Ah, ok, in that case the only additional insight is that if finance is regulated by law, that any agent of the state if its a judge or a police officer, they can only proceed their work (and accept payment) if they prove their steps are valid, both parties can directly point at proofs or lemmas it has constructed and those will be automatically accepted when valid.

Also this would mean you can verify your claims at home and have the same software running locally verify if the verifier-as-a-judge will accept or reject your proof before you even submit it.