No nontrivial Syracuse cycle of period at most one hundred and ninety-nine

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 $1 \le a \le 199$, then $m = 1$.

Equivalently, every Syracuse cycle of length at most $199$ is the trivial cycle $\{1\}$. The return time $a$ need not be minimal.

The proof is uniform across the $199$ 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 $6725$ is available because no nontrivial cycle contains a value below $6725$. At $a = 199$ these are comparisons of $857$-digit integers.

The bound $199$ is exactly where this threshold stops: the inequality holds for every $a \le 199$ and fails first at $a = 200$. Enlarging the verified small-value range moves the boundary further, but never to all periods — $2^{K_a}/3^a$ comes arbitrarily close to $1$ along the convergents of $\log_2 3$, and at $a = 306$ no threshold whatsoever suffices.