I'd like to add a hidden term to a Google site search box (like the
bottom one at ://www.google.com/searchcode.html.)

For example, I might want the "search mysite.com" button to always
search with the additional term "restricted". Thus, the user would
enter into the site search box a normal search like

   bubble gum

but would get the effect of searching mysite.com with the search

  bubble gum restricted

I'd also like this to work for -restricted, to ~exclude~ pages with
the word "restricted".

I've tried various ways of just adding the hidden term "restricted" to
"mysite.com" in the value= clauses, but no matter how I do that Google
pulls it into the site name, like "mysite.com_restricted" or
"mysite.com,restricted" or "mysite.com%20restricted" and of course
gets no results.