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Comment by Cu3PO42 | Hacker News Reader

Just two days ago, a preprint by Julia Stadlmann went up on arXiv [0] improving the prime gap from 246 to 240. Now OpenAI announces Astra has shown a gap of 186 [1]. That must really blow.

[0] https://arxiv.org/abs/2608.31126

[1] https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16...

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Based on her comments in the paper it sounds like she was aware that an AI result was coming and rushed to release her work beforehand. 240 was not a tight bound from her methods.

With very little review: https://github.com/openai/PrimeGaps186/blob/main/formalizati...

"No independent human semantic review. Whole-file sorry counts and a complete auxiliary-declaration audit are not established; separate declaration lint has not been run."

I think this builds straight upon her method, which she said could be improved herself so...

It cites to her at: [19] J. Stadlmann, On primes in arithmetic progressions and bounded gaps between many primes, Adv. Math. 468 (2025), Art. 110190. Numbered references use arXiv:2309.00425v3.

Though that's not her latest paper.

This one is her latest paper: [20] J. Stadlmann, Bounded gaps between primes, Forthcoming

What's just as interesting is this morning Axiom Math announced 212 and OpenAI then appears to have rushed out their 186 announcement just 1-2 hours later followed by Astra. Did they accelerate the release of Astra itself? Not necessarily, but it definitely looks like they ended up pushing much harder and faster than planned on their 186 result. X activity suggests Anthropic had a similar result as well but wasn't as fast as OpenAI in packaging it up and sharing it in response to Axiom, so they mostly just bolted onto OpenAI's messaging.

The reason I think this is interesting is that Axiom is a tiny lab in comparison that wouldn't have had access to Astra at all. I'd be curious to learn how Axiom is able to effectively compete at this frontier with vastly fewer resources.

Don't think everything is just "who can produce the biggest/smallest number": https://mathstodon.xyz/@tao/117208619314517025.

Such a result should be considered worthless: the proof is 10MB of Lean. (https://github.com/openai/PrimeGaps186).

I can't think of a single mathematical proof being anywhere close to ten million characters. For all you know, 90% of the proof could be useless, 8% would be writing out Shakespeare, and 1% abusing another bug in Lean. Humanity gets zero value from that, aside from "some bot seems to think it's 186". Unusable by anyone.

You talk about modern math and worthlessness at the same time? That’s brave.

You can have your opinions about modern math, its usefulness in the world as it is, whether or not knowing if hairy balls can divide by three is actually going to be beneficial for anything but just obscure knowledge's sake. You may even say it's useless.

Needless to say, a useless result that absolutely no mathematician will ever read, confirm, understand, agree with or even consider to solve their "useless" problems is an impressive waste of resources.

Worthless is a pretty good description IMO in the context of what Lean is trying to achieve: "enable correct, maintainable, and formally verified code". Tens of millions of lines of LLM vomit may be many things, but it often turns out to not be correct and certainly not maintainable. Formally verified remains as a thin fig leaf covering the uncomfortable truth that formal methods only provide assurances under assumptions (your toolchain, libraries, compiler, OS, and hardware are "correct" and don't expose some exploitable flaw).

It doesn't mean that it cannot improve over time, maybe the proof can be "minified" to a state where human reviewers are able to comprehend it; but as it stands there isn't really much insight or confidence to be gained from the artifact itself.

There's a branch of mathematics called "pointless topology" [1].

[1] https://en.wikipedia.org/wiki/Pointless_topology

Yeah. Unless human can verify it, not sure if it is certain or useful.

Wasn’t the proof of Fermatt’s Last Theorem proof similar in complexity?

It's probably not 10MB, but famously the groundwork to prove the statement 1+1=2 is nearly 400 pages in to principia mathematica. That's not even proving 1+1=2, it's just the set-theoretic proofs you need to EVENTUALLY get there.

Saying "proving 1+1=2" is pretty misleading though. The book deals with all the foundational things needed to set up a mathematical universe where 1+1=2 actually has meaning and is consistent. That setup took 400 pages.

Yes. But I think that misses the point.

In 1799, Paolo Ruffini published a 500 pages long proof showing that there is no closed algebraic solution for the roots of a polynomial of degree five or higher. The proof is extremely verbose and brute-force, essentially enumerating and checking hundreds of cases by hand. It is by today’s standards insignificant.

About 25 years later, Evariste Galois proved the same result in about 95% less space by describing the first general theory of groups and fields. It is considered one of the greatest contributions to mathematics of that century, not because of the result, but because its approach opened up a whole new universe of questions, methods and insight. There would be no AES encryption without Galois.

To me, Astras proof looks like Ruffinis proof.

Touché!

Iirc some mainstream physycists never acknowledged quantum theory because they couldn’t accept that universe was that unintuitive and hard to understand.

Ditto ones that opposed Einstein’s general relativity.

The human-written https://github.com/AxiomMath/PrimeGapsLib adds up to 4MB of Lean so it's that far off.

Is this human-written? Axiom Math is a company building AI theorem provers, one would think this would also be heavily AI-generated.

Terence Tao says something surprisingly similar in a recent talk (https://news.ycombinator.com/item?id=49056620 ) Not that the proof is worthless but that the value comes after it's revised into a cleanly understandable form and then canonicalized so that other mathematicians can use it.

He also made a video on the same topic for Big Think: https://news.ycombinator.com/item?id=49551848

Tao is saying that there is very little insight from something like an LLM counterexample (e.g. Jacobian conjecture counterexample he investigated further on his blog) - you don't learn much about the subject and _why_ a conjecture was true or false from an LLM giving a counterexample. That's why he wrote the blog post - to analyse what the counterexample says about the subject.

Tao does not disbelieve the counterexample (it's seemingly easy enough for him to verify it is a counterexample).

Parent is saying something very different - they're saying they literally don't have any faith that this is a proof. Given its size, it could just be a bunch of completely useless statements that do pass the type checker.

You're putting a lot of words in my mouth. What I'm saying is that whether or not it's a proof, it's useless: it does not improve human knowledge, because the only thing able to consume 10MB of Lean to build upon it is another LLM that's going to build a 50MB piece of shit.

It's very much likely a proof. It's also completely useless.

You said:

> For all you know, 90% of the proof could be useless, 8% would be writing out Shakespeare, and 1% abusing another bug in Lean.

So you were implying the possibility of there not actually being a proof at all.

Anyway, I disagree. I'd refer you to Tao's blog post about the Jacobian conjecture counterexample.

The existence of a proof is something you can use, with an LLM, to derive insight, just as Tao did with the existence of the counterexample.

A counterexample (at least the jacobian conjecture one) is a lot easier to manually verify than 10MB Lean proof

If we accept that it is a proof then it does improve human knowledge, even if no one can understand how to get there.

If you were navigating a pitch dark cave, wouldn't you find it useful to be able to see the light of the cave opening even if it's not bright enough to illuminate your path to it?

I'd like to note that we should remember a formalized Lean proof does have value in that it enters the pantheon of true things other Lean proofs can rely on. Agreed that for the humans, descriptions and being able to 'grok' the proof / assess it for new tools and concepts is extremely helpful.

The proof of the classification of finite simple groups is bigger than that.

It's like hitting a local minima, it was given a technique, it brute forced it for a lot of money and slightly improved the result. But nothing new was discovered, no new mathematical tool was built, only a huge file that no one will read or build on.

Ok, so the rumour was exactly true: there was a withheld prime gaps improvement, that "an AI company" was holding onto until release of a model.

https://github.com/openai/PrimeGaps186

https://news.ycombinator.com/item?id=49555257

I'm surprised the OpenAI employee who pushed this didn't take the minute or two to format README.md to use GitHub-supported LaTeX (https://docs.github.com/en/get-started/writing-on-github/wor...)

edit: my comment was on the submission for https://github.com/openai/PrimeGaps186 but seems to have been moved to the main Astra submission

> OpenAI employee

Why would you think it was an employee who did the push, instead of a random GPT agent?

Where did you get the link to the pdf? Was it announced somewhere?

Happened to multiple people I know.

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