Minimum Steps to reach to a Smaller Number in 3n+1/Collatz Process

Abstract

We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for the associated stopping and cycle numbers. These bounds imply that $F_{q}(m)/m \to 1$ as the sequence length increases, while equality is impossible for any finite sequence. Consequently, no finite nontrivial cycle is compatible with the iteration, and the trivial cycle at $1$ is the only admissible periodic orbit.