alt Hacker NewsThree new discoveries in mathematics, computer science, and physics
Related:
The Exponential Time Hypothesis Is False [pdf] - https://news.ycombinator.com/item?id=50251661
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ETH is false! I really would not have believed it.
[0]:
> We give a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula. For every fixed k ⩾ 2, on a formula with n variables and m clauses, our algorithm uses $poly_k (n, m)$ space and runs in time $(n + m)^{O_k(1)} 2^{n/(\log n)^{1/2 + o(1)}}$
The dominant term there is (roughly) $2^{n / (\log n)^{0.5}}$, so still superpolynomial but less than exponential. Fucking wild.
[0] https://github.com/openai/math/blob/main/preprints/The-Expon...
ETH is false! This is one of the big long-standing hypotheses that I actually remember from university, and afaik, it was reasonable to assume it would be true; reasonable enough for a lot of papers and results to assume this for bounds or hardness proofs of common problems in parameterized complexity, anyways.
Also glad to see that one of the authors of the paper stating it is also in the loop here.
2026 is weird.
There's a large logical leap from "we have very useful raw research data" to "we should give this data only to a small group of people." They explain that "the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community" but we are left to wonder why the experts need to be empowered to foster anything. In the case of immediate release the experts will have their hands on the same data as the rest of us peasants, and then they can go right on fostering.
In 2025, Hong Wang won the Fields Medal for proving the Kakeya conjecture in three dimensions. Here, OpenAI is claiming to have solved a stronger problem that proves the Kakeya conjecture in all dimensions!
That's so insane to me. This top mathematician (and colleagues) at her pinnacle spent years proving it in three dims and now OAI proves it in every dimension...
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?
> Our models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and then subsequently produced a proof for QCD on September 18th, 2026. Given the length and complexity of the work, we postponed release until the Lean formalization completed. We finished our formalization of Yang–Mills’ existence on October 16th, 2026, which led us to develop the collaboration described here
absolutely insane
Think openai should start including visual and interactive explainers in these where the viewer can toggle level to get a more accessible/technical explanation as needed. Would also be a great demo, and perhaps increasingly essential for us to know that explainability is not lagging behind provability.
Lots of progress recently in formally verifiable domains. Really curious if this is a product of the machine learning folks behind most large language models being much more likely to have a background in math, physics, etc. Or if it is that large language models have some fundamental limitations in the more "scruffy" sciences like biology.
My guess is that in formally verifiable domains you can use RL-esque processes to get lots of good high-quality training data with reward signals (does it verify?), but the same does not apply to "scruffier" work. I would also imagine that the data volume of high-quality scientific work in "scruffy" domains is too low for the large language models to have a good manifold for "high-quality biology research". Would love to be proven wrong if anyone knows of anyone doing great scientific work with large language models in non-verifiable domains though.
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]
The title buries the lede quite a bit:
> The Exponential Time Hypothesis is false
Capital allocation to advance mathematics and sciences is undoubtedly a good thing. But this does beget a few questions:
1. How much did it cost?
2. Would an equivalent allocation to incentivize human effort have borne the same results?
3. Post IPO, will Open AI continue allocating similar levels of capital to further these causes?
#2 is particularly important to answer, given research grants have been getting cut elsewhere: see U.C.L.A's research grant cuts from last year and Terrence Tao's attempt to raise funds (about $60k) by collaborating with Open AI earlier this year [1].
[1] https://www.nytimes.com/2026/10/16/us/math-ai-terence-tao.ht...
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.
Not enough chatter about the QCD result here! I'm of the opinion that whatever the correct representation of the universe is, the GUT (Grand Unified Theory) will be authored not by humans but by LLMs.
Well, this is certainly different, can't say they ignored the criticism. Wonder if this is a monkey paw curls moment; if the findings continue to accelerate, most of it just isn't public now.
It is mind blowing that these long-standing conjectures and open questions are just falling by the dozen every other week. What a time to be alive.
Thurston wrote about this in “On Proof and Progress in Mathematics,” including how his own approach changed from proving theorems to helping a community understand the mathematics.
Nicolas Le Roux, a Lagrange Prize winner / prof / strong mathematician, turned me onto that around the time I finished my PhD in ML last year. I have been thinking about it a lot, lately, for obvious reasons.
At first, it felt silly to grapple with what science is only after finishing a PhD. This publication from OpenAI finally pushed me over the line to publishing an essay about this struggle between AI science and human science: https://news.ycombinator.com/item?id=50253759
Regarding Yang-Mills, as of a few weeks ago there has been some expectation that this (specifically completing Balaban's program for the existence part without the mass gap) could happen, though from the Anthropic side: https://nitter.meowing.monster/ElliotGlazer/status/211115394...
(Elliot Glazer seems in general to be a pretty good source to follow, however he does that...)
This is an exciting time, but also there are very real changes at hand. Game changing discoveries will likely not happen at universities but at private AI frontier labs. Moreover, the models that are used on this are not always open to the broader community today.
What will it be like in two years? Will such models that help enable paradigm shifts in science be only accessible to employees of those private companies? Still thinking about the ramifications ... amazing and discombobulating at the same time.
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.
It uses sparseness. I noticed a simple thing way back that for a n-boolean variables, the average size of a satisfiable boolean expression must be 2^n bits - basically every binary number that satisfies it listed out. So if a boolean expression of 4 variables is satisfied by 1111, 1110, and 0000, the number denoting it is 1100000000000001, and for every such number there is a boolean expression of 4 variables and for every expression one number of 16 digits. So it’s a fool’s errand to make a universal solver when average inputs scale at 2^n, if a solution exists it has to leverage the small size of the problems we actually care about, sparsity. Based on the AI description that’s what they did, something with polynomials, but it’s too soon. Looking forward to finding out.
> In our view, these manuscripts are best understood as a form of raw research data.
Much better!
> For the computer science collaboration, the group has decided to make the natural language proof and Lean formalization public immediately, and we are publishing them directly to our GitHub repository. For the physics collaboration, the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves.
> All of these approaches involve tradeoffs. On the one hand, the collaborations empower experts to foster unpacking the results and contextualizing them in the broader academic community. On the other hand, choosing initial shepherds necessarily shapes who gets involved first and creates asymmetry of information. We welcome feedback on how to broaden participation and make this process more useful.
Very good!
I was highly critical of their approach with the earlier mathematical publications, but this is much improved.
Most importantly, they aren't releasing slop papers that would never have been published in any self-respecting journal or made it past a deep round of peer review questioning.
> We are committed to ensuring funding for researchers to independently examine, explain, and build on the results above.
> We continue to invite feedback on the best way to disclose results and engage with scientists to accelerate discovery.
Well done!
Now you need to actually LISTEN to the community and the feedback, so you aren't done yet, but this is a much better start than last time. Keep it up!
I thought it was interesting that in the original drop of papers, they announced much more progress on most other Millenium problems than on Yang-Mills existence. At the time I thought this might mean that it was a more stubborn problem, but it turns out that they had made a lot progress after all.
The Yang-Mills Millenium Problem has two parts though. The second part is to show that Yang-Mills theories have a mass gap, meaning that the lightest particle has non-zero mass. This is important because experimentally we know that there is a mass gap, so the theory must have one too to in order to actually be the correct model of nature. I wonder if they again have unannounced progress here.
I wonder why the physicists are the only group to not want the full information released upfront? As a casual fan of physics[0] it seems like their completion of the standard model is possibly the biggest deal of all these breakthroughs, am I wrong?
[0]: PBS space time watcher
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
These resolve 6 entries in my top 500 open problems list (LLM ranked, https://www.proofatlas.ai/open-problems/) and there is partial progress on #3 overall (Yang–Mills Existence and Mass Gap):
Kakeya conjecture — #38
Fourier restriction conjecture — #46
Local smoothing for wave equations — #187
Bochner–Riesz conjecture — #190
Exponential Time Hypothesis — #43
Strong Exponential Time Hypothesis — #63
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?
The proof of the Kakeya conjecture is amazing. Only 54 pages. Will probably lead to a lot of theory.
As a non abelian harmonic analysis PhD I am very happy
this is the first time one of the big labs has published anything related to my field. and i'm actually pretty excited? im more on the applications side of QFD than the theory side of things, but gosh i actually cant wait for the results to be released to see if i can use it for anything! i kinda hope more results come out, in more ways relevant to what i do, its so cool to see any sort of progress! wow!
I'm a little surprised the article stresses Lean programs so much, without ever stating what that is or why it somehow proves something. Maybe that's expected knowledge for this sort of thing.
The QCD and Yang-Mills results don't seem to go beyond what the experts know/expect. Basically, they're saying there is a path integral measure, but shrugging when asked what functions are integrable with respect to it. The latter question is key for mass gap and confinement, and existence of the measure only means that the question of whether a mass gap exists makes sense.
I'm incredibly skeptical about everything these frontier labs put out. Results are results though.
However, could someone with a firmer grasp on the problems weigh in on how much are these recent results are function of cutting a virtually unlimited budget for some very smart people? I believe the LLMs certainly help iterate on ideas faster... but I am aware of very impressive work that goes on in quantitative finance. Stuff that would not really be published. That stuff is the result of paying smart people millions to produce results which produce $$$.
I have to wonder, would this have been solved if we simply gathered tons of smart people and paid them a ton of money to solve it? Or is it exclusively because they had access to LLMs?
Could they have achieved this result with previous versions of ChatGPT?
They resolved the Kakeya conjecture in every dimension! This is very big news; the resolution in dimension 3 was just recently.
I do not like this at all. At best, we find that there are classes of problems that even AI cannot help much in solving, in the sense that the rationals are an infinitesimally small part of the reals. At worst, a fundamental result trivializes a problem whose complexity is a key component of a critical system, and there’s societal upheaval. If PhDs ran the government, and thought the latter, we might have already seen military attacks on data centers.
The question is what are these discoveries going to mean in the long run.
Dont even care anymore. Its an AI discovery.
If it's human, at this point, we will see it in the title
Billy Connolly quote: Aye, did yer
Someone should take the sub-exponential proof & write something executable.
> In our view, these manuscripts are best understood as a form of raw research data. To convert this raw research data into scientific understanding, in this release we are experimenting with a new approach
Mathematics and computer science are turning into an empirical science
Scott Aaronson of Shtel Optimzed and now at OpenAI has a great podcast episode discussing this -worth watching
OpenAi needs to stop trying to please everyone. Just release the findings and give all the credit to your models/collaborators.
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.
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.
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.
I'm too dumb for this.. but does this mean PKI is doomed?
In my opinion its okay for them to work on science and math problems and release the results.
I think OpenAI deserve credit for adjusting their approach after the last math drop. I think Terrance Tao shares some credit here for galvanizing the response.
Including the relevant mathematicians and scientists before dumping this onto the public is sure to go a long way to relaxing fears in those communities. I'm not sure how they will deal with accusations of favoritism but at least they are demonstrating a willingness to participate in a process.
I hope the community can express some positivity that will encourage OpenAI and other foundation labs to continue with this collaborative approach.