If the brain does something that is not computable, that's a direct challenge to some of our most established science. It is possible, but I think very unlikely.

Could you? That's sort of begging the question. We do not know if something "Turing complete" can be used to build a brain like the human brain. That's precisely the point.

> If the brain does something that is not computable, that's a direct challenge to some of our most established science.

A challenge for computational neuroscience maybe. Otherwise I don't see the challenge for neither neuroscience nor computer science. If someone wants to make the claim you can build a human brain out of something Turin-machine-like, that's an extraordinary claim, not established science.

If a brain cannot be produced in a turing machine, it must perform some non-computable activity. That would mean physics cannot be accurately simulated in a computer, which I believe would be earth-shaking in that world. That brains can be reproduced in a simulation is a default assumption, that something composed of molecules can produce outcomes that cannot be computed is an extraordinary claim, for which, I believe, there is no evidence.

The point is, I think, people from the steam engine era had similar reasons why the mind/soul was exactly like a steam engine. I won't try to reproduce them here, but I'm sure there were convincing arguments at the time. Who has the awareness to claim, before the current fashionable technology becomes unfashionable, that maybe no, the brain is not a close match for an information processing machine? ;)