Their fine, those first two leaves always die shortly after the real foliage begins to get big enough. That's normal..

First, those are the cotyledon leaves and it is normal for those to shrivel and fall off. They contain everything your plants need (nutrient wise) to survive until it gets real leaves and can start the photosynthesis process.

Second, I'm assuming your setup is an aeroponic system? What media are you using? Just hydroton or do you have a rapid rooter in there as well?

It's aeroponic but it's a small pump which makes the droplets pretty large compared to true aeroponic system.

I'm using Hydroton with Rapid Rooters.

I just realized the other day I chose an indeterminate tomato variety so sadly, I'm going to have to thin these out very soon. Since posting those pictures a week ago, they have grown into a dense mass of plants.

If they are growing good, I'd leave it on the schedule that you're on. With the rapid rooter in there it may be a little excessive since that will hold the moisture a lot more than just plain hydroton but like I said, if its working, go with it. Also, I believe that whether you are indeterminate or determinate won't matter on your setup. You are going to want no more than four plants in that bucket. Probably just two though.