> The idea that humans are physical systems with Turing-computable behavior would mean that any problem not solvable by Turing machines is also not solvable by humans.

But there are real-world instances that contradict this hypothesis already. Humans don't exhibit Turing-computable behaviour because they did indeed solve problems that aren't solvable by Turing Machines.

I can give a very concrete example as well. Consider the following program (in Python for simplicity's sake):

  def nat(x):
    yield x
    yield from nat(x+1)

  for a in nat(2):
    for b in nat(2):
      for c in nat(2):
        if a*a*a + b*b*b == c*c*c:
          print(f"{a}^3 * {b}^3 = {c}^3")
          exit()

There is no Turing Machine capable of proving that this program won't halt (technically it will, due to practical limitations).

Yet it has has been proven centuries ago that the above program won't halt.

The mathematicians couldn't have used a Turing Machine-compliant computation, so something else must be involved.

Now this of course doesn't mean that humans are generally capable of solving every problem that's not solvable by an algorithm, but it means that they can solve problems that algorithms can't solve.