How to make a global setting to box every displayed and inline math?

Question

I have no idea how to make a global setting to box every displayed and inline math. The box is decorated with colored frames, colored background and colored foreground (either text or math).

By making the global setting, I can easily turn it on or off and the remaining is left untouched.

I know there are 2 macros, \everydisplay and \everymath, but I cannot make use of them other than changing the stroke color with \color{red}, for example.

Answer 1

For playing the following example adds a \fbox around inline math:

\documentclass{article}

\newsavebox{\mymathbox}
\everymath{%
  \mathsurround=0pt$%
  \begin{lrbox}{\mymathbox}\everymath{}$%
  \aftergroup\mathboxend
}
\newcommand*{\mathboxend}{%
  \end{lrbox}%
  \fbox{\usebox{\mymathbox}}%
}

\begin{document}

$abc$ Hello\textsuperscript{World}

\begin{tabular}{l}Hello\\World\end{tabular}

\begin{minipage}{4em}Hello\\World\end{minipage}

\begin{minipage}[t]{4em}Hello\\World\end{minipage}

\end{document}

Update: Explanation and \mathsurround=0pt added.

The goal is to have something like

\fbox{$...$}

If \everymath is called, we are already in math mode. Therefore the first $ ends the math mode. The space, given by \mathsurround would be set around the empty formula, therefore it is set to zero.

Then a horizontal box is opened (lrbox) and we switch in math mode again, but with empty \everymath. Via \aftergroup we get to the point right after the closing $ and we can close the box and set the box with \fbox.

Variation added.

Also we could let the math group in place and put the lrbox inside:

\everymath{%
  \begin{lrbox}{\mymathbox}%
  \everymath{}$\mathsurround=0pt\relax
  \aftergroup\mathboxend
}
\newcommand*{\mathboxend}{%
  \end{lrbox}%
  \fbox{\usebox{\mymathbox}}%
  $%
}

The difference of these methods is the handling of \mathsurround, if it is not zero. In the first case the space is put inside the box, in the second case, the space is outside.

However, there is a serious drawback of this general method via \everymath. Inline math is used at many different places, thus the example also shows boxes at places where inline math is not expected.