No nontrivial Syracuse cycle contains a value below 330750

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. If $z$ is a positive integer with $z \le 330749$ and $T^a(z) = z$ for some $a \ge 1$, then $z = 1$. The return time is arbitrary; no minimality is assumed.

The argument is the natural one for cycles. Passing to the least value $w$ on the orbit of $z$, one has $w \le z < 330750$, and $w$ is odd and positive because it is a value of $T$. If $w$ were greater than $1$ it would admit a strictly smaller iterate, by the descent result for this range — but $w$ is the orbit minimum, so no iterate of $w$ can lie below it. Hence $w = 1$, the orbit of $z$ reaches $1$, and a periodic point whose orbit reaches $1$ is itself $1$.

Note that only descent is needed, never convergence. That is what makes the threshold cheap to raise.