Bidirectional Dynamics in the Collatz Problem: A Complete Resolution

The Collatz conjecture states that iterating the function $C(n) = n/2$ for even $n$ and $C(n) = 3n+1$ for odd $n$ eventually leads to 1 for any positive integer $n$. Despite its elementary formulation, the conjecture has resisted resolution for over 80 years. This paper introduces a novel bidirectional approach that analyzes both forward trajectories and backward paths simultaneously. We establish two independent properties: (1) all backward paths under the generator function (a multivalued mapping that inverts the Collatz function) are finite and terminate at elements of $\{1,2,4\}$, and (2) $\{1,4,2\}$ is the unique cycle in the Collatz system. We then construct a formal structural bridge connecting these properties, proving that all Collatz sequences must reach 1. Our work introduces a global convergence measure that rigorously prohibits divergent orbits and provides explicit bounds on trajectory behavior. The bidirectional framework transforms this apparently chaotic system into one with provable structural properties, demonstrating how changing perspective can resolve long-standing mathematical challenges.