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Which Basis Is More Fundamental?

As you have explained things so far, Schroedinger's equation is local in configuration space, not necessarily physical space. You seem to be claiming locality in physical space as well.

You can compute the rate of change of amplitude in a configuration from derivatives of amplitude over similar configurations. Are you claiming you can also compute that rate of change from a much lower-dimensional neighborhood of configurations that only have changes in a local patch of space?

Even more, I don't see how to slice the wavefunction locally. You could fix the state of a small patch of state, but that leaves you with a function giving amplitudes to every configuration of the rest of the universe, which doesn't seem very local. How should you locally model an EPR experiment?