I find it very confusing that you seem to use "real number" and "floating point number" interchangeably. Like this: > The DFT over the reals is not lossless.

I find it very confusing that you seem to use "real number" and "floating point number" interchangeably. Like this: > The DFT over the reals is not lossless. Using floats, for example, and applying it to max_float overflows, so cannot be inverted. I don't see why you make statements about the reals based on what happens with floats. They aren't the same, and the DFT exists independently of actually computing it.