Every odd number below 12825 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 $1 \le m \le 12824$ there is a $k \ge 0$ with $T^k(m) = 1$.

This is a finite verification. The $6412$ odd starting values below $12825$ have orbits meeting $10179$ distinct odd numbers in total, the largest being $9038141$, and every one of those 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 no nontrivial Syracuse cycle contains a value below $12825$, which in turn excludes every cycle of period at most $252$.