The mathematical underpinnings of superconductivity are still in flux, but there's still every reason to believe that it's an emergent phenomenon from known physics. I don't see any obvious reason why consciousness can't be something like that.
In both examples you made we do in fact have good mathematical descriptions that allow us to describe the system and the way it works, and moreover we can infer/predict non obvious things about it. Do you know of anything similar that can be applied to human/animal behavior?
Emergent behavior can be totally opaque, even when we have perfect understanding of the elements of the system. Just because the behavior is totally inscrutable doesn't mean there's anything we are missing about the underlying mechanisms. There's no hidden principle of Turing machines that would let us predict them reliably either- their underlying mechanisms are totally understood, the behavior is still essentially unknowable.
Perhaps living systems are like that. It seems quite likely, since their underlying mechanisms are actually much more complicated than toy models like the Game of Life, that they can manifest totally opaque behavior without requiring any new physics at all.