> The choice quote from part 1: "[our] final interpretation says that every spacetime is locally approximately flat in the sense that near any point of any spacetime (or near sufficiently small segments of a curve), there exists a flat metric that coincides with the spacetime metric to first order at that point (or on that curve) and approximates it arbitrarily well," [emphasis mine].
This statement is a mathematical conclusion from GR of a similar nature to noting that for any point on a sphere, there is a map projection onto the plane where distances on the sphere coincide to first order to distances on the plane.
This no more or less means that space-time is locally flat than it means that a sphere is locally flat. To a first order approximation, it is. But when we calculate the curvature tensor, we find that it isn't flat at all.
How do you propose to measure the ultralocal Riemann curvature? I agree with you that it can be there in an exact solution, but I don't really agree that it's necessarily physical (there are many many exact solutions which aren't, for starters). Even with a broader definition of local, a Synge 5-point like process is not going to find a failure of parallel transport in a sufficiently small region. Would you be satisfied with "effectively flat" (in an EFT sense) or even "FAPP flat"?
This statement is a mathematical conclusion from GR of a similar nature to noting that for any point on a sphere, there is a map projection onto the plane where distances on the sphere coincide to first order to distances on the plane.
This no more or less means that space-time is locally flat than it means that a sphere is locally flat. To a first order approximation, it is. But when we calculate the curvature tensor, we find that it isn't flat at all.