Paper II: Refinement Towers, Weighted Transfer, and Integral Separation of Operator Packets

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round]
2 \node[anchor=east] at (-0.15000,2.70000) {$b=0$};
3 \draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,2.30000) rectangle ++(1.15000,0.80000);
4 \node[] at (0.57500,2.70000) {$0$};
5 \draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,2.30000) rectangle ++(1.15000,0.80000);
6 \node[] at (1.72500,2.70000) {$9$};
7 \draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,2.30000) rectangle ++(1.15000,0.80000);
8 \node[] at (2.87500,2.70000) {$6$};
9 \draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,2.30000) rectangle ++(1.15000,0.80000);
10 \node[] at (4.02500,2.70000) {$3$};
11 \node[anchor=east] at (-0.15000,1.85000) {$b=1$};
12 \draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,1.45000) rectangle ++(1.15000,0.80000);
13 \node[] at (0.57500,1.85000) {$4$};
14 \draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,1.45000) rectangle ++(1.15000,0.80000);
15 \node[] at (1.72500,1.85000) {$1$};
16 \draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,1.45000) rectangle ++(1.15000,0.80000);
17 \node[] at (2.87500,1.85000) {$10$};
18 \draw[fill=black!85,draw=black!45,line width=.35pt,] (3.45000,1.45000) rectangle ++(1.15000,0.80000);
19 \node[text=white] at (4.02500,1.85000) {$7$};
20 \node[anchor=east] at (-0.15000,1.00000) {$b=2$};
21 \draw[fill=black!9,draw=black!45,line width=.35pt,] (0.00000,0.60000) rectangle ++(1.15000,0.80000);
22 \node[] at (0.57500,1.00000) {$8$};
23 \draw[fill=black!9,draw=black!45,line width=.35pt,] (1.15000,0.60000) rectangle ++(1.15000,0.80000);
24 \node[] at (1.72500,1.00000) {$5$};
25 \draw[fill=black!9,draw=black!45,line width=.35pt,] (2.30000,0.60000) rectangle ++(1.15000,0.80000);
26 \node[] at (2.87500,1.00000) {$2$};
27 \draw[fill=black!9,draw=black!45,line width=.35pt,] (3.45000,0.60000) rectangle ++(1.15000,0.80000);
28 \node[] at (4.02500,1.00000) {$11$};
29 \node[] at (0.57500,3.55000) {$a=0$};
30 \node[] at (1.72500,3.55000) {$a=1$};
31 \node[] at (2.87500,3.55000) {$a=2$};
32 \node[] at (4.02500,3.55000) {$a=3$};
33 \node[] at (2.30000,4.15000) {$a=n\bmod4$};
34 \node[] at (7.50000,3.20000) {$b=n\bmod3$};
35 \node[] at (7.50000,2.25000) {$(a,b)=(3,1)\ \Longleftrightarrow\ n=7\pmod{12}$};
36 \node[] at (7.50000,1.25000) {$7,\ 19,\ 31,\ldots$};
37 \node[] at (7.50000,0.15000) {$P_{12,7}=P_{4,3}P_{3,1}$};
38 \end{tikzpicture}

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