A new analysis of OpenAI's AI-proved mathematics says the interesting part is what is missing. Researcher Daniel Paleka, in a newsletter post on October 16, 2026, reports that statistics, machine-learning theory, optimization and numerical analysis make up about 20% of arXiv theorems but under 1% of OpenAI's more than 700 released results. Cryptography, which social media noticed first, is only part of a wider pattern.
We have already covered the release itself in OpenAI's 722 math manuscripts and the eight results that matter, with expert reactions. This post is about the gap, and about how far a gap like this can be pushed as evidence.
TL;DR: the claim, the numbers, the caveats
| Question | Answer |
|---|---|
| Who made the claim? | Daniel Paleka, in his newsletter, October 16, 2026 |
| What is the finding? | Statistics, ML theory, optimization and numerical analysis: about 20% of arXiv theorems, under 1% of OpenAI's released results |
| Does it hold for open problems only? | Paleka says the gap shrinks but stays statistically significant |
| Is cryptography covered? | Not in the release; commentators noticed this first |
| Does it show withholding? | No. It is one hypothesis among several, and Paleka does not test it |
| Has anyone replicated it? | Not that we found; the method is described only in outline |
| Why does it matter? | It hints at what labs consider valuable to keep, and at where AI math may go next |
What did OpenAI actually release?
On October 6, 2026, OpenAI published Sharing AI progress in mathematics and a GitHub repository, openai/math, with manuscripts produced mostly by an unreleased internal model. OpenAI says about 4,000 problems were posed and that an average result used compute equal to roughly three hours of ChatGPT Pro thinking. Many manuscripts come with Lean formalizations, but OpenAI warns that some unformalized ones could have issues. Paleka describes the dump as "more than 700 mathematics results."
The release set off the usual argument about verification, and our posts on the Lean RL environments built from it and the mathematicians' association statement track those threads. Paleka's angle is different: instead of asking whether the proofs are right, he asks which proofs we were given.
Why did people think cryptography was missing?
Two X posts frame the opening of his piece. Ben Podgursky wrote on October 7 that it was a "lucky coincidence" that labs are "only successfully cracking esoteric math problems and not the problems where a solution would give the winner an asymmetric commercial advantage." The post had 588K views at the time Paleka embedded it. The next day, cryptographer Matthew Green replied to another account: "I think we might lose public key cryptography." Paleka's embed shows about 1.99M views.
A key sliding into a lock, illustrating public-key cryptography and the worry that AI mathematics could weaken it
Those are opinions, not findings. Our own post on what is claimed and what is verified about AI breaking cryptography records that Scott Aaronson relayed an anonymous claim that labs have begun testing models against cryptographic protocols, and that no break has been reported. Nothing in Paleka's piece changes that. What he adds is a measurement that cryptography is not alone.
What does the field-by-field gap look like?
Paleka says he used a coding agent and a Semantic Scholar API key to compare the topics of the OpenAI results against arXiv. He reports that the underrepresented areas are:
- Statistics
- Machine-learning theory
- Optimization
- Numerical analysis
Together these are roughly 20% of arXiv theorems and under 1% of the OpenAI set. He then tests the obvious objection, that these fields simply have fewer open problems. If the baseline is restricted to arXiv papers that solve open problems, he says, "the gap reduces, but is still statistically significant."
He also considers whether the OpenAI team simply does not know open problems in optimization or ML theory. He points to a counterexample: in August 2025, OpenAI researcher Sebastien Bubeck posted on X (embedded in Paleka's piece) that GPT-5 Pro proved a better bound than a published convex-optimization paper, a claim Bubeck said he checked. So familiarity with the area existed at OpenAI a year ago.
One caution on the numbers: they are reported by one author in a newsletter, with no code or table in the post that we could rerun. They are plausible and specific, but treat them as a lead, not a result. A simple check anyone can do is to open the openai/math contents file, tag each manuscript by arXiv category, and compare with a sample of recent arXiv math papers.
What are the possible explanations?
Paleka's own explanation for the pattern is that these fields are "closest to what all labs (not just OpenAI) might want to use internally; hence all labs will generally abstain from publishing, to preserve comparative advantage." He states this as a possibility, and a commenter on the post, Soroush Pour, wrote that the piece never answers its opening question about whether labs are holding back categories of math.
Here are the main readings, ordered by how much evidence the post offers:
| Hypothesis | Evidence in the post | What would test it |
|---|---|---|
| Labs withhold math useful for training | Pattern fits, nothing direct | Leaks, lab statements, later releases in those fields |
| Those fields have fewer clean open problems | Weakened by the open-problem baseline | A better baseline, such as problem lists by field |
| The prompting team picked problems it knew | Partly countered by the 2025 Bubeck example | Which team members chose the roughly 4,000 problems |
| Proofs there are harder to formalize in Lean | Not discussed | Count Lean coverage by field |
| The model is weaker there | Not discussed | Release the model or run the same problems elsewhere |
The fourth and fifth rows are our additions, not Paleka's. Optimization and statistics lean on real analysis and probability libraries that are harder to formalize than algebra or combinatorics, and an unreleased model cannot be tested by outsiders. Either could produce a gap without any strategic withholding, which is why the data alone cannot settle the question.
On Hacker News, where the post got a handful of points, commenter rluna828 made a different argument: if a prompt can solve a useful proof, someone else will find that prompt, so nobody is hiding secrets for advantage. That is one commenter's view, but it matches a real tension: withholding only pays if rivals cannot reproduce the result.
What is the one result Paleka found personally interesting?
He says only one result in the dump would have interested his 2020-21 self, who worked on statistical learning theory and mixed-integer optimization before drifting away: that Max-Cut cannot be approximated better than 0.878 assuming P is not equal to NP. He adds that it "does not really count," because it follows from a complexity-theory result in the same batch, described as a Unique Games Conjecture theorem.
That UGC claim is itself the headline item in the release as Aaronson described it in "The Mathocalypse," as we noted in our cryptography post. It remains a claimed proof, with a Lean certificate reported by Aaronson, and independent expert review is the thing to watch. If you want the broader reaction, our expert-reactions post lists who has weighed in.
Why does this matter beyond mathematics?
The second half of Paleka's piece is an argument about trajectory, which he labels his belief. His reasoning:
- Mathematics has shown the fastest capability growth across domains of human expertise.
- The usual bottlenecks do not apply: you can create synthetic data, verification is cheap, and curriculum learning is easy. "Compute in, math out," he writes.
- So the right conclusion is not that labs focused on math for short-term gain, but that AI improves faster there than elsewhere, and more compute produces more math.
- Math may then return to feed LLM training, even though pre-2026 deep-learning theory was "far too weak."
He also says recursive self-improvement, "widely dismissed as science fiction in polite society about two years ago, is now openly the plan of multiple trillion-dollar companies." That is his characterization of lab strategy, not a quote from a company. His practical advice to would-be researchers: do not prove small theorems about deep learning, but "think much bigger," and prepare questions in advance, because he expects a short critical phase in which AI is superhuman at wielding mathematical objects but not at everything else.
You can disagree with the forecast and still use the observation. If verification is cheap and proofs can be machine-checked, then fields where results can be Lean-certified will see the first automated wins. For the funding side of the same trend, see our post on the Simons Foundation's $15 million AI-math initiative.
A magnifying glass resolving scattered shapes into an orderly green grid, illustrating verification of AI math proofs
What should readers do with this?
- Do not cite the percentages as settled. Link Paleka and say "reported."
- Check the raw list. The openai/math contents file is public; categorizing it takes an afternoon.
- Separate three questions: Are the proofs correct, are the topics representative, and are labs withholding. This analysis speaks only to the second, and only weakly to the third.
- If you maintain cryptography, keep doing the boring work: inventory public-key uses and make algorithms swappable. Nothing here changes that advice, as our cryptography post spells out.
- If you research optimization or ML theory, treat the absence as an open flank: the first lab to publish convincing results there would break the pattern, and you would want to know quickly.
- For agent builders, note the safety angle: if AI math capability grows fast while other skills lag, evaluations built around general tasks may miss it.
What we do not know
We did not rerun Paleka's analysis, and OpenAI has not commented on field coverage in anything we found. We cannot say whether other labs hold unreleased math, because none has made a comparable release. We also cannot say how OpenAI chose the roughly 4,000 problems, so a skewed starting list could explain much of the gap. Until OpenAI discloses its selection or someone replicates the counts, the finding is best read as a sharp question rather than an answer.
Related reading
- OpenAI's 722 math manuscripts: what is in the repo and what to check
- OpenAI's math results that matter, with expert reactions
- Can AI break cryptography? Claimed vs verified
- Are AI labs hoarding solved math problems? The Aaronson rumor
- Lean RL environments built from OpenAI's math results
- The mathematicians' association statement on working with OpenAI
- Discussion: Hacker News thread
Figures and quotes are from Paleka's October 16, 2026 post and OpenAI's public release, accurate as of October 17, 2026.