No nontrivial Syracuse cycle contains a value below 27114

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. If $z$ is a positive integer with $z \le 27113$ 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 $[1, 27113]$ is the trivial cycle $\{1\}$.

It follows from the finite verification that every odd number below $27114$ 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 $6724$, and is the input that lets the margin criterion exclude every cycle of period at most $305$.