No nontrivial Syracuse cycle of period at most ninety-three

Let $T$ be 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 $1 \le a \le 93$, then $m = 1$.

Equivalently, every Syracuse cycle of length at most $93$ is the trivial cycle $\{1\}$. The return time $a$ need not be minimal.

The proof is uniform across the ninety-three periods. For each $a$ one checks the single numerical inequality

where $K_a$ is the least exponent with $3^a < 2^{K_a}$, and feeds it to the margin criterion; the threshold $1193$ is available because no nontrivial cycle contains a value below $1193$.

The bound $93$ is exactly where this method stops: the inequality holds for every $a \le 93$ and fails first at $a = 94$. Raising the verified small-value range would move the bound further — but never to all periods, since $2^{K_a}/3^a$ comes arbitrarily close to $1$ along the convergents of $\log_2 3$, most sharply at $a = 306$.