Don't close that book, you might eventually find either that it can make more sense to you, or that it shouldn't need to. I think we mistake familiarity for understanding-- we are all familiar with the concept of a trajectory and much fuzzier about the concept of a wave, but if you think about it, trajectories are a much more bizarre concept than a wave. A wave constantly recreates itself, as everywhere it passes becomes a source for the next stage in its propagation and a sink that cancels where the wave has been-- out with the old, in with the new, makes perfect sense. A trajectory, on the other hand, asks us to believe that somehow a single particle can "know" its speed and direction, so it can manifest an inertia. How exactly does it do that anyway, and how did such a bizarre notion ever get to be thought of as more sensible than wave propagation? Was Galileo any less a genius of his day to understand this strange concept of inertia than was Schroedinger for understanding that the time evolution of a wave function was the key concept?

That's interesting, it speaks very much to the way solutions are emergent of theories, rather than explained by theories.

I didn't see it, but it sounds like something I'd agree with heartily. I always laugh at the concept of a "theory of everything", because I'm reminded of Galileo's comments to the effect that any time you truly discover something, you are not rewarded with a sense of how well you understand the universe, but rather, you are awed by the realization of how much more there is to understand. The ever-present "yes, but why", I imagine.I don't know how well we know that, many of us know that the surface area of a black hole is to be associated with having entropy so you can get one to form despite the loss of entropy everywhere else. But connecting that to a ball rolling down an incline isn't terribly obvious! Still, I may see where you are going with that.
Yes, maybe can we do gravity in QM like the way the energy levels of hydrogen are calculated without QED, and maybe not. The problem would be ever testing the result. My point is merely that we might imagine we can do this by analogy with electric forces, but if very strong gravitation isn't unifiable with such forces, maybe very weak ones aren't either, and they only overlap "in the middle".
It's possible that I'm using the term "correspondence" in a nonstandard way, or a way that is not well known outside the physics community. I think of a "correspondence principle" as the statement that one theory is in effect a strict subset of another, superceded as it were, with no new insights not available in the more fundamental theory, but possibly some computational advantages based on approximations or idealizations suggested by the superceded theory. When quantum mechanics was first formed, it was a point of some note that it did exhibit a correspondence to classical mechanics, in that with the appropriate simplifications applied at the QM end of things, all results of CM were consistent with QM predictions. CM at that point became a superceded theory that was only used for computational convenience when quantum accuracy was not needed.

The reason such correspondence is relevant is that if you think the goal is not to find a set of axioms that works in some new situation, but rather a more fundamental theory that subsumes all the old ones with which it overlaps, then you have to have a correspondence principle. But it seems to me that a crucial violation of such a correspondence is what one might call "emergent properties" of some particular theory. Like the way irreversibility is emergent of thermodynamic treatments of a gas, whereas the statistical mechanical equations, or the quantum mechanical ones for that matter, are entirely reversible. The reversibility of the "more fundamental" theory gets supplanted by the irreversibility of the "more practical" simplified over-theory. The property of determinism that exists in CM is lost by any real application of its uncertainties, so it is kind of an illusory propery compared to reality, and the over-theory sees through the illusion and replaces it with something that works better. In that sense the over-theory does not exhibit correspondence to the fundamental theory that is supposed to give rise to it-- the whole is greater than the parts, and contains a new concept that is not present in the purportedly more fundamental theory.
What I'm suggesting is, since GR is compatible with any imaginary physical interactions that don't violate its axioms, whether they are possible or not, finding a self-consistent GR solution to any particular physical interactions you select would seem to involve interplay between GR and those interactions. So the EFE seem more like a prescription for unifying any cockamamie physics you like into a holistic entity that includes gravity, and as such, it is not built on the concept of correspondence, it simply borrows whatever correspondences exist in the cockamamie physics and then in a single flourish calculates the dynamical evolution of such a system. If so, it has no general concept of correspondence built into it, it is an entirely "emergent" phenomenon in general.

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