This really is the key, of course. I didn't. When I started work on this I thought 'as long as I don't try to do anything the rest of the program isn't trying to do, I'll be okay', but that has turned out not to be the case at all.
I queried a much more experienced friend on this and now have a better understanding of CGI; however, I am now uncertain how to solve the problem at all.
I can make the desired 'captcha' image display correctly on the first of a number of screens. However, I don't know how to have the generated captcha object 'persist' to a 'place' where I can run the check of the user's guess against it: I can't shove it into a hidden field -- or at least I don't know how to correctly. So after the user hits submit on that first screen, I have the user's input in a parameter, but the generated captcha object is gone.
I'd write more but I'm practicing succinctness. :)
UPDATE: problem solved. It was a stupid problem, having nothing to do with making objects persist -- but everything to do with making my cool headspace persist so I don't miss stupid things. Thanks!!