Is there a proof that collections of human neurons are capable of carrying out proofs in first order logic, in full generality?

Is anyone trying to prove that all humans can?

Can a cat reason? It knows nothing of FOL or math, but can be very smart when figuring out how to catch a mouse.

The term 'reason' is being overloaded here.

The type of 'reason' a cat uses is different from the 'reason' used in math

The information a cat uses is incomplete whereas the information used in math and logic is theoretically all accessible.

The reasoning a cat uses allows for inconsistencies with its model because of its use of incomplete information whereas no inconsistencies are permissible in math and logic.

Formally speaking, math uses deductive reasoning whereas the cat uses abductive reasoning.

Transformers with chain of thought can use FOL: https://arxiv.org/abs/2310.07923

"Turing complete as long as you only use a polynomial number of steps in the size of the input" is another way of saying "not Turing complete" (doubly so when you look at the actual polynomials they're using). In fact, that paper just reaffirms that adding context doesn't improve formal expressive power all that much. And this is not equivalent to "well, Turing Machines are fake anyway because of finite space"--I can pretty trivially write programs in Coq, a total and terminating language, that will take far longer than the age of the universe to normalize but can be described in very little input.