Is there an associated machine-checked proof of this?
We're in full vibe-code mode at work, so I understand both how powerful frontier models can be and how often they can over-confidently state subtly (or not so subtly) wrong things, even when you're taking great efforts to try to keep that from happening.
So without a Lean development or extensive human verification, I guess I'm a little bit skeptical, and even sort of hoping this is wrong - not just because of my not so positive feelings about AI, but by my disposition towards beauty in math. n log n is an awful lot nicer than what we have here.
Agree 100% on wanting machinr verification of AI generated math.
But in regards to beauty, i feel like multiplication already has a lot of non beautiful exponents. Best known matrix multiply is O(n^2.371). For integer factorization, the inverse of this problem, general number field sieve is a crazy subexponential.
If factorization is just barely subexponential, is it really that surprising that multiplication is just barely sub n lg n ?
Me too. This is the sort of thing that would traditionally be hidden as n lg n ^ (1 - eps) for some eps > 0, but it's much more amusing this way.
No way this is the correct upper bound, and I imagine it'll get refined fairly quickly. IIRC, the GapCVP results were released with a 1/n^400 complexity term, but people quickly got it down to 1/n^8 by more careful accounting.
This also reaffirms my (wishful) thinking that if there’s a way to do FTL communication it’ll be something with an absurdly tiny factor like 2^-182 with a slight asymmetry in a probability somewhere.
Then you’re not violating FTL, just gaining a very slight chance that you might know something FTL – probably.
Given that c is the speed of causality itself, FTL communications would effectively be like predicting the future.
From that angle, beating light speed by some absurdly tiny factor would probably correspond to a means of predicting the future at some almost absurdly tiny factor better than random guessing.
Edit: Actually...it doesn't make sense to call this FTL communication, it's just predicting the future state of a system given some previous state. FTL comms would have to be predicting the future state of a system without information about the previous state.
Practically speaking predictive modeling would be a means of compensating for light speed comms, kind of like branch prediction in processors or speculative decoding in LLMs, but that wouldn't actually be FTL comms.
Exactly! It’s not forbidden, just very unlikely and would be very strange.
It’d likely involve exponentially more energy as well. It’d be a good sci-if plot point if FTL communications required machines the size of Jupyter to get a few milliseconds of prescience.
You can bootstrap a tiny duration of prescience into arbitrary durations by passing back the same message over and over as many times as you like.
If you know what will happen in one minute, write down the message you see yourself writing down in one minute. In a minute, do the same thing. Now you can pass messages back two minutes.
What if I see myself writing a message in one minute, and thus I dont need to send that message back in in time for me from the future, coz ive already seen it, so I wont write it down for me in a minute?
Ah that's no fun. Maybe something like a quant sending a message to his mistress to get out of his vacation home because his wife is on the way, but the wife is cheating on the quant with the CEO of one of his investments which is the firm that's doing the FTL communication, and the signals get mixed up so the CEO the quant both arrive at the office, ready to go, and meet each other there and it's really awkward.
I think it's more likely that modern physics are wrong than that all of those weird things happen. We might simply discover a preferred reference frame, instead.
I mean, being pedantic a little, we don't actually know if c is constant, since measuring c is rather difficult. If c is not in fact constant in some medium or environment, then a huge number of things get very weird very fast.
So, yes, we could've measured c wrong. We just would have no idea if we did.
Source: Veritasium did a very fascinating video explaining this problem.
> So, yes, we could've measured c wrong. We just would have no idea if we did.
We know we've got it pretty well close to accurate, for the bulk of the observable universe.
If c changes, then chemistry changes, and stuff like hydrogen absorption lines shift. This is how scientists have looked for changes in c in the early universe.
If we have variable c, it would be indistinguishable.
We even use the Lorentz transform, developed for post-Michelson aether theories, precisely because you cannot tell whether you’re in a varying aether or a varying geometry.
We define distance and time relative to c, so we cannot measure it in the usual sense.
Ok. I liked how in this thread some people mentioned instead "faster than causality" instead. Which is both obviously non sense and more relevant than "light".
Causality is conceptually out of time, so it's not traveling. Instead we except causality to operate everywhere anytime uniformly, and all physical dimensions to be bound by causality.
Can causality even conceptually exist outside of time? That sounds like nonsense to me, whereas "faster than causality" seems like a pretty straightforward idea.
The relative difference is so absurdly small to be irrelevant at any realisable input size. At least that's my read; e.g. even at n=10^80 (~number atoms in universe), the relative difference is ~0. That's still probably underselling how similar this is to n log n.
this is how all the proofs today are looking, they seems so minimal even when compared to minor improvements in these fields from that last 10 years im wondering why even publish these and not just make research notes public
O(n lg n) is a bit of a threshold value. For a lot of algorithms, this is the best you can do, even in theory (similar to how O(n^2) is also a threshold for many algorithms). So for many algorithms, people stop trying when they get close to O(n lg n) on the belief that you'll never do better than that.
The fact that you can in principle go faster than n lg n, even if just by an almost imperceptible amount, is kind of surprising. It raises the question of, if n lg n isn't the limit, what is? How far down can we get the speed? If we can get it a little past n lg n, maybe we can go a lot further.
[or at least that is my understanding. not a theoretical computer scientist]
It's potentially very interesting because there are a number of critical algorithms that are all related and share asymptotic lower bounds as a result. My first thought on seeing this was whether a proven result here would also open the door for FFTs to go below O(n log n).
"We give a deterministic algorithm that computes the discrete Fourier transform at every length n in O ( n ( log n ) ** (1 − 10 ** −13)) operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included."
It's like when Tony Hawk did a 900 for the first time. Now 900s in skateboarding aren't a big deal, kids can do it now. It was proving to the world what was possible was the mental hurdle that inspires others to actually try at the problem harder.
It shows that nlogn is not the limit, how much better we can go? Not sure, probably not much, but breaking the barrier is important. Like in marathon for years nobody thought human could break the 2 hour barrier, then someone did and now it's become normal occurrences.
Multiplication can be done by FFT, which is an exceptionally efficient algorithm that has nlogn complexity, you'd be called crazy if you claim you have something more efficient than FFT.
If you view them as "theories of computational limits" instead of "proposed practical speedups" they can be a lot more interesting.
It's most interesting when the lower bound can actually be proven. In lack of that, we have to guess what the best possible algorithm might yield (generalized or not). This tells us that need not be O(n log n) and we have the opportunity to still find better algorithms than we typically thought would be possible. This does the latter, which is interesting, but it just leaves us to hunger more for what the real limit must be :).
i understand this, but it always feels like we are being tricked when they say "integer multiplication below nlogn" because we intuit that that must mean "faster integer multiplication below nlogn EVERYWHERE!". but in reality it comes with 15 asterisks about the conditions that must be true for their statement to hold true.
Your issue is that I am viewing this proof as what it really is in terms of progressing the field and not from an imaginative perspective. I think that it is important to ground our selves somewhat in reality when discussing research like this because at the end of the day open ai is not doing for fun either.
openai wants to show the world what their product can do and i am simply not impressed
I can respect your opinion if it's consistent- i.e. it's not just directed at OpenAI's results. But this seems to be an example of a common phenomenon with AI discourse: while disparaging LLM achievements, you indirectly insult the careers/accomplishments of 99% of mathematicians for whom this would easily be the crown jewel of their CV.
This progresses the field a great deal, just perhaps not the field you're interested in? There's nothing wrong with a "and what can I apply that to in my life tomorrow" approach but it's certainly not the only approach worth having in the world.
Incidentally, the previous most efficient known algorithm for integer multiplication was co-discovered by the chief author of GNU TeXmacs, a typesetting word processor.
I mean, cracking anything below the nlogn bound implies that there might be much more room for improvement. Often a very minor win over the theory opens up enough extra attention to later truly move the needle.
It's 50 pages and cites this other paper in the same repo:
OpenAI. An explicit power saving for the exact discrete Fourier transform.
Here's a random excerpt:
8.3 The middle transform and the final permutation
The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.
There are several of these "exponent used to be 1.0, we reduced it to 0.9999999" results in the "catalog".
There are also a bunch of other stinkers, like building a Turing machine out of Navier-Stokes fluids -- except that it only works if you can encode literally infinite amounts of data in the relative positions of two particles. I.e. assuming physics is based on set-theoretic real numbers, something we've known is wildly false for over a century: https://en.wikipedia.org/wiki/Banach-Tarski_paradox
Just like vuln reporting, the AI industry has put zero effort into triage here, and the models are really good at making their findings sound more important than they really are.
The cynic in me suspects this is a smokescreen for the Navier-Stokes tokenstream plagarism fiasco.
It’s not meant to be an engineering improvement, it’s an important theoretical result because it casts doubt on things that were previously thought to be impossible.
It is surprising, but physicists don't care much for set theory. Heck in their daily work they still use infinitesimals!
So yeah, among the few physicists who understand enough set theory to comment on this, most of them are totally fine with admitting that our reality is not based on set-theoretic real numbers.
For mathematicians, sets are the base reality and everything else is constructed in terms of those. For physicists experimental validation is the base reality, and if it disagrees with set theory that's okay with them; it just means that the set-theoretic real numbers aren't the correct real numbers for doing physics. They're very pragmatic when it comes to mathematical foundations, and that's probably the right approach to take for doing physics.
There have been a few attempts to formalize real numbers that more closely match our observed reality; look into measurable sets. Most of these require a large cardinal, so they go beyond ZFC.
I love this, entirely separate from any applications or even understanding. It's incredible that we needed this trillion-dollar technology to learn about a faster way to multiply two numbers!
Math is incredibly rich, and even the simplest things have insanely complicated structure when you zoom in. However this all ends up, math is bigger than LLMs, and the people who claim it is getting "solved" and we are running out of open problems haven't stared into the abyss enough.
Sure, all of this might end up being very unpleasant, and I'm glad not to be a mathematician right now. But that's still better than a future where there aren't even any questions left that we can understand and an AI can't solve.
Also, it still seems that AI has a much different style from humans, with more brute force and using obscure literature results, and the future might still end up human/AI complementary. We aren't in an AlphaZero situation where the AI learns everything through self-play. (Yet? But we don't even seem to be moving that way much? Can anybody qualified help out?) Things are just moving really fast now and it's hard to process everything.
What I meant is e.g. the unit distance construction, which (per mathematicial comments) needed a combination of distant fields, so nobody had the necessary expertise. That's different from working within one obscure field.
We shaved a whole: 1/6129982163463555433433388108601236734474956488734408704 off the nlogn
I thought this was hyperbole, but no.
Is there an associated machine-checked proof of this?
We're in full vibe-code mode at work, so I understand both how powerful frontier models can be and how often they can over-confidently state subtly (or not so subtly) wrong things, even when you're taking great efforts to try to keep that from happening.
So without a Lean development or extensive human verification, I guess I'm a little bit skeptical, and even sort of hoping this is wrong - not just because of my not so positive feelings about AI, but by my disposition towards beauty in math. n log n is an awful lot nicer than what we have here.
Agree 100% on wanting machinr verification of AI generated math.
But in regards to beauty, i feel like multiplication already has a lot of non beautiful exponents. Best known matrix multiply is O(n^2.371). For integer factorization, the inverse of this problem, general number field sieve is a crazy subexponential.
If factorization is just barely subexponential, is it really that surprising that multiplication is just barely sub n lg n ?
Matrix multiplication is <=O(n^2.25) actually... [1]
1. https://github.com/openai/math/blob/main/preprints/Matrix-Mu...
We can just wait for whomever they stole THIS proof from to come forward with threatening emails sent by OpenAI.
I laughed out loud at the n lg n ^ (1 - 2^{-182}). It is so funny.
2^-182 is very funny but it's bigger than 0 and that's going to shatter a lot of people's conjectures.
Me too. This is the sort of thing that would traditionally be hidden as n lg n ^ (1 - eps) for some eps > 0, but it's much more amusing this way.
No way this is the correct upper bound, and I imagine it'll get refined fairly quickly. IIRC, the GapCVP results were released with a 1/n^400 complexity term, but people quickly got it down to 1/n^8 by more careful accounting.
> IIRC, the GapCVP results were released with a 1/n^400 complexity term, but people quickly got it down to 1/n^8 by more careful accounting.
sqrt(n) now https://github.com/Mira-acc/cvp
Wowzers!
This also reaffirms my (wishful) thinking that if there’s a way to do FTL communication it’ll be something with an absurdly tiny factor like 2^-182 with a slight asymmetry in a probability somewhere.
Then you’re not violating FTL, just gaining a very slight chance that you might know something FTL – probably.
Given that c is the speed of causality itself, FTL communications would effectively be like predicting the future.
From that angle, beating light speed by some absurdly tiny factor would probably correspond to a means of predicting the future at some almost absurdly tiny factor better than random guessing.
Edit: Actually...it doesn't make sense to call this FTL communication, it's just predicting the future state of a system given some previous state. FTL comms would have to be predicting the future state of a system without information about the previous state.
Practically speaking predictive modeling would be a means of compensating for light speed comms, kind of like branch prediction in processors or speculative decoding in LLMs, but that wouldn't actually be FTL comms.
I think that is entirely expected from what we know of modern physics. It doesn't say you can't do it, just things are very weird if you can.
Exactly! It’s not forbidden, just very unlikely and would be very strange.
It’d likely involve exponentially more energy as well. It’d be a good sci-if plot point if FTL communications required machines the size of Jupyter to get a few milliseconds of prescience.
You can bootstrap a tiny duration of prescience into arbitrary durations by passing back the same message over and over as many times as you like.
If you know what will happen in one minute, write down the message you see yourself writing down in one minute. In a minute, do the same thing. Now you can pass messages back two minutes.
What if I see myself writing a message in one minute, and thus I dont need to send that message back in in time for me from the future, coz ive already seen it, so I wont write it down for me in a minute?
First, Don't Panic! and consult your copy of Dr. Dan Streetmentioner's Time Traveler's Handbook of 1001 Tense Formations.
reveal at the end of the story: the mysterious purpose for which all of that was built? high frequency trading.
Ah that's no fun. Maybe something like a quant sending a message to his mistress to get out of his vacation home because his wife is on the way, but the wife is cheating on the quant with the CEO of one of his investments which is the firm that's doing the FTL communication, and the signals get mixed up so the CEO the quant both arrive at the office, ready to go, and meet each other there and it's really awkward.
I think it's more likely that modern physics are wrong than that all of those weird things happen. We might simply discover a preferred reference frame, instead.
If there was a way to do FTL communications you’d expect that Jane Street would have found it already
!did ydaerla eW
Are time and distance, and consequently velocity, even well defined enough at that level of precision on any physical scale?
That would only mean we calculated c wrong
I mean, being pedantic a little, we don't actually know if c is constant, since measuring c is rather difficult. If c is not in fact constant in some medium or environment, then a huge number of things get very weird very fast.
So, yes, we could've measured c wrong. We just would have no idea if we did.
Source: Veritasium did a very fascinating video explaining this problem.
> So, yes, we could've measured c wrong. We just would have no idea if we did.
We know we've got it pretty well close to accurate, for the bulk of the observable universe.
If c changes, then chemistry changes, and stuff like hydrogen absorption lines shift. This is how scientists have looked for changes in c in the early universe.
After his "fascinating" video on speed of electricity no one should put science (even pop-sci) label anywhere near Veritasium.
If we have variable c, it would be indistinguishable.
We even use the Lorentz transform, developed for post-Michelson aether theories, precisely because you cannot tell whether you’re in a varying aether or a varying geometry.
We define distance and time relative to c, so we cannot measure it in the usual sense.
What's FTL?
faster than light
Ok. I liked how in this thread some people mentioned instead "faster than causality" instead. Which is both obviously non sense and more relevant than "light".
Causality is conceptually out of time, so it's not traveling. Instead we except causality to operate everywhere anytime uniformly, and all physical dimensions to be bound by causality.
Can causality even conceptually exist outside of time? That sounds like nonsense to me, whereas "faster than causality" seems like a pretty straightforward idea.
Dangit! I was betting on -183.
You didn't believe!
This is better
Why is that funny?
The relative difference is so absurdly small to be irrelevant at any realisable input size. At least that's my read; e.g. even at n=10^80 (~number atoms in universe), the relative difference is ~0. That's still probably underselling how similar this is to n log n.
this is how all the proofs today are looking, they seems so minimal even when compared to minor improvements in these fields from that last 10 years im wondering why even publish these and not just make research notes public
The -182 feels highly arbitrary.
For the uninitiated, why is this interesting given it doesn't seem to be so much below the threshold?
O(n lg n) is a bit of a threshold value. For a lot of algorithms, this is the best you can do, even in theory (similar to how O(n^2) is also a threshold for many algorithms). So for many algorithms, people stop trying when they get close to O(n lg n) on the belief that you'll never do better than that.
The fact that you can in principle go faster than n lg n, even if just by an almost imperceptible amount, is kind of surprising. It raises the question of, if n lg n isn't the limit, what is? How far down can we get the speed? If we can get it a little past n lg n, maybe we can go a lot further.
[or at least that is my understanding. not a theoretical computer scientist]
It's potentially very interesting because there are a number of critical algorithms that are all related and share asymptotic lower bounds as a result. My first thought on seeing this was whether a proven result here would also open the door for FFTs to go below O(n log n).
https://github.com/openai/math/blob/main/preprints/An-explic...
"We give a deterministic algorithm that computes the discrete Fourier transform at every length n in O ( n ( log n ) ** (1 − 10 ** −13)) operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included."
[dead]
It's interesting because people wondered if it was possible to go below the threshold at all, that's all. Many suspected it was not possible.
It's like when Tony Hawk did a 900 for the first time. Now 900s in skateboarding aren't a big deal, kids can do it now. It was proving to the world what was possible was the mental hurdle that inspires others to actually try at the problem harder.
It shows that nlogn is not the limit, how much better we can go? Not sure, probably not much, but breaking the barrier is important. Like in marathon for years nobody thought human could break the 2 hour barrier, then someone did and now it's become normal occurrences.
Multiplication can be done by FFT, which is an exceptionally efficient algorithm that has nlogn complexity, you'd be called crazy if you claim you have something more efficient than FFT.
I would like a lean formalization of this result - this looks like like an extremely strong result.
this is perfect for when i have an array of at LEAST 2^118000 items
i will NEVER care about proposed multiplication speedups unless they are truly generalized
If you view them as "theories of computational limits" instead of "proposed practical speedups" they can be a lot more interesting.
It's most interesting when the lower bound can actually be proven. In lack of that, we have to guess what the best possible algorithm might yield (generalized or not). This tells us that need not be O(n log n) and we have the opportunity to still find better algorithms than we typically thought would be possible. This does the latter, which is interesting, but it just leaves us to hunger more for what the real limit must be :).
i understand this, but it always feels like we are being tricked when they say "integer multiplication below nlogn" because we intuit that that must mean "faster integer multiplication below nlogn EVERYWHERE!". but in reality it comes with 15 asterisks about the conditions that must be true for their statement to hold true.
Your issue is that I am viewing this proof as what it really is in terms of progressing the field and not from an imaginative perspective. I think that it is important to ground our selves somewhat in reality when discussing research like this because at the end of the day open ai is not doing for fun either.
openai wants to show the world what their product can do and i am simply not impressed
I can respect your opinion if it's consistent- i.e. it's not just directed at OpenAI's results. But this seems to be an example of a common phenomenon with AI discourse: while disparaging LLM achievements, you indirectly insult the careers/accomplishments of 99% of mathematicians for whom this would easily be the crown jewel of their CV.
This progresses the field a great deal, just perhaps not the field you're interested in? There's nothing wrong with a "and what can I apply that to in my life tomorrow" approach but it's certainly not the only approach worth having in the world.
Well, that’s true for a lot of TCS algorithms. The n log n algorithm prior was also not very practical for any numbers relevant to humans.
I wonder if the AI spent extra time on this without being told to
Why is an openai release in .pdf? Isn't all ai in .md now?
Equations.
Incidentally, the previous most efficient known algorithm for integer multiplication was co-discovered by the chief author of GNU TeXmacs, a typesetting word processor.
I mean, cracking anything below the nlogn bound implies that there might be much more room for improvement. Often a very minor win over the theory opens up enough extra attention to later truly move the needle.
This is pretty remarkable, IF someone can understand it :)
I only skimmed the paper but it doesn’t see particularly dense, mostly just relying on college math?
It's 50 pages and cites this other paper in the same repo:
OpenAI. An explicit power saving for the exact discrete Fourier transform.
Here's a random excerpt:
8.3 The middle transform and the final permutation The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.
Sure, I don’t see what’s horrible about this? It is a lot to read, sure, but it doesn’t seem unreasonably advanced
There are several of these "exponent used to be 1.0, we reduced it to 0.9999999" results in the "catalog".
There are also a bunch of other stinkers, like building a Turing machine out of Navier-Stokes fluids -- except that it only works if you can encode literally infinite amounts of data in the relative positions of two particles. I.e. assuming physics is based on set-theoretic real numbers, something we've known is wildly false for over a century: https://en.wikipedia.org/wiki/Banach-Tarski_paradox
Just like vuln reporting, the AI industry has put zero effort into triage here, and the models are really good at making their findings sound more important than they really are.
The cynic in me suspects this is a smokescreen for the Navier-Stokes tokenstream plagarism fiasco.
It’s not a stinker.
It’s not meant to be an engineering improvement, it’s an important theoretical result because it casts doubt on things that were previously thought to be impossible.
Wait is that the consensus interpretation of banach-tarski for physicists? That reality must not encode reals? I’m surprised by this.
It is surprising, but physicists don't care much for set theory. Heck in their daily work they still use infinitesimals!
So yeah, among the few physicists who understand enough set theory to comment on this, most of them are totally fine with admitting that our reality is not based on set-theoretic real numbers.
For mathematicians, sets are the base reality and everything else is constructed in terms of those. For physicists experimental validation is the base reality, and if it disagrees with set theory that's okay with them; it just means that the set-theoretic real numbers aren't the correct real numbers for doing physics. They're very pragmatic when it comes to mathematical foundations, and that's probably the right approach to take for doing physics.
There have been a few attempts to formalize real numbers that more closely match our observed reality; look into measurable sets. Most of these require a large cardinal, so they go beyond ZFC.
Related:
Sharing AI Progress in Mathematics
https://news.ycombinator.com/item?id=49984923
I love this, entirely separate from any applications or even understanding. It's incredible that we needed this trillion-dollar technology to learn about a faster way to multiply two numbers!
Math is incredibly rich, and even the simplest things have insanely complicated structure when you zoom in. However this all ends up, math is bigger than LLMs, and the people who claim it is getting "solved" and we are running out of open problems haven't stared into the abyss enough.
But who is going to be the one solving them? Just query an LLM no?
Sure, all of this might end up being very unpleasant, and I'm glad not to be a mathematician right now. But that's still better than a future where there aren't even any questions left that we can understand and an AI can't solve.
Also, it still seems that AI has a much different style from humans, with more brute force and using obscure literature results, and the future might still end up human/AI complementary. We aren't in an AlphaZero situation where the AI learns everything through self-play. (Yet? But we don't even seem to be moving that way much? Can anybody qualified help out?) Things are just moving really fast now and it's hard to process everything.
> with more brute force and using obscure literature results,
Is that true? Look at the average math paper on arxiv. Of course it's obscure to those not in the exact sub-field - math has become very specialized.
What I meant is e.g. the unit distance construction, which (per mathematicial comments) needed a combination of distant fields, so nobody had the necessary expertise. That's different from working within one obscure field.
I never said its unpleasant, what I'm meant math in its current (social) form / interface is "solved"