The Potato Box Experiment was a failure. Well, it was a failure in that it didn’t work but not that I didn’t learn anything from it.
Only one potato of any size grew in the box above the soil within the mulch/compost mix. There were a few pea-sized tubers as well. It turns out I was wrong about the whole burying the stem thing. That works with tomatoes to get them to grow more roots so I assumed since potatoes are related the same thing would happen. Not so! The stolons that potatoes form on only grow from the first, short segment of the stem that emerges from the seed potato. To keep it elongating and growing more stolons in the box or whatever container you use, the compost needs to be added gradually as the plant grows. I’m going to research this more and perhaps try it again next year.