OpenAI's 722 maths papers explained: what Lean checks, and what it can't

On 6 October 2026 OpenAI put 722 mathematics manuscripts on GitHub, written by a model that nobody outside the company can run, and said they resolve or make substantial progress on hundreds of open questions. For 185 of the main results, a computer has checked the proof line by line. The real story is not whether a machine can do research mathematics. It is what "checked by a computer" buys you, and what it leaves to a community of humans who were already overwhelmed.

The plain-English version

Imagine you commission a bridge. The builder hands you a finished bridge and a thick blueprint, and an independent inspector certifies that every beam, bolt and weld matches the blueprint exactly. That certificate rules out sloppy construction. But it says nothing about whether the blueprint describes the bridge you ordered. If it quietly specifies a footbridge where you asked for a railway crossing, the inspector will still sign.

That is roughly the situation with OpenAI's release. The company has an unreleased model that it described at its Navier–Stokes announcement, as reported by Unite.AI, as "significantly more capable" than GPT-6 Astra, a model the public can use. It posed that model about 4,000 open problems, kept the results it judged significant, and published them as 722 manuscripts in 372 "families" (a main result plus its companion arguments, consequences and alternative proofs). The average result, OpenAI says, used roughly three hours of ChatGPT Pro thinking.

Many results come with a second file written in Lean, a programming language for mathematics. A Lean proof is the inspector's certificate: a small, well-tested program checks that every logical step follows from the one before, with no gaps. A human reading a 60-page proof can be fooled by confident prose. Lean cannot be, because it does not read prose at all. What Lean does not check is whether the formal statement matches the question mathematicians care about. Someone still has to compare the blueprint with the order, and OpenAI's own "scope pages", which say what each formal proof leaves out, show that the two do not always match.

The claims are big. Four that OpenAI's chief executive Sam Altman singled out, as reported by OfficeChai: that the Riemann zeta function and all Dirichlet L-functions have no zeros with real part above 7/8 (a "quasi-Riemann hypothesis", far short of the real thing, but a kind of result nobody had managed before); a proof of the Unique Games Conjecture, a central open question in computer science since 2002; the Hodge conjecture for a special class of geometric objects; and a question about "free group factors" open since the 1940s. Three of the four have Lean proofs. Altman himself said none had yet been confirmed by outside mathematicians.

Mathematicians are split. On Hacker News, one self-described analytic number theorist called the zeta result a bigger deal than the 1896 proof of the prime number theorem, and another number theorist in the same thread said that was hard to argue. Twenty-five Fields Medallists had signed a declaration in September saying AI labs' race for famous problems was harming mathematics. An independent advisory group of senior mathematicians, which OpenAI helped set up, said on 6 October that its involvement should not be read as an endorsement. OpenAI told Scientific American it is not bound by the group's recommendations.

How it actually works

The pipeline, as far as OpenAI has described it, has five stages. Only the fourth is automatic.

The problem. Large language models produce fluent, confident text, and in mathematics a confident wrong proof is the normal failure, not the exception. Even a right proof may be unreadable: Oxford's James Maynard told NPR that it had been "very difficult to really extract any human understanding" from OpenAI's earlier Navier–Stokes paper. If you cannot read a proof and cannot trust its author, you need a third option.

The old way. Peer review: two or three experts read the paper over months. That works for a few hundred papers a year in a subfield. It does not work for 722 papers landing in one day across 17 areas of mathematics, by one outside tally.

The new idea. Make the computer the referee, but pin down what it is refereeing. Lean is a proof assistant: mathematics is written as code, and a small kernel program verifies that each step is a valid application of the rules. Mathlib, the community library that Lean proofs build on, already defines zeta functions, group algebras and so on. A Lean proof that compiles is correct relative to the definitions and axioms it uses. No charisma, no hand-waving.

The pinning down is the part most coverage misses. OpenAI used Comparator, a tool built by the Lean Focused Research Organization for AI proving competitions. Each result has a "challenge" file: the theorem statement with the proof replaced by the placeholder sorry. A separate "solution" file supplies the proof. Comparator confirms that the proved theorem is identical to the challenge statement, that the proof contains no sorry and uses only the axioms listed in a configuration file, and that the Lean kernel accepts it. Compilation runs inside a sandbox so a misbehaving proof cannot cheat by touching the filesystem or the network. The goal is to make it impossible for a prover, human or machine, to win by quietly weakening the statement.

Why it works. The trust moves from the 60-page paper to a few lines of formal statement. If you can read the challenge file and agree that it says what the paper's title says, the kernel does the rest. For the claim that the irrationality exponent of pi is exactly 2 (family 017), the whole formal statement fits in a dozen lines, reproduced below.

What it costs. Choosing the formal statement is where the risk now lives. The scope page for family 017 says the paper's claim about the Flint–Hills series is "outside this selected statement"; the page for the zero-free region excludes the paper's "later applications". Honest disclosures, but they mean "has a Lean proof" is true of a chosen core, not of every sentence in the PDF.

The deep dive

The release by the numbers

The counts do not line up neatly: 162 formalised manuscripts is 22% of 722, but formalisation targets principal results, so coverage by family is higher (Kingy counted 235 scope pages, about 63% of families). The manifest marks its own state as scope: Partial progress and review.status: unchecked.

The trust chain, end to end

Here is the complete Comparator challenge for family 017, as published in the repository:

theorem main :
  (∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧
    ∀ p q : ℤ, Q ≤ q →
      (q : ℝ) ^ (-ν) ≤ |Real.pi - (p : ℝ) / (q : ℝ)|) ∧
  sSup {ν : ℝ | 0 < ν ∧
    Set.Infinite {r : ℚ | 2 ≤ r.den ∧
      0 < |Real.pi - (r : ℝ)| ∧
      |Real.pi - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}} = 2 := by
  sorry

In words: for every exponent $\nu > 2$, all fractions $p/q$ with large enough denominator satisfy $|\pi - p/q| \ge q^{-\nu}$, and the supremum of exponents for which infinitely many good approximations exist is exactly 2. That is the textbook definition of "the irrationality exponent of $\pi$ is 2", and since every irrational number has exponent at least 2, it is the smallest possible value. A reader who accepts that this code says what the title says needs nothing else from the paper to believe the theorem.

The configuration file for the quasi-Riemann challenge names one theorem to check, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re, and permits exactly three axioms: propext, Quot.sound and Classical.choice, the standard axioms of classical mathematics in Lean's library. Compilation runs under landrun, a sandbox, and lean4export dumps the checked environment so Comparator can inspect which axioms were actually used.

So the chain is: paper, then scope page, then challenge file, then solution, then kernel, then three axioms. The first two links are prose. The last four are mechanical.

Where the chain can still break

The scope pages are where to read carefully. Family 003's page formalises: the zeta function and every Dirichlet L-function are zero-free for $\Re s > 7/8$, uniformly in the modulus; the same for finite-order Hecke L-functions over $\mathbb{Q}(\sqrt{-3})$; and a uniform real-zero gap $1 - \beta \ge c/\log q$ for primitive real characters, with "no explicit value of c". The paper's later applications are excluded. That core is still strong enough to matter: the best previously known zero-free regions, due to Vinogradov and Korobov, shrink towards the line $\Re s = 1$ like $\sigma \ge 1 - c\,(\log|t|)^{-2/3}(\log\log|t|)^{-1/3}$, and no fixed half-plane left of 1 was known to be zero-free. A companion paper proves the weaker region $\Re s > 11/12$; that is the one writeup OpenAI says was "human edited for readability".

Family 197 is the cautionary example. The catalogue headline is "A torsion-free group algebra that is not directly finite", and the entry says the construction disproves Kaplansky's direct-finiteness conjecture, which asks whether $ab = 1$ forces $ba = 1$ in a group algebra, "even without torsion". The Lean scope page lists four challenge files. Of the three direct-finiteness statements it describes, one uses a finitely presented group "with an element of odd prime order" and another "a finitely generated group $G$ with torsion"; none is described as torsion-free, and the page says the further conclusion that the group is nonsofic "is outside these statements". Kingy AI flagged the discrepancy. A counterexample with torsion still contradicts the conjecture as usually stated, which covers every group and every field, but the torsion-free headline is not what the listed challenges pin down.

There is precedent. After OpenAI's first ten-result batch in August, a reviewer argued, according to 36Kr, that one of the two groups constructed in the Connes-rigidity paper, whose formalisation ran to about 37,000 lines of Lean, did not satisfy the required conditions, and that the formal proof concerned a transformed object rather than the original group. A human audit posted to arXiv in August and revised in September found "no confirmed substantive mathematical error in a principal result" remained in that batch, and that independent work had since reused the same mechanism to confirm the conjecture is false. Both can be true: the mathematics held, and the formal statement was not the one some readers assumed.

The four headline claims

Family and manuscript counts are from the repository's CONTENTS.md. If the Unique Games proof holds, the Goemans–Williamson 0.878 approximation for Max-Cut is optimal unless P = NP, and Vertex Cover cannot be approximated within $2 - \varepsilon$. The catalogue describes the proof as a polynomial-time reduction from 3SAT, with companion manuscripts deriving those hardness results.

Theoretical computer science: galactic bounds

The largest section of the catalogue is also the strangest.

• Family 109, integer multiplication. Two $n$-bit integers in $O(n(\lg n)^{1-\kappa})$ worst-case time with $\kappa = 2^{-182}$. No Lean. Harvey and van der Hoeven reached $O(n \log n)$ in 2019, the bound Schönhage and Strassen had conjectured in 1971 was both achievable and optimal; the catalogue frames the new paper as a disproof of that optimality conjecture. The improvement factor is indistinguishable from 1 for any $n$ a human could write down, a point made repeatedly on the dedicated Hacker News thread, where one commenter recalled that an absurd exponent elsewhere (a $1/n^{400}$ term in a lattice result) was "quickly" brought down to $1/n^8$ "by more careful accounting". What matters is the sign of $\kappa$, not its size.
• Family 107, matrix multiplication. The exponent $\omega \le 9/4$ over the complex numbers, three manuscripts, with Lean. The best human-proved bound stands at 2.371177 as of August 2026; the trivial lower bound is 2, and most researchers believe $\omega = 2$. A jump to 2.25 would be more than 0.12 in an exponent that human work has recently moved by thousandths.
• Family 103. Exact derandomisation of logarithmic space, $L = RL = BPL$. No Lean.

The pattern: existence proofs with enormous constants (one scheduling result discussed on Hacker News carries an exponent of 150,020), which is what a prover optimising for "a proof exists" rather than "a proof a person would find" produces.

What the reasoning summaries show

OpenAI published ten abridged reasoning summaries as PDFs, one with LaTeX source. The Mézard–Parisi one opens with the prompt, which asks the model to "prove or disprove" that for every diluted spin-glass model satisfying stated hypotheses the thermodynamic limit exists and equals an infimum over finite replica-symmetry-breaking hierarchies. The summary runs in three sections: structural obstacles (the model explores candidate counterexamples before settling on a many-site cavity argument), a shift of "branching depths" to avoid an earlier tree structure, and verification. One exclamation survives the abridgement: "Aha recursive structure: overfitting shared variable can simulate original model at smaller scale". No tool calls, Lean interactions or numerical experiments appear. It reads like a compressed chain of thought, not a transcript.

The advisory group had asked for the model name, prompts, a summarised chain of thought, time and estimated cost for every result. OpenAI provided summaries for 10 of 372 families and, per Scientific American, prompts for none.

How this run differs from Navier–Stokes

Navier–Stokes figures are OpenAI's, as reported by Unite.AI; the model began training on 28 August 2026. Scientific American reports that OpenAI describes the new batch as 372 results from a single prompt to a single agent, with a spokesperson acknowledging some needed multiple attempts.

The compute unit is OpenAI's own. "Three hours of ChatGPT Pro thinking" on an internal model is neither a token count nor a price, no total is given for the roughly 4,000 attempts, and nothing is said about the failures, which the advisory group had specifically asked for. On Hacker News the question was put bluntly: "What was the cost to OpenAI in dollars?"

Why it matters

For everyday users there is nothing to try: no product, no API, no model name. What will reach them is the idea. "Machine-checked" is about to become a label on AI output the way "peer-reviewed" is on science, and this is the first large-scale demonstration of what the label does and does not promise.

For developers and builders, the Comparator pattern is the transferable part. Whenever an AI output can be reduced to a formal statement, pin the statement in a challenge file, run the check in a sandbox, whitelist the axioms, and publish a scope note saying what was left out. The repository is Apache-2.0, built on Lean 4.34.1 and Mathlib, so the formal proofs can be studied, reused and eventually upstreamed regardless of who wrote them.

For companies and labs, the advisory group's September guidelines are now the scorecard. OpenAI met some items: a public repository with version history, Lean proofs with Comparator challenge files and a formalization.yaml, an average compute figure, a count of attempted problems, reasoning summaries for a few results. It did not meet others: a model name, per-result prompts and costs, failure accounting, a repository not controlled by an AI lab (OpenAI says it is "exploring community-hosted repositories"), and the headline request that labs stop testing advanced problems on proprietary models at all. Every lab that releases AI mathematics will now be measured against that list.

For the field, if even a modest fraction of the 372 families holds up, the map of open problems in 17 areas is redrawn at once. The bottleneck then moves from discovery to understanding, which is what the advisory group meant by calling the release "the beginning, not the completion" of the process. Second-order effects are already visible. Thomas Bloom announced on 6 October that the Erdős problems website will remove its open/solved status labels and pause proof claims, to discourage people "copying problems into their AI to get an OPEN->SOLVED dopamine hit". And the Fields Medallists' central complaint, that open problems have become a benchmark for model releases, is echoed by OpenAI's README, which says the exercise is part of how it evaluates models during development.

What to be skeptical of

• None of it is peer reviewed, and OpenAI says so. The README states that "some of the unformalized results could have issues". That covers every manuscript without a Lean file, including the Hodge conjecture for CM abelian varieties, Hilbert's tenth problem over the rationals and the integer multiplication paper.
• The formal statement is not the headline. Scope pages exclude consequences (family 017) and applications (family 003), leave constants unspecified, and in family 197 formalise torsion-bearing constructions under a torsion-free title. The August Connes-rigidity dispute is a reminder that a kernel-accepted proof of a statement other than the intended one is a live failure mode.
• Selection is OpenAI's. About 4,000 problems in, 372 families out, chosen and judged significant by the company. That is not a success rate, and 722 manuscripts are not 722 independent results. The advisory group asked how many comparable problems failed and how problems were chosen; neither has been published.
• The process cannot be verified. No prompts, no model. MIT's Andrew Sutherland told Scientific American: "We should ask for receipts." Toronto's Daniel Litt, in the same piece, argued the other way on disclosure: if we want the answers to these questions, there is no reason to ask the company to keep them secret.
• Attribution is unresolved. The Navier–Stokes announcement came with a credit dispute, in which Tristan Buckmaster alleged OpenAI's approach followed his and Levent Alpöge's unpublished work and OpenAI's Sébastien Bubeck replied "We did not use their prompt or proofs to prompt our models". The new release promises better citations in future, which is an admission about the present.
• Conflicts of interest are structural. OpenAI selected the problems, judged significance and hosts the repository, which has no Issues tab, so there is no public place to challenge a proof in situ. The advisory group's members are unpaid and independent, but the Institute for Advanced Study stresses it has no decision-making power at any company, and OpenAI excluded the pace of its research from the group's remit.
• Readability is poor. One writeup needed human editing; a Hacker News reader noticed the two quasi-Riemann papers carry inconsistent disclaimers about human assistance.
• Early reactions cut both ways. One Hacker News commenter who said they had worked on Barnette's conjecture (family 180) on and off for 24 years called the model's proof "approachable at first glance" and "quite tractable". Another predicted that "way more than just one of these is wrong". Both can be right.

What to watch next

1. Formalisations for the unformalised headliners. OpenAI says it will add Lean proofs as it obtains them and record corrections as new versions. Watch the repository's history for the Hodge/CM result, Hilbert's tenth over the rationals, $L = RL = BPL$ and integer multiplication, and for the first scope correction.
2. Human reconstructions. Number theorists on the 7/8 zero-free region, complexity theorists on Unique Games and $\omega \le 9/4$, operator algebraists on the free group factors. The first arXiv preprints that re-derive or refute a family will set the tone.
3. The advisory group's next move. Whether OpenAI migrates to a community-controlled repository, publishes prompts and failure counts, and funds the workshops, conferences and special programs it has promised, with details due "in the near future".
4. The model itself. OpenAI says it is "working to responsibly release" it. Google's Gemini 4 Argon launched behind a gate to vetted partners; see our explainer on how staged access works.
5. Incentives in the community. The Erdős problems site changes take effect now, and the Fields Medallists' declaration drew thousands of further signatures within days of its 11 September release. Watch whether other labs keep treating open problems as evaluations.

How this was made: researched and written by an AI model (Claude) from the primary sources listed above, then checked claim-by-claim against those sources in a separate AI fact-check pass. Spotted an error? Email [email protected] and we correct it publicly. Our process.