How does \everyhbox work?

Question

Knuth in the TexBook Chapter 24, gives a short list of the every commands. The \everyhbox command inserts a token list - as its name implies- before a horizontal box.

Here is a short example:

\documentclass[11pt]{article} 
\begin{document}
 \begingroup
     \everyhbox{a} % change this to $a$ to see the issue!
     \def\oneLineBox#1#2%
     {%
          \hfuzz=0pt
          \overfullrule=0.25pt
          \setbox0=\hbox spread#2{#1}%
          \setbox1=\hbox{\the\badness}% 
          \setbox2=\hbox to 5cm{\box0\hfil\box1}%
          \box2
     }
     \oneLineBox{Badness of line }{-1em}
     \oneLineBox{Badness of line }{-0.50em}
     \oneLineBox{Badness of line }{-0.39em}
     \oneLineBox{Badness of line }{0em}
     \oneLineBox{Badness of line }{1em}
     \oneLineBox{Badness of line }{2em}
     \oneLineBox{Badness of line }{3em}
 \endgroup

\end{document}

The oneLineBox macro above set ups some boxes and using \everybox we add the letter a in each horizontal box. But when we change the letter to a math letter say $a$, the badness is shown as zero, why? Surely the badness of the line did not disappear? Similar results can be obtained by changing \badness to \hbadness, this time TeX displays 1000 for all the lines. The overfullrule also seems to go a bit awry!

Answer 1

This doesn't really have anything to do with \everyhbox. The real issue is that constructing a math list involves constructing hboxes and the \badness reflects that.

For example,

\setbox0\hbox spread -1em{X}
\the\badness

\setbox0\hbox spread -1em{X}
$a$\the\badness
\bye

The first box is infinitely bad because the glue shrunk beyond its limit. In the second case, the box is again infinitely bad, but the \badness reflects that of the a and its italic correction, I believe.

Okay, I finally tracked down where this happens in tex.web. mlist_to_hlist converts a math list to a horizontal list as per the rules in Appendix G. In particular, it calls hpack which contains

last_badness := badness(...)

and this global variable is exactly the badness reported by \badness.

I'm not sure if this is ever actually mentioned in The TeXbook—I looked through texbook.tex (my copy of the book is in my office) and didn't see any mention of it. It doesn't appear to be in TeX by Topic either.

\hbadness and \vbadness are parameters that control the "amount of tolerance before TeX reports and underfull or overfull horizontal/vertical box" (TeX by Topic). That it doesn't change is not surprising.