One generally cannot prove the determinism of a system from within that system, due to "unpredictable" and "random" being effectively indistinguishable in such a case.
My personal opinion is that the brain more closely resembles a quantum computer. A computer generates a result, and examines it for validity. If it's valid, then good. If not, toss and find a new possible answer. However, when you solve a problem that way, I think your brain generates many answers simultaneously, which is more closely related to quantum computing than classical computing.
Quantum computing is equivalent to classical computing, and there is nothing that can be done on quantum computer that cannot be done on a classical computer in theory. The only thing quantum computing buys you is that it changes the computational complexity class of some algorithms (notably from exponential to polynomial). In other words, it makes certain types of deterministic computations vastly more efficient than they are on vanilla silicon.
(BTW, non-axiomatic computational models generally work by implicitly sifting all possible answers they are capable of expressing in parallel, selecting the best answer in a given context from the entire set. This is what you are essentially talking about. The details of how this works I will leave to literature.)