Every odd number below 1193 reaches one under the Syracuse map

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. For every odd $m$ with $1 \le m \le 1192$ there is a $k \ge 0$ with $T^k(m) = 1$.

This is a finite verification: the $596$ odd starting values below $1193$ have orbits meeting only $961$ distinct odd numbers in total, the largest being $83501$, and every one of these orbits terminates at $1$.

Its role is to supply the small-value input to the margin criterion for excluding Syracuse cycles. Combined with the fact that a periodic point whose orbit reaches $1$ must equal $1$, it shows that no nontrivial Syracuse cycle contains a value below $1193$, which in turn excludes every cycle of period at most $93$.