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, 2, 4\}. Despite extensive computations and probabilistic results indicating convergence for almost all integers, no deterministic proof is known. We propose a novel conjecture for the Syracuse (Collatz) sequence. This conjecture is non-trivial and, under its assumption, allows for a conditional proof of the uniqueness of the trivial cycle. Extensive numerical tests support the conjecture, although it remains unproven. While a full proof of non-divergence is not yet established, it introduces structural constraints limiting possible divergent behaviors. This single, well-defined condition differs from prior probabilistic or partial analyses, providing a framework to guide further exploration of the Collatz problem. The work formalizes a conditional result that may shape future investigations into the dynamics of these sequences.