The recently proposed o(n log n) integer multiplication algorithm [pdf]

> It would be good to have a formalised proof of the o(n log n) result, for example in Lean. As far as I can tell, one does not yet exist.

The result is fully formalized in Lean by @platypii and myself (on Mathlib's TM2 machine), with ongoing improvements to the exponent (currently at 1-2^-14.38): https://github.com/platypii/integer-mult-bounds-lean

Contributions are welcome!

> The new result does not directly imply new complexity bounds for polynomial multiplication over a fixed finite field, say multiplication in F2[x]. The state of the art for this problem is an algorithm that multiplies polynomials of degree n in time O(n log n 4^log*(n)) [HvdH19].

@platypii and I obtained a Lean proof of the same improvement for DFT over prime-order finite fields (yielding polynomial multiplication via the convolution theorem): https://github.com/platypii/integer-mult-bounds-lean/blob/ma...

Note in this case the title says "O(n log n)", but the important part is this is "o(n log n)", not just "O(n log n)'. The former is a strict subset of the latter. We already knew how to multiply two n-bit integers in O(n log n) time, via the FFT.

> Executive summary (longer version) > The bottom line is that I think the claimed result is plausible. I have not actually read the paper.

> I have not actually read the paper. In fact, there are three closely related papers [TC26a], [TC26b], [TC26c], and I have not read any of them. I tried, but soon gave up. The math(s) dump does not appear to be strong on exposition. Instead, I asked Fable to explain [TC26a] to me. Fable seems to understand the argument and thinks that it is correct.

this does not feel like a healthy place to be

Awesome write up and intro by a fellow Sydney-sider, kept me hooked! As someone who's not a maths wizard, the technical details flew by my head but the prose allowing me to empathise deeply with those in the field was very enlightening. I never realised how big of a deal the maths dump was and how much it has shaken up the field. I guess we are all just trying to figure out this whole "intelligence on tap" thing together.

I really enjoy the writing style here.

edit: I mean this seriously. This kind of thing

> Some of my colleagues believe that this is a nefarious Company, whose only goal is profit, if necessary at our community’s expense. Perhaps there is some truth to this, but I think it is missing the larger point. The fact is that there now exists an automated system that can perform mathematical research — at least the part that involves “solving hard problems” — far better than any human. Now that such a system has been demonstrated, it is clear that results like these were bound to start appearing soon anyway. I do not blame the Company for getting there first. Who got there first is an irrelevant detail, a historical footnote.

is very pleasantly honest and well-thought-out.