01_three_questions.tex

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

01_three_questions.tex

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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 \draw[black!25] (4.55,0) -- (4.55,4.45);
4 \draw[black!25] (9.25,0) -- (9.25,4.45);
5 \node[font=\small,align=center] at (2.1,4.35) {A residual root};
6 \node[font=\small,align=center] at (2.1,3.55) {$x^2+1=0$ over $\mathbb F_2$};
7 \path[draw,fill=black!12] (2.1,2.55) circle (0.24);
8 \node[font=\small,align=center] at (2.1,2.55) {$1$};
9 \node[font=\small,align=center] at (2.1,1.45) {one point mod $2$};
10 \node[font=\small,align=center,font=\footnotesize] at (2.1,0.55) {no point mod $4$};
11 \node[font=\small,align=center] at (6.9,4.35) {A coherent branch};
12 \node[font=\small,align=center] at (6.9,3.55) {$x^2+1=0$ over $\mathbb Z_5$};
13 \path[draw,fill=white] (5.3,2.5) circle (0.29);
14 \node[font=\small,align=center,font=\scriptsize] at (5.3,2.5) {$2$};
15 \node[font=\small,align=center,font=\scriptsize] at (5.3,1.9) {$5^{1}$};
16 \path[draw,fill=white] (6.38,2.5) circle (0.29);
17 \node[font=\small,align=center,font=\scriptsize] at (6.38,2.5) {$7$};
18 \node[font=\small,align=center,font=\scriptsize] at (6.38,1.9) {$5^{2}$};
19 \draw[->] (6.07,2.5) -- (5.61,2.5);
20 \path[draw,fill=white] (7.46,2.5) circle (0.29);
21 \node[font=\small,align=center,font=\scriptsize] at (7.46,2.5) {$57$};
22 \node[font=\small,align=center,font=\scriptsize] at (7.46,1.9) {$5^{3}$};
23 \draw[->] (7.15,2.5) -- (6.69,2.5);
24 \path[draw,fill=white] (8.54,2.5) circle (0.29);
25 \node[font=\small,align=center,font=\scriptsize] at (8.54,2.5) {$182$};
26 \node[font=\small,align=center,font=\scriptsize] at (8.54,1.9) {$5^{4}$};
27 \draw[->] (8.23,2.5) -- (7.77,2.5);
28 \node[font=\small,align=center,font=\footnotesize] at (6.9,0.55) {reductions agree};
29 \node[font=\small,align=center] at (11.65,4.35) {An occupied packet};
30 \path[draw=black!30,fill=black!8] (10.8,2.32) rectangle (11.63,3.15);
31 \node[font=\small,align=center,font=\small] at (11.215,2.735) {$1$};
32 \path[draw=black!30,fill=white] (11.63,2.32) rectangle (12.46,3.15);
33 \node[font=\small,align=center,font=\small] at (12.045,2.735) {$0$};
34 \path[draw=black!30,fill=white] (10.8,1.49) rectangle (11.63,2.32);
35 \node[font=\small,align=center,font=\small] at (11.215,1.905) {$0$};
36 \path[draw=black!30,fill=white] (11.63,1.49) rectangle (12.46,2.32);
37 \node[font=\small,align=center,font=\small] at (12.045,1.905) {$0$};
38 \node[font=\small,align=center] at (11.65,1.05) {$\operatorname{rank}E=1$};
39 \node[font=\small,align=center,font=\footnotesize] at (11.65,0.4) {positivity needs a carrier};
40 \end{tikzpicture}

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