We present a comprehensive resolution of the Collatz conjecture through two complementary approaches. The primary approach develops a novel framework based on inverse function properties and generative completeness, establishing that all trajectories in the Collatz dynamical system converge to the cycle {1 ,4 ,2}. A second, independent proof using measure theory provides explicit computational bounds for convergence times. Together, these approaches offer both structural understanding and quantitative characterization of the system’s behavior, providing a complete resolution of this long-standing problem.