Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:
> We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
1. Local smoothing and the Kakeya conjecture
2. The Exponential Time Hypothesis is false
3. Existence of quantum chromodynamics and completions of the Standard Model
Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.
ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.
QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.
I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
>We give a deterministic algorithm for exactly counting the satisfying assignments of
an arbitrary k-CNF formula [in superpolynomial but subexponential time]
They do that to make it easier for readers to verify the statement (which is entirely contained in that file) separately from the proof (which is spread over thousands of files and github can't even show them all).
I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
Honestly, I’m a bit scared but also a bit excited.
I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.
The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.
I honestly at this point don’t know anything about what could be possible and what couldnt.
I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.
I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."
The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.
Treating every result from the AI like a gift from god is getting tiring.
In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."
(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)
I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.
$5 million might sustain 5-10 PhD students for 5 years.
Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.
Title is no doubt crafted to get ahead of the usual online backlash, but here’s the actual announcement:
> We are announcing three new discoveries in mathematics, computer science, and physics about harmonic analysis, complexity theory, and quantum field theory.
1. Local smoothing and the Kakeya conjecture
2. The Exponential Time Hypothesis is false
3. Existence of quantum chromodynamics and completions of the Standard Model
To give some context:
Kakeya / local smoothing (math): The result would resolve the Kakeya conjecture and Stein’s restriction conjecture across all dimensions together. Mathematicians had been making progress one dimension at a time, so this would settle a whole family of central harmonic analysis problems at once.
ETH is false (computer science): ETH has been used as a computational hardness premise in hundreds of papers, so a disproof would undercut a large body of existing complexity results. It would also change what researchers think is achievable for efficient computation.
QCD existence (physics): It would give a rigorous mathematical foundation to the theory of the strong force, which physicists have used for decades without proving it’s well-defined. It also bears on a Millennium Prize Problem.
I understand that the Exponential Time Hypothesis being false doesn't necessarily mean P=NP, but for people who are a lot more knowledgeable about complexity theory, does this update your priors significantly in the direction of P=NP?
The title buries the lede quite a bit:
> The Exponential Time Hypothesis is false
https://github.com/openai/math/tree/main/preprints/The-Expon...
>We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula [in superpolynomial but subexponential time]
What a time to be alive honestly. Can’t know what one will wake up to
Oh and also:
>Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026
And presumably they are just missing the "mass gap" part of it.
Very clearly these were the results of taking big swings at all millenium prize problems.
This is just one of the three ledes, really!
https://github.com/openai/math/blob/main/lean/docs/378.md
by fucking sorry written at the end of the proof
https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050...
looks like the real proof is in this file
awesome, thanks!
They do that to make it easier for readers to verify the statement (which is entirely contained in that file) separately from the proof (which is spread over thousands of files and github can't even show them all).
Proof is at https://github.com/openai/math/tree/main/lean/OAI/Algorithms...
I was confused too, but it looks like https://github.com/openai/math/blob/main/lean/ComparatorChal... (with the sorry at the end) is just the problem statement; the actual proof is a big pile of lean in the Algorithms folder, with the final theorem stated here https://github.com/openai/math/blob/7ef5c0cc5ca4b0e072b7a050...
A newbie here, does it mean that SETH is false too?
Yes it means both SETH and ETH are false.
I am not sure if I side with Tao on this, or 3B1B. It's already causing quite a kerfuffle (to say the least) in the academic communities. This is the new norm now.
Honestly, I’m a bit scared but also a bit excited.
I’m currently reading Africa by John Reader which covers a lot of humanity storyline. From HN I get the humanity achievements that are happening right now.
The feeling digesting both daily is in order of magnitudes bigger than what any really exciting tv show did to me.
I honestly at this point don’t know anything about what could be possible and what couldnt.
I understand that this is really freightening, but also especially ETH being false gives even more hope that some fundamental problems could be solved in our lifetime, for example understanding universe, aging, colonization of planets and also our limits. I know I’m the minority, but these answers really interest me.
OpenAi needs to stop trying to please everyone. Just release the findings and give all the credit to your models/collaborators.
They have released them, they are just also hiring mathematicians to clean them up in parallel.
I wonder how much compute they are throwing at problems in ML like continual learning and sample efficiency. Also how successful they are with these. Results like in the post make me hopeful we'll see some type of so singularity soon
Terrible title. I was hoping to see more interesting comments on the story, haha!
The proof appears to be a sub-exponential algorithm for #3-SAT not just 3-SAT, I think this is considered a much stronger result?
Isn't 3-SAT NP-complete?
Up until now, it was unknown whether containment of NP in EXPTIME was strict or not.
It is the canonical NP complete problem.
I'm going to be the cynical one and offer this translation: "Our results are too important to wait for the pesky peer review, so we are dumping a whole ton of money in a fund and giving some elites a breathless first look so our buzz keeps getting jacked up."
The reality that Science is a grind and there are thousands of new results every year discovered by regular humans who use a regular human process to cross-check and review.
Treating every result from the AI like a gift from god is getting tiring.
In the Star Trek Voyager capstone episodes titled "Scorpion" parts 1 & 2, Chakotay explains to Janeway the behavior of the Borg through the parable of the fox and the scorpion making a transactional deal to cross the river together. After reaching the shore, the scorpion stings the fox, and the dying fox asks why. The scorpion replies, "Because it is in my nature."
(Upside of the story is Captain Janeway's trusted crew works together to formulate a trick up their sleeve precisely in case of predicted contingency...)
The fable of the scorpion and the frog is not original to Star Trek.
https://en.wikipedia.org/wiki/The_Scorpion_and_the_Frog
"Working with PR to make us look less like cheating mind leeches".
Working with academic communities lol.
I looked over the announcement for any substantial contribution, and found they're donating $5 million to a research foundation.
$5 million might sustain 5-10 PhD students for 5 years.
Maybe it's just me but I think they should contribute more in exchange for forcing the scientific community into paying for tokens to generate slop and then pouring their time into verifying or falsifying the slop.
No one "forces" anyone to verify it.
I wonder how many prompts for proof approaches they stole for these. Nothing on how many tokens were burned.
Proud to funnel money through the Simons Foundation, which is an AI booster itself and completely aligned with OpenAI.
They make it very hard to enjoy a single result with the business and the politics being so dirty.