Paper X: Universal Residual Seeds and Conditional Occupied-System Completion

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \node[font=\small,align=center] at (2.2,4.3) {Scalar over $\mathbb F_3$};
4 \node[font=\small,align=center] at (2.2,3.25) {$x=0,1,2$};
5 \node[font=\small,align=center] at (2.2,2.35) {$x^2+1=1,2,2$};
6 \node[font=\small,align=center] at (2.2,1.25) {no solution};
7 \draw[black!25] (4.45,0.3) -- (4.45,4.45);
8 \node[font=\small,align=center] at (7,4.3) {Matrix over $\mathbb F_3$};
9 \path[draw=black!30,fill=white] (6.2,2.65) rectangle (7,3.45);
10 \node[font=\small,align=center,font=\small] at (6.6,3.05) {$0$};
11 \path[draw=black!30,fill=black!8] (7,2.65) rectangle (7.8,3.45);
12 \node[font=\small,align=center,font=\small] at (7.4,3.05) {$-1$};
13 \path[draw=black!30,fill=black!8] (6.2,1.85) rectangle (7,2.65);
14 \node[font=\small,align=center,font=\small] at (6.6,2.25) {$1$};
15 \path[draw=black!30,fill=white] (7,1.85) rectangle (7.8,2.65);
16 \node[font=\small,align=center,font=\small] at (7.4,2.25) {$0$};
17 \node[font=\small,align=center] at (7,0.85) {$A^2+I=0$};
18 \draw[black!25] (9.3,0.3) -- (9.3,4.45);
19 \node[font=\small,align=center] at (11.7,4.3) {Scalar over $\mathbb F_9$};
20 \node[font=\small,align=center] at (11.7,3.25) {$u^2=-1$};
21 \node[font=\small,align=center] at (11.7,2.2) {roots $u,-u$};
22 \node[font=\small,align=center] at (11.7,1.15) {coefficient extension};
23 \node[font=\small,align=center,font=\footnotesize] at (7,-0.05) {Fix the coefficient field, dimension, and point category before declaring an occupancy failure.};
24 \end{tikzpicture}

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