No nontrivial Syracuse cycle of period at most sixteen

Let $T$ denote the Syracuse map, sending $n$ to the odd part of $3n+1$. If $m$ is a positive integer with $T^a(m) = m$ for some $a$ with $1 \le a \le 16$, then $m = 1$.

Equivalently: every Syracuse cycle whose length is at most $16$ is the trivial cycle $\{1\}$. Note that no minimality is assumed of $a$ — it is any return time, not necessarily the least one.

This subsumes and extends the individually established exclusions for return times $1$ through $7$. The proof is uniform: for each of the sixteen periods one checks a single numerical margin inequality $103^a < 2^{K_a}\,34^a$, where $K_a$ is the least exponent with $3^a < 2^{K_a}$, and feeds it to the margin criterion. The value $34$ enters because no nontrivial Syracuse cycle contains a value at most $33$.

The bound $16$ is exactly where this method stops: the margin inequality holds for every $a \le 16$ and fails first at $a = 17$, where $2^{27}/3^{17} \approx 1.0393$ is smaller than $(103/34)^{17}/3^{17} \approx 1.181$.