> the algorithm requires you to multiply complex numbers whose magnitudes grow superpolynomially with n.

Err... doesn't that completely invalidate the big-O analysis!?

I suspected that these Open AI "proofs" of sub-nlogn times were cheating somehow, by sweeping something under a rug. This seems to be it!

I can't speak for the FFT claim, but least the sub-O(n log n) multiplication claim is for the multitape Turing machine model with a fixed number of tapes, which if anything is more restrictive than actual random-access memory. Of course, it's still galactic just like the O(n log n) algorithm, but it definitely seems like there's something there.

You can be linear in the bit length of the input number, but exponential in the magnitude.

This is a common thing in complexity, and I think even the abstract says it is talking about “length n” and not “magnitude n”