A Proof of the Collatz Conjecture via Boundedness and Cycle Uniqueness

We prove the Collatz Conjecture by demonstrating that all Collatz sequences are bounded and converge to the 4→2→1 cycle. Our proof employs a two-step approach: first, we establish boundedness through novel \textbf{asymptotic analysis} and refined congruence restrictions modulo 4 and modulo 12, revealing deterministic residue class transitions that preclude unbounded growth. Second, cycle uniqueness is rigorously demonstrated through two independent number-theoretic proofs: one utilizing a novel product equation and prime factorization analysis, and the other employing a minimality argument. These complementary approaches provide a definitive characterization of Collatz cycles.