> In other words, you could theoretically have no knowledge and just get astronomically lucky to an arbitrary degree (whatever degree it would take to cross that proof threshold).
Even if you remove all the statistical nature from the system: They're trying to prove they have a piece of knowledge of finite size. If astronomical luck is a real concern, then you have to worry that even a non-probabilistic prover could have just guessed the knowledge.
Or as an analogy, even if you had a perfect and magically irreversible hash, someone could still guess the password first try.
In many applications of zero knowledge proofs, you prove knowledge of solution to a puzzle that need not have a solution. E.g. graph isomorphism, a 3-coloring, a Sudoko puzzle. So the probability of guessing the solution can be 0, while there's a nonzero probability of convincing a verifier.
Even if you remove all the statistical nature from the system: They're trying to prove they have a piece of knowledge of finite size. If astronomical luck is a real concern, then you have to worry that even a non-probabilistic prover could have just guessed the knowledge.
Or as an analogy, even if you had a perfect and magically irreversible hash, someone could still guess the password first try.