The Collatz Conjecture: A Graph-Theoretic Structural Proof

This paper presents a structural proof of the Collatz Conjecture by demonstrating that the Collatz map operates within a finite structural framework. We formalize this framework as a \textbf{17-state Finite State Machine (FSM)} derived directly from the modular properties and integer partitions intrinsic to the Collatz function. The core of the proof lies in a rigorous graph-theoretic analysis of this FSM. We prove that all states representing integers not in the terminal cycle $\{1, 2, 4\}$ form a single \emph{Strongly Connected Component} (SCC), and that this SCC possesses a \emph{unique exit transition} leading irreversibly to the states corresponding to the terminal cycle. By the fundamental properties of finite directed graphs, these structural invariants guarantee that any trajectory---represented as a path through the FSM---must reach the unique terminal cycle in finitely many steps. Thus, convergence emerges as a \emph{structural consequence} of the FSM’s topology, independent of descent-based reasoning or cycle-elimination techniques.