# How do we know a single differential/linear trail dominates the others?

I have read the proof of AES's resistance against differential cryptanalysis. In the proof the authors show that there is no single differential trail with prop ratio higher than $$2^{-300}$$ over 8 rounds. I understand that part but it does not prove that there is no differential with prop ratio higher than $$2^{1-n} = 2^{-127}$$ because we could end up with the same differential using many different trails. I have read in Jon Daemen's dissertation that in a well constructed cipher a single trail should dominate all the other trails. Is it proven for AES that, in fact, no different differential trails combine to give a significantly higher prop ratio of a differential? If yes, how do they prove that? If no, why is AES assumed to be proven resistant against linear and differential cryptanalysis?