A Sieve Method for Non‑Trivial Cycles of the Collatz Conjecture Based on a Ratio Inequality

We present a sieve criterion for hypothetical non-trivial cycles of the Collatz iteration, based on the classical ratio inequality of Eliahou (1993) and the computational lower bound a_1 ≥ 2^71 of Barina (2023). For a given number of odd steps S, we prove that any candidate cycle must have total even-division count M = ceil(S log_2 3), subject to a size restriction on S. This yields an explicit arithmetic condition on the fractional part of S log_2 3 that rules out certain values of S. We also prove an improved ratio upper bound via the pigeonhole principle, sharpening the bound on 2^M from (3 + 1/a_1)^S to (3 + 1/(1.16 a_1))^S, which effectively raises the lower bound on a_1 to approximately 1.16 × 2^71. Numerical sieve results up to S ≈ 2.3 × 10^11 are included, identifying several surviving candidates, including the known lower bound S = 72,057,431,991 and a new independent candidate S = 222,759,114,643. Keywords: Collatz conjecture, non-trivial cycles, sieve method, Diophantine approximation, ratio inequality.