Although I have reached the conclusion that we have no useful predictive information, despite a possible `perfect' understanding, I have not yet addressed the issue of the root of emergence and the loss of information. Indeed, can we justify that emergence truly exists?
Many complex systems have been shown to be capable of universal computation (e.g. CA with as few as 14 states). Therefore many questions about their infinite time behaviour are formally undecidable. In the case of finite size or time, what happens to analogues of these undecidable propositions? I posit that they are emergent - the computations retain their irreducibility (see [15][17][16] for a heuristic discussion of irreducibility). Therefore the root of emergence and the commensurate loss of information lie in the foundations of decidability and complexity in computation.
Note that these complex systems are often only universal for very particular, rare initial conditions. Following [17], I would suggest that emergence is far more common and occurs in systems which are not computationally universal.
Can we determine, for a given system, whether or not it is emergent? This question seems rather more subtle, now that we have reduced an apparent dichotomy to a continuous parameter range. The answer may be extremely difficult to determine for systems near the transition.