It's good to see some responses that agree this is a good idea.  I can't say I can exactly say how everything should work or what everyone can do. I was hoping This thread could be a good place for everyone to be allowed to brainstorm about their ideas and arriving at some initial plan.

It seems like the tokens earned by proving theorems are going to need to be maintained separate from the currency. People could prove theorems, then the system gives them "proof tokens" as a result. these could then be used the for mining and (as a result) converted to get currency via the block reward. People could get more tokens by proving more important theorems than less important.

This opens up a question: is it entirely clear why one theorem is more important than another? The problem I can see with the "no theorems are inferior but all are the same" setup is that computers could generate proofs of "theorems" that a supermajority of people would Consider useless (and worthless?).