The same way we distinguish the shadows cast in Plato's cave from the objects occluding the light. The more situations in which a theory or model makes accurate predictions, the more correct it is. Epicycles are much, much more wrong than mathematically perfect elliptical orbits.
> Epicycles are much, much more wrong than mathematically perfect elliptical orbits.
Not if you define "wrong" as "inaccurate predictions". You can approximate ellipses with circles and epicycles to any desired degree of accuracy by putting in more epicycles. So you can match the predictions of ellipses to any desired accuracy with epicycles.
Also, as I noted, the actual orbits of the planets are not perfect ellipses once GR effects are taken into account. Have you proven mathematically that it is impossible to construct an epicycle model that makes more accurate predictions than perfect ellipses, based on the actual data (which confirms the GR predictions to within current observational accuracy)?
You're being intentionally obtuse. "Just add more epicycles!" isn't building a better model (for a sensible definition of "better"). It's just overfitting. You are the reason why regularization exists. https://en.wikipedia.org/wiki/Regularization_(mathematics)
How do you distinguish "an entirely imaginary math artifact" from the "principles" that "reality works on"?
(Hint: planetary orbits are not ellipses once you take GR effects into account.)