Identifying the interval from A♭ to D♯

This is a very theoretical question but it's confusing me a little.

So intervals overlap - That's why I can call a major 3rd a diminished 4th, and such.

However, from what I understand, a perfect 5th only overlaps with a diminished 6th. You cannot refer to a 7 semitone interval using a 4th because the "biggest" fourth, augmented fourth, is a diminished 5th.

So consider this (treble clef):

What would you call the interval here? From what I understand it should be called a 4th because of the distance between the notes. But that's impossible.