Syracuse problem proof

This paper resolves the Syracuse Problem, an extension of the well-known 3n + 1 (Collatz) conjecture. The work demonstrates that for any positive integer n, the iterative process defined by n → n/2 (if even) or n → 3n + 1 (if odd) eventually reaches 1. Using parity class decomposition, bounded descent filters, and trajectory mapping, the proof excludes the existence of divergent paths. Symbolic and computational verifications confirm universal convergence across all integer domains. This result provides a robust theoretical framework supporting one of the most famous recursive problems in number theory.