I had another thought: the surface I've made is rigid enough for my purposes, but is it rigid enough to protect a polyurethane coating from cracks? I did a little research on the properties of polyurethane and concluded that it is not. So, the question is: what should I use instead?

I want something that will permeate the surface of the cardboard, seal it against stains and dirt, and ideally resist dents and scratches at least a little. I don't expect it to form a coating on the surface that won't crack and flake off. I don't think it needs to.

This certainly could be argued either way, but I have reasons. One: I need some additional parts to hold the surface in place, and for reasons that aren't important here, it would be much less troublesome right now to cut them with a box knife than with a table saw. Another: as I said, this is a _semi_permanent fixture; I want something I won't feel bad about putting in the trash when I'm done using it.

I've finished making it now. I glued four sheets of cardboard together with alternating orientations, and the resulting surface is admirably stiff, although it will nick and dent easily. I decided to coat it with Thompson's Water Seal, which should protect it from ground-in dirt, although it won't make the surface any tougher.

I would have glued on a plastic cloth.

Finish both sides. It'll warp otherwise.

Mmm... good advice, as a precaution. But I did a test on a single thickness of cardboard, and it didn't warp.

I applied the water seal with a paper towel, not a brush, so it probably didn't penetrate completely. But that's probably good -- if I let it penetrate completely, cardboard, unlike wood, probably would fall apart.

My experience is stuff like that will start flat, then humidity changes and then it warps.

I'm not sure what you mean by "stuff like that." Four sheets effectively laminated together, with a total thickness of about an inch, should warp very little. If the sheet warps the inside layer must shrink or the outside layer must stretch, or both, and cardboard is not renowned for that property.

Plywood warps with only one side finished. Everything made of wood or paper expands and contracts with humidity change. The side that's finished changes slower than the exposed side, resulting in warp.

I would have wrapped it in contact paper. Or shrink wrapped it using the plastic from a gift basket-with a heat gun. Will it be left outdoors? Man-curious as to what this is?

I like the contact paper idea! It will add a little resistance to dents and holes as well as soiling. I'll see if I have any around.

Regarding warping: I mentioned that I performed an experiment with Thompson's Water Seal on a single sheet of cardboard. I applied it to only one side, and when I re-inspected it I found no warping at all. As a bonus, the ugly dark color of the treated side has lightened over time so that it's almost exactly the same as the untreated side.