Conditional Proof of the Uniqueness of the Trivial Cycle in the Collatz (Syracuse) Sequence

The Collatz (Syracuse) conjecture asserts that every positive integer under the iteration n → n/2 if even, n → 3n+1 if odd, eventually reaches the trivial cycle {1, 4, 2}. Despite extensive computations and various heuristic and probabilistic results, no deterministic proof is known. We introduce a novel conjecture for the Collatz sequence that imposes a dynamical constraint via a normalized sequence $(U_n)$ and does not appear to be trivially equivalent to the classical Collatz conjecture. Assuming this conjecture, we prove the uniqueness of the trivial cycle and derive structural limitations on potential divergent trajectories. Extensive numerical tests for integers up to $10^8$ support the conjecture, although it remains unproven. This framework formalizes a conditional approach to study the dynamics of the Collatz sequence, providing a new perspective on growth constraints beyond classical probabilistic or heuristic analyses.