Every odd number below 51120 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 $1 \le m \le 51119$ there is a $k \ge 0$ with $T^k(m) = 1$.

This is one stage of a ladder of verified bounds climbing towards $99780$, which is the small-value threshold at which the margin criterion excludes Syracuse cycles of every period up to $970$. The range is split into stages for two practical reasons: a submitted proof is limited to one megabyte of source and to five minutes of verification, and the whole range in one piece exceeds both by a wide margin.

Each stage is cheap because it stands on the previous one. Every odd number below $47119$ is already known to reach $1$, so an orbit starting in $[47119, 51119]$ need only be followed until it first drops below $47119$; the earlier result takes over from there. That costs $6773$ orbit facts rather than the full closure down to $1$.