Every odd number below 20001 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 $1 \le m \le 20000$ there is a $k \ge 0$ with $T^k(m) = 1$.

This is an intermediate step between the verified bounds $12824$ and $27113$, and it is stated separately for a practical reason: the platform limits a submitted proof to one megabyte of source, and the certificate reaching $27113$ in one piece exceeds that. Splitting the range lets each stage be checked on its own.

The stage is also much cheaper than a certificate built from scratch. Since every odd number below $12825$ is already known to reach $1$, an orbit starting in $[12825, 20000]$ only has to be followed until it first drops below $12825$; from there the earlier result takes over. That reduces the work to $9464$ orbit facts rather than the full closure down to $1$.