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

The Collatz Conjecture, a long-standing open problem in number theory, asserts that every positive integer sequence generated by the Collatz function eventually reaches the 4-2-1 cycle. This paper presents a rigorous proof by modeling the Collatz dynamics using a 17-state finite state machine (FSM) derived from a structured partition of the integers. This FSM comprises a two-state precursor stage (for multiples of 3), a 12-state transient core (for other numbers outside the cycle), and a 3-state terminal stage (representing the 4-2-1 cycle). We analyze the deterministic transitions within this FSM and prove that every state in the precursor and core stages has a finite path leading inevitably to the terminal cycle stage, guaranteeing convergence for all starting integers. Our approach resolves the conjecture through deterministic finite-state analysis, demonstrating the inevitable collapse of any Collatz sequence into the unique 4-2-1 attractor.