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

We present a deterministic and fully constructive proof of the Collatz Conjecture by modeling its dynamics as a \textbf{17-state Finite State Machine (FSM)} defined over modular arithmetic. The FSM partitions the positive integers $\mathbb{Z}^+$ into five disjoint classes based on parity and residue modulo 9, providing a complete finite representation of all possible trajectories. Within this framework, every sequence ultimately follows a canonical trajectory through three inevitable stages: an initial stage of multiples of 3, a finite transient stage of 12 coprime states, and the terminal cycle $\mathcal{C}=\{1,2,4\}$. The arithmetic refinement of the map stabilizes modulo $M = 2^3\!\cdot\!9 = 72$, producing a closed deterministic residue system. Within this finite modular structure, the Collatz map admits \textbf{no additional cycles or unbounded trajectories}: every path is funneled through the \textbf{unique gateway state $S_{11}$} (residue 8 mod 9) into the invariant cycle $\mathcal{C}$. The result establishes global convergence of the Collatz map, supported by computational verification up to $10^{7}$.