Will sleeping on a mattress without a box spring harm the mattress?”
As long as you give it support, such as on top of *painted* plywood on top of the slats, you should be OK. I say painted plywood, and I mean plywood finished on at least the side facing the mattress, so you don’t ruin the mattress with splinters when you turn it. I also have a pillow mattress on top of box springs on top of a frame, and the matress is at least 3 feet off the floor – there is nothing wrong with that.
