Yeah its ridiculously small, but any improvement on n log n is wild.

Like there is somehow redundancy in a fourier transform that makes it sub Linearithmic?

Which low and behold ->

130. Fourier transforms below n log n.

They also separately give algorithm for Fourier transform over complex number faster than O(n log n)

Wikipedia just told me there's a galactic algorithm for integer multiplication in O(n log n) based on FFT so I'm guessing those two proofs are related.