What do you mean "refutation"? This is making you look as much of a crank as they are. There is no imaginary requirement to have a direct counterpart in the physical world, in "matter/energy".
Infinity, axiom of choice, .. are all tools. We use them because their presence gives us other helpful results. None of it has anything to do with "lumpiness of matter", "matter/energy", "experiental 4-space", "real line".
Refutation as in "you overlooked this standard point that undermines your position", like maybe "polarised light describes a perfect circle", or "the gravitational field of a black-hole (when shaved) is a sphere".
AFAIK there's nothing truly infinite, nor perfectly geometric, that we can experience in our physical spacetime -- I'd like to hear from those who can show me that's wrong.
The world itself isn't axiomatic, if certain theories/systems that are unfalsified show parts of it are infinite/etc. -- despite that just being a model -- that's an interesting result.
Infinity, axiom of choice, .. are all tools. We use them because their presence gives us other helpful results. None of it has anything to do with "lumpiness of matter", "matter/energy", "experiental 4-space", "real line".