CLUES FOR SABINE
Having previously read your article, Minimal Length Scale Scenarios for Quantum Gravity, and with the intro saying you have the ability to shoot down theories, let me offer a few suggestions/ clues/ pointers, with the best of good wishes on the FQXi grant award...
1. You say, "We know the theories we have right now are inconsistent--when you combine them the answer is nonsense", ...and further, "It is clearly not how nature works, there has to be a better answer".
Quantum Mechanics separates space and time, while Special and General relativity welds them together. There are then two probable options, it is either Space and time are separate or they are one entity, spacetime. This being the case, it is rational to apply the welded entity to Quantum mechanics and see if it also works, and in turn apply space and time as separate entities to Relativity and also see if it works. Where they don't work, why not? Are there alternative explanations that would not require welding of space and time?
2. They said, you have been working on models involving flat spacetime because it is easier to do the math. And you are quoted to have said, "We don't live in flat spacetime -- (...)".
I therefore put this question to you. If we don't live in flat spacetime, where were the experiments conducted on which basis the theory for spacetime rests its validity? If an experiment is conducted underwater, can it be used to substantiate or make theoretically infallible claims of what exists above water? You may have heard of the claim made severally that the domain of applicability of Special relativity is flat spacetime, and you have now said that we do not live in such a spacetime, how then can the NULL experimental findings of Michelson and Morley, 1887 be said to be due to the effect of Special relativity on the behavior of light in the experiment, a theory that applies only in flat spacetime which you have said we are not living in.
If you measure the speed of light in curved spacetime, can it have the same value in flat spacetime? Einstein says No, but some physicists insist Yes. More on this later, if necessary.
3. You say, "People like to talk about 'atoms of spacetime'".
In the intro, it is said that Spacetime is the four dimensional fabric conceived of by Albert Einstein in his theories of relativity. It is therefore clear that talking of atoms of spacetime applies only to the conceptions (or misconceptions) of Special and General relativity and does not apply to Quantum Mechanics, where space and time are separate. When looking therefore through a Quantum Mechanical lens, what people will expect you to look out for are 'atoms of space'. It is noted that in the introduction to the Living Reviews article linked above, you have yourself stated and asked, "For one, this is because Democritus' search for the most fundamental constituents carries over to space and time too. Are space and time fundamental, or are they just good approximations that emerge from a more fundamental concept in the limits that we have tested so far? Is spacetime made of something else? Are there 'atoms' of space?".
The study of space is basically geometry, and the definitions of concepts in geometry recognize the atom of space as the 'point'. The next question then is whether this atom has a minimal length or is of zero dimension? Look at this. I will urge that you shoot down the theory that says the atom of space is of zero dimension. You will thereby be removing infinities that are plaguing our physics.
Finally, the greatest clue of all is to find out whether these 'atoms of space' are eternally existing objects. That is, can they perish and cease to exist, and can they arise out of nothing? If they can, would this not be the most fundamental event underlying all other phenomena in our physics? Would such annihilation and emergence of its atoms create 'wrinkles in space'? Would the disparate lifetimes of the atoms not interfere with the otherwise continuous nature of space, thereby causing discreteness with 'time' being the separator since space cannot do its separation into discreteness? If the universe itself can be conceived in some cosmological theories as arising out of nothing in a Big bang, and annihilating to nothing in a Big crunch, can an atom of space then be eternally existing and be incapable of being extinguished?
All the best as you confront these profound theoretical physics problems.
Akinbo