1. The Cycle Inequality
\[0 < m\ln 2 - k\ln 3 \le k\ln\bigl(1+\tfrac{1}{3B}\bigr).\]
2. Computed Lower Bounds
| Verified bound \(B\) | Min. odd \(k\) | Min. total length |
|---|---|---|
| \(2^{30}\) | 47 468 | \(\ge 122\,703\) |
| \(2^{36}\) (this work) | 190 537 | \(\ge 492\,531\) |
| \(2^{68}\) | \(8.96\times 10^{9}\) | \(\ge 2.3\times 10^{10}\) |
| \(2^{71}\) | \(7.17\times 10^{10}\) | \(\ge 1.85\times 10^{11}\) |
Consequence for \(B=2^{36}\)
Any nontrivial cycle with elements \(>2^{36}\) must contain at least 190 537 odd terms (total length \(\ge 492\,531\)).
3. Limitations
The bound pushes counter-examples far away; it does not prove absence of cycles or of divergent trajectories.