A use-case could be optimising compilers. These need to search for alternative (faster) series of statements that are provably equivalent to the original given some axioms about the behaviour of the underlying machine code and basic boolean algebra and integer mathematics.
This could be monetised: Theorems along the shortest path from a desired proof to the axioms are rewarded. New theorems can be added by anyone at any time, but would generate zero income unless they improve the state-of-the-art. Shortest-path searches through the data structure would remain efficient because of this incentive.
Client tools such as compilers could come with monthly subscriptions and/or some other mechanism for payments, possibly reusing some existing crypto coin. These tools advertise desired proofs -- just like how blockchain clients advertise transactions they like to complete along with a fee -- and then the community can find new theorems to reach those proofs, hoping not just for the one-time payment, but the ongoing reward if the theorems are general and reusable for other purposes.
Imagine you're a FAANG and there's some core algorithm that uses 1% of your global compute. You could advertise a desire to improve the algorithm or the assembly code to be twice as efficient for $1M. Almost certainly, this is worth it. If no proof turns up, there's no payment. If a proof does turn up, a smart contract debits the FAANG's crypto account and they receive the chain of theorems proving that there's a more efficient algorithm, which will save them millions of USD in infrastructure costs. Maths geeks, AI bots, and whomever else contributed to the proof get their share of the $1M prize.
It's like... Uber for Fields medals, used for industrial computing.
Fully automated gig work for computer scientists and mathematicians.