4.5.13 * For any real numbers $a, b$, the inequality $\max \{|a+b|,|a-b|,|2006-b|\} \geqslant$ $C$ always holds, find the maximum value of the constant $C$. (Note: $\max \{x, y, z\}$ represents the maximum of $x, y, z$.)
Parse the maximum value of the constant $C$ is $\max \{|a+b|,|a-b|,|2006-b|\}$. Let $\max \{|a+b|,|a-b|,|2006-b|\}=M$, then $M \geqslant|a+b|, M \geqslant|a-b|$, $M \geqslant|2006-b|$, so
$$
\begin{aligned}
4 M & \geqslant|a+b|+|b-a|+2|2006-b| \\
& \geqslant|(a+b)+(b-a)+2(2006-b)|=4012,
\end{aligned}
$$

Therefore, $M \geqslant 1003$. When $a=0, b=1003$, $M=1003$, hence the minimum value of $M$ is 1003, and thus the maximum value of $C$ is 1003.