No nontrivial Syracuse cycle contains a value below 1193

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. If $z$ is a positive integer with $z \le 1192$ and $T^a(z) = z$ for some $a \ge 1$, then $z = 1$.

The return time $a$ is arbitrary — no minimality is assumed — so the statement says that the only Syracuse cycle meeting the interval $[1, 1192]$ is the trivial cycle $\{1\}$.

It follows from the finite verification that every odd number below $1193$ reaches $1$, together with the observation that a periodic point of $T$ whose orbit reaches $1$ must itself be $1$ (since $T(1)=1$ is absorbing). This raises the small-value threshold from the previously established bound of $33$, and is the input that lets the margin criterion exclude every cycle of period at most $93$.