Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules.

I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.

So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.

Mathematicians make new discoveries by building and applying mathematical tools in new ways. It is tons of iterative work, following hunches and exploring connections. While true that LLMs can't truly "make discoveries" since they have no sense of what that would mean, they can Monte Carlo every mathematical tool at a narrow objective and see what sticks, then build on that or combine improvements.

Reading the article, that seems exactly how the discovery was made, an LLM used a "surprising connection" to go beyond the expected result. But the result has no meaning without the human intent behind the objective, human understanding to value the new pathway the AI used (more valuable than the result itself, by far) and the mathematical language (built by humans) to explore the concept.

I agree with one of the mathematician's responses in the linked PDF that this is somewhat less interesting than proving the actual conjecture was true.

In my eyes proving the conjecture true requires a bit more theory crafting. You have to explain why the conjecture is correct by grounding it in a larger theory while with the counterexample the model has to just perform a more advanced form of search to find the correct construction.

Obviously this search is impressive not naive and requires many steps along the way to prove connections to the counterexample, but instead of developing new deep mathematics the model is still just connecting existing ideas.

Not to discount this monumental achievement. I think we're really getting somewhere! To me, and this is just vibes based, I think the models aren't far from being able to theory craft in such a way that they could prove more complicated conjectures that require developing new mathematics. I think that's just a matter of having them able to work on longer and longer time horizons.

A difficult part was constructing a chess board on which to play math (Lean). Now it's just pattern recognition and computation.

LLMs are just the beginning, we'll see more specialized math AI resembling StockFish soon.

The more I read about these achievements the more I get a feeling that a lot of the power of these models comes from having prior knowledge on every possible field and having zero problems transferring to new domains.

To me the potential beauty of this is that these tools might help us break through the increasing super specialization that humans in science have to go through today. Which in one hand is important on the other hand does limit the person in terms of the tooling and inspiration it has access to.

> The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.

Can someone please elaborate on this?

Without knowing all this model has been trained on though, it is pretty hard to ascertain the extent to which it arrived to this "on its own". The entire AI industry has been (not so secretly) paying a lot of experts in many fields to generate large amounts of novel training data. Novel training data that isn't found anywhere else--they hoard it--and which could actually contain original ideas.

It isn't likely that someone solved this and then just put it in the training data, although I honestly wouldn't put that past OpenAI. More interesting though is the extent to which they've generated training data that may have touched on most or all of the "original" tenets found in this proof.

We can't know, of course. But until these things are built in a non-clandestine manner, this question will always remain.

We can argue about recombination/interpolation of training data in LLMs, but even if this was an interpolation, the result was contrarian rather than a confirmation. Any system that can identify an error in Erdős's thinking seems very useful to me (though perhaps he did not spend much time thinking about or checking this particular conjecture).

For those in academics, is OpenAI the vendor of choice?

- Does anyone know if this was a 1 minute of inference or 1 month?

- How many times did the model say it was done disproving before it was found out that the model was wrong/hallucinating?

- One of the graphs say - the model produced the right answer almost half the times at the peak compute??? did i understand that right? what does peak compute mean here?

woah.

Many of my colleagues and I have been experimenting with LLMs in our research process. I've had pretty great success, though fairly rarely do they solve my entire research question outright like this. Usually, I end up with a back and forth process of refinements and questions on my end until eventually the idea comes apparent. Not unlike my traditional research refinement process, just better. Of course, I don't have access to the model they're using =) .

Nevertheless, one thing that struck me in this writeup, was the lack of attribution in the quoted final response from the model. In a field like math, where most research is posted publicly and is available, attribution of prior results is both social credit and how we find/build abstractions and concentrate attention. The human-edited paper naturally contains this. I dug through the chain-of-thought publication and did actually find (a few of) them. If people working on these LLMs are reading, it's very important to me that these are contained in the actual model output.

One more note: the comments on articles like these on HN and otherwise are usually pretty negative / downcast. There's great reason for that, what with how these companies market themselves and how proponents of the technology conduct themselves on social media. Moreover, I personally cannot feel anything other than disgust seeing these models displace talented creatives whose work they're trained on (often to the detriment of quality). But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though.

What is preventing AI from continuing to improve until it is absolutely better than humans at any mental task?

If we compare AI now vs 2022 the difference is outstandingly stark. Do you believe this improvement will just stop before it eclipses all humans in everything we care about?

- It does not show an example of the new best solution, nor explain why they couldn't show an example (e.g. if the proof was not constructive)

- It does not even explain the previous best solution. The diagram of the rescaled unit grid doesn't indicate what the "points" are beyond the normal non-scaled unit grid. I have no idea what to take away from it.

- It's description of the new proof just cites some terms of art with no effort made to actually explain the result.

If this post were not on the OpenAI blog, I would assume it was slop. I understand advanced pure mathematics is complicated, but it is entirely possible to explain complicated topics to non-experts.

1. Erdos 1196, GPT-5.4 Pro - https://www.scientificamerican.com/article/amateur-armed-wit...

There are a couple of other Erdos wins, but this was the most impressive, prior to the thread in question. And it's completely unsupervised.

Solution - https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...

2. Single-minus gluon tree amplitudes are nonzero , GPT-5.2 https://openai.com/index/new-result-theoretical-physics/

3. Frontier Math Open Problem, GPT-5.4 Pro and others - https://epoch.ai/frontiermath/open-problems/ramsey-hypergrap...

4. GPT-5.5 Pro - https://gowers.wordpress.com/2026/05/08/a-recent-experience-...

5. Claude's Cycles, Claude Opus 4.6 - https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cyc...

The underlying model may still effectively be a stochastic parrot, but used properly that can do impressive things and the various harnesses have been getting better and better at automating the use of said parrot.

I find this hyperbolic, but ya gotta juice up the upcoming IPO. I hate that they took an interesting announcement and reminded me why I hate tech and our society at the end.

For example, these machines, if scaling intellect so fiercely that they are solving bespoke mathematics problems, should be able to generate mundane insights or unique conjectures far below the level of intellect required for highly advanced mathematics - and they simply do not.

Ask a model to give you the rundown and theory on a specific pharmacological substance, for example. It will cite the textbook and meta-analyses it pulls, but be completely incapable of any bespoke thinking on the topic. A random person pursuing a bachelor's in chemistry can do this.

Anything at all outside of the absolute facts, even the faintest conjecture, feels completely outside of their reach.

Why would anyone believe this to be true even for a split second?

can we please put these ground breaking AIs to work on actual problems humans have?

What was discovered were numerous mistakes in the published literature on the subject. “New math! AI!” No, just mechanical application of rules, human mistakes.

There were things that were theorized, but couldn’t be exhaustively checked until computers were bigger.

Once again, a tool is applied, it has the AI label - its progress! But it isn’t something new. It’s just an LLM.

There’s a consistent under appreciation of AI (and math, honestly), but watching soulless AI mongers declare that their toy has created the new is something of a new low; uninspired, failed creatives, without rhyme or context; this is a bigger version of declaring that your spell checker has created new words.

The result is more impressive than what was done with tables of integrals and SAINT in 1961, sure.

Apparently if you add a “temperature” knob to a text predictor, otherwise sane individuals piss themselves and call it new.

Then again I thought NFTs, crypto, and the Metaverse were stupid, so what do I know.