English is an amazingly complex language. Sometimes when words seem to coincide, it's just a coincidence (see what I did there!).
When speaking of evidence of a claim, proof is a noun, and is related to the verb prove. When allowing the microbial cultures in a dough to develop before baking, proof is a verb. The gerund is proofing, and what you have made is a proofing box.
Your proofing box is excellent and I am envious. On the other hand, now I really want a proving box. I imagine that I can put ideas in there, wait some time, and the see if they are proved or not!