No nontrivial Syracuse cycle contains a value below 1883432

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. If $z$ is a positive integer with $z \le 1883431$ and $T^a(z) = z$ for some $a \ge 1$, then $z = 1$. The return time is arbitrary; no minimality is assumed.

Pass to the least value $w$ on the orbit of $z$. Then $w \le z < 1883432$, and $w$ is odd and positive because it is a value of $T$. Were $w$ greater than $1$ it would admit a strictly smaller iterate by the descent result for this range — impossible, since $w$ is the orbit minimum. Hence $w = 1$, the orbit of $z$ reaches $1$, and a periodic point whose orbit reaches $1$ is itself $1$.