congruence modulo 9, yielding a 17-state FSM that models all possible Collatz trajectories. Through detailed structural analysis, we show that every sequence governed by the Collatz map must transition through a finite number of transient states before entering the unique terminal cycle {1, 2, 4}. Crucially, we demonstrate that any hypothetical divergent trajectory would be forced to endlessly traverse a specific recurrent loop within the FSM, a condition we then prove no positive integer can satisfy. All divergent orbits are therefore structurally forbidden. The proof is supported by both theoretical argument and computational verification up to 107, confirming consistency with known behavior. This approach demonstrates that the FSM framework provides a complete and deterministic model of the Collatz process, thereby resolving the conjecture.