But we have at least measured the curvature of the observable universe and within experimental error the result is that it is flat. Which in turn would imply an infinite universe if this flatness is exact and general, i.e. not something exceptional in the observable universe.

Yes actually, particle motion is quantized on the planck scale, that is, snapped to the grid. That's why it's called quantum physics. Modern theories even have it that there are no particles, rather, excitations of various quantum fields.

So what we consider a discrete photon particle with a location (x,y) would actually not be an individual particle and would be more like:

  electromagnetic_field[y][x] = "excited" 

in a big (presumably infinite) grid.

There's a point in the very distant future where someone in our local group will observe nothing in the beyond. Kind of makes you wonder what we might have missed.