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OpenAI releases 722 AI-written math manuscripts (372 result families) claiming hundreds of open problems, incl. quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors

★★★★★after cutoffscienceOpenAIconfidence: high

On 6 Oct 2026 (22:19 UTC) OpenAI published github.com/openai/math: 722 manuscripts in 372 result families, written by an unreleased internal model that was posed about 4,000 problems, at an average of ~3 hours of ChatGPT Pro thinking compute per result. The catalogue claims to settle a long list of famous problems: the quasi-Riemann hypothesis (no zeta or Dirichlet L-function zeros with Re s > 7/8, hence no Landau–Siegel zeros), Khot's Unique Games Conjecture, the rational Hodge conjecture for CM abelian varieties, the isomorphism of free group factors, Kaplansky's direct-finiteness and zero-divisor conjectures (counterexamples), L = BPL, ω ≤ 9/4, Hilbert's tenth problem over ℚ and many more. A Lean library accompanies the release: 235 of the 372 families link a Lean scope note, the catalogue lists 162 manuscripts with a formalized main result, and OpenAI marks its review status 'unchecked'. OpenAI warns that unformalized results 'could have issues'. The release is real and checkable; the mathematics is mostly not yet independently verified.

Key facts

Science result

Field
mathematics / multiple (number theory, algebraic geometry, analysis, TCS, combinatorics, operator algebras, topology, PDE, mathematical physics)
Problem
≈4,000 open problems posed to an internal model; 372 result families kept
Result
722 manuscripts claiming proofs or counterexamples for hundreds of open problems, including the quasi-Riemann hypothesis (Re s > 7/8), the Unique Games Conjecture, the Hodge conjecture for CM abelian varieties, the free group factor problem, Kaplansky's conjectures, L = BPL and ω ≤ 9/4.
AI system
OpenAI internal model (unreleased, unnamed)
Human role
Autonomous by a fixed procedure for the vast majority (≈3 h ChatGPT Pro compute per result); stated exceptions: the zeta zero-free region work (11/12 write-up human-edited) and the Hodge conjecture for CM abelian varieties
Verification
Mixed: Lean formalizations for 162 manuscripts / 235 family scope notes (OpenAI-produced, review status 'unchecked'); the rest are unrefereed preprints
Status
pending
Why surprising
No previous release, human or AI, has claimed this many famous open problems at once; several (Unique Games, quasi-RH, free group factors) come with Lean proofs that can be checked by machine.

What happened

Late on 6 October 2026, OpenAI published the list it had promised on 21 September, when it said an internal model had "resolved" more than 100 open problems. The list turned out to be far larger: a public GitHub repository with 722 manuscripts in 372 "result families", each family grouping a main result with companions, consequences or alternative proofs, classified into 17 subject areas from number theory to PDE.

The README says the model was posed about 4,000 problems during evaluation, "after performance on our existing mathematical evaluations saturated". Each kept result used on average about three hours of ChatGPT Pro thinking compute. OpenAI names two exceptions to this fixed procedure, the zeta zero-free region and the Hodge conjecture for CM abelian varieties, without saying exactly how they differed. It also publishes abridged "summarized chain of thought" documents for ten families. These show the model trying and discarding approaches; the free-group-factor summary, for example, lists failed attempts with rigidity, Gaussian crossed products and free entropy before the final construction.

Verification is uneven, and OpenAI says so. A Lean 4 library ships with the release. formalization.yaml lists 162 manuscripts with a formalized main result and 185 Comparator challenges, and 235 of the 372 families in CONTENTS.md link a Lean scope note. Each challenge states a theorem, and a checker confirms that OpenAI's proof proves exactly that statement using only Lean's standard axioms. Whether each formal statement faithfully captures the famous problem is still a question for human reviewers, and the catalogue's own review field reads "unchecked". The other families, among them the Hodge, BSD, L = BPL and Hilbert's tenth claims, are unrefereed manuscripts.

The release followed advice from the independent Advisory Group on Mathematics and AI (AGMAI) at IAS, whose 29 September guidelines asked labs to disclose compute, prompts and reasoning, formalize proofs and fund human understanding. AGMAI's statement the same evening called it "an important event for mathematics" but said its role "should not be interpreted as a judgment of the impact of these results or an endorsement of the process".

The most famous claims

Status below: Lean means OpenAI's catalogue gives a formal proof of the main statement (machine-checked against a published statement, not yet independently reviewed); paper means manuscript only.

  • Quasi-Riemann hypothesis: ζ and every Dirichlet L-function have no zeros with Re s > 7/8, plus uniform exclusion of Landau–Siegel zeros (Lean). Separate entry: 2026-10-06-quasi-riemann-hypothesis-openai.
  • Khot's Unique Games Conjecture, plus direct optimal Max-Cut and Vertex Cover hardness (Lean). Separate entry: 2026-10-06-unique-games-conjecture-openai.
  • Rational Hodge conjecture for all CM abelian varieties, which would give the Tate conjecture for abelian varieties over finite fields (paper). Separate entry: 2026-10-06-hodge-conjecture-cm-abelian-varieties-openai.
  • Free group factors: L(F₂) ≅ L(F₃), so all interpolated free group factors are isomorphic (Lean). Separate entry: 2026-10-06-free-group-factors-isomorphism-openai.
  • Kaplansky's conjectures: a torsion-free group algebra over 𝔽₂ with zero divisors (196); direct finiteness fails in characteristic 2 and odd characteristic, through a finitely presented nonsofic group, which also refutes Gottschalk's surjunctivity conjecture and the group-ring Determinant Conjecture (197). All Lean.
  • Irrationality exponent of π is exactly 2, which implies the Flint–Hills series converges (Lean for the exponent); Catalan's constant is irrational (Lean).
  • Thompson's group F is nonamenable; Cannon's conjecture; Artin K(π,1) conjecture; a torsion-free hyperbolic group that is not residually finite (all Lean).
  • Matrix multiplication ω ≤ 9/4 (Lean); integer multiplication and the DFT below n log n (109, 130); L = RL = BPL (paper); NP-hardness of 3-colouring with any constant number of colours (Lean).
  • Combinatorics: Erdős's reciprocal-sum conjecture, i.e. any set of integers whose reciprocals diverge contains arbitrarily long arithmetic progressions (Erdős's best-known prize problem; Lean); the plane is not 5-colourable, Borsuk fails in dimension 9, Sidorenko's conjecture false, Seymour's second-neighbourhood conjecture, Barnette's conjecture, the circulant Hadamard conjecture (all Lean); Hadwiger's conjecture false in fractional form (paper; Lean covers only the companion linear list-Hadwiger bound, compare 2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra).
  • Analysis and geometry: Falconer distance conjecture in all dimensions (Lean); Kakeya maximal conjecture in 3D (paper); symmetric and general Mahler conjectures (Lean); Hilbert's sixteenth problem, uniform bound on limit cycles in each degree (paper; Lean covers only the quintic Liénard case); hot-spots conjecture for simply connected planar domains (Lean); global regularity for 3D relativistic Vlasov–Maxwell (Lean).
  • Number theory and arithmetic geometry (paper): full BSD formula for Selmer corank ≤ 1 and Goldfeld's conjecture; Hilbert's tenth problem over ℚ (undecidable); modularity over imaginary quadratic fields; abelian Zilber–Pink; the p-adic section conjecture; infinitude in Artin's primitive root conjecture.
  • Operator algebras: Kadison similarity, strong Kadison–Kastler, the generator problem for II₁ factors, a counterexample to the hyperinvariant-subspace problem on Hilbert space (Lean); counterexamples to Baum–Connes (coefficient-free reduced) and Kadison–Kaplansky (paper).

Reactions (first ~7 hours; X view counts from fxtwitter, early 7 Oct)

  • Alex Kontorovich (number theorist): "Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked… They got a zero free strip!!!! Insane" (X, ~420k views). A follow-up: "Yeah, no Siegel zeros either. So I guess two Fields medals…"
  • Levent Alpöge (Anthropic): "Big, big, big, big props for quasiriemann and no Siegel zeroes". He called it "obviously the most significant moment in mathematical history", and also wrote: "There are some sad stories related to their users getting scooped / conflicts of interest (kakeya maximal comes to mind)" (X, ~112k). The Kakeya scooping allegation is unverified.
  • Daniel Litt (Toronto): "Looks like mathematicians have a lot of exciting work to do". He called some framings of "progress" on famous conjectures "a stretch", and flagged that the Bloch's-conjecture paper "seems to cite some wrong existing work without mentioning it's wrong… but AFAICT doesn't meaningfully rely on it" (X, X).
  • Jared Duker Lichtman: "This output is absolutely historic… However, it is not the Millenium problem rumor". He said the cubic family of L-functions "appear to be essential for the proof" of the zero-free strip (X).
  • Steven Strogatz, on ω ≤ 9/4: "This leap in progress is like Bob Beamon's long jump" (X, ~118k).
  • Joe Bebel (TCS): the release "includes a claimed proof and Lean formalization of the Unique Games Conjecture, which I have been personally working on for the last year and a half" (X, ~99k). Chris Peikert: "Unique Games Conjecture, proved. L=RL, proved." (X).
  • Kiwamu Watanabe (algebraic geometer, in Japanese) called the algebraic-geometry list (Hodge for CM abelian varieties, Fujita, Nagata, log abundance, MMP termination) "quite amazing", but said he cannot tell whether it is correct (X, ~70k). Isaac Kim called the quantum-information claims "a shocking list of problems" (X).
  • Ken Ono: "AI-assisted mathematical discovery at the highest level is no longer hypothetical" (X). Gil Kalai: "Certainly this is an amazing milestone for mathematics, and the results will needs to be verified" (blog).
  • Terence Tao, about 30 minutes after WIRED reported the coming release (before it went live), on Mathstodon: solutions "are now being harvested at large scale in an unsustainable fashion", and a "Math 2.0" should "decenter the role of raw problem solving" (Mathstodon). He had not commented on the content by early 7 Oct.
  • Andrew Sutherland (MIT), to Scientific American: treat one-shot claims as unverified "until and unless they release the model… We should ask for receipts."
  • OpenAI side: Sam Altman, "We are entering a new era of discovery now" (X, ~520k); Greg Brockman (X); Codex lead Thibault Sottiaux (X, ~356k). Former OpenAI contributor @AcerFur on family 109: "we can do integer multiplication faster than n log n" (X, ~264k).
  • Commentary: Zvi Mowshowitz, "simultaneously the biggest day in mathematical history, and also felt kind of slow"; François Chollet asked whether the jagged frontier is "mainly math + code" driven by RL with verifiable rewards; Gary Marcus said it "uses symbolic AI (Lean, etc)". High-reach posts by Dan McAteer (388k), Will DePue (282k) and AISafetyMemes ("90 of the 500 most important open problems", based on an LLM-ranked list at proofatlas.ai, not a mathematicians' ranking).
  • Independent Lean re-check: within hours, a public repo (davegoldblatt/openai-zeta-proof-check, created 7 Oct 01:36 UTC) re-ran the Comparator check of the 7/8 zeta theorem. It reports acceptance by two kernels (Lean's and nanoda) with only the three standard axioms; the proof spans ~2,900 files and ~500k lines. This is one person's run; the paper itself was not reviewed.
  • No expert had reported a mathematical error in a main proof by early 7 Oct.

Press

  • WIRED (6 Oct, before the release): "OpenAI is pissing off a bunch of mathematicians again". It reports that mathematicians felt their advice was ignored (Bryna Kra: "Apparently, that input was ignored"; Nestor Guillen: "there's a perception of mobster behavior").
  • NYT (Kenneth Chang, Siobhan Roberts): "OpenAI Releases Findings on 377 Math Problems, Further Roiling Field". Paywalled; we have only the preview, and the 377 figure is the NYT's count. Gizmodo also says 377; the repo itself says 372 families.
  • Scientific American (Joseph Howlett), The Verge (Robert Hart; quotes AGMAI asking labs not to use releases as "marketing vehicles"), Gizmodo (Mike Pearl; notes AGMAI's 29 Sep request that labs "stop testing advanced mathematical problems on proprietary models"), WSJ (not read).

Why it matters

If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks. Machine-checked Lean proofs for famous statements such as Unique Games and the quasi-Riemann hypothesis make parts of it verifiable far faster than refereeing. The scale also creates a new problem, the one AGMAI flagged: the community must now absorb, check and understand hundreds of results it did not choose. Until that happens, treat each individual claim as unverified unless a Lean proof exists and experts have confirmed that its formal statement matches the problem.

Changelog

  • 2026-10-07: added first reactions (X, Mathstodon, blogs), press and the independent Lean re-check
  • 2026-10-07: created from the repository README, overview.tex, CONTENTS.md, lean/formalization.yaml and Lean scope notes, the OpenAI blog post and the AGMAI statement
  • 2026-10-07: added Claude Opus 5.5's read of the repo (@ereliuer_eteer) and our own checks of the comparator files (405 challenges; HarmonicGrowth axiom; FinalAssemblyUnconditional)

People

Daniel Litt François Chollet Gary Marcus Greg Brockman Levent Alpöge Sam Altman Steven Strogatz Terence Tao Thibault Sottiaux Zvi Mowshowitz

Related posts (31)

Related events

  1. OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts) ★★★
  2. Mathematicians' AGMAI publishes norms for AI labs releasing AI-generated results; Simons Institute TCS group issues 12 actions ★★★
  3. OpenAI model claims the quasi-Riemann hypothesis: no zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized) ★★★★★
  4. OpenAI model claims a proof of Khot's Unique Games Conjecture, with Lean formalization ★★★★★
  5. OpenAI release claims the rational Hodge conjecture for all CM abelian varieties, which would give the Tate conjecture for abelian varieties over finite fields (no Lean proof) ★★★★★
  6. OpenAI model claims the free group factor problem: L(F₂) ≅ L(F₃), so all free group factors are isomorphic (Lean-formalized) ★★★★★
  7. OpenAI claims a Millennium Prize problem: 10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts ★★★★★
  8. OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs ★★★★★
  9. Quanta's math editor asks 'Is AI the End of Math As We Know It?'; young mathematicians form an Association for Human Mathematics ★★★
  10. Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
  11. Artin–Davenport conjecture proved: every integral cubic form in 10+ variables has a nontrivial zero (Bonolis, Browning, Glas, Wang; key lemma via ChatGPT 6.0, Codex Lean auto-formalisation) ★★★★★

Sources (23)

id: 2026-10-06-openai-math-release-722-manuscripts · updated 2026-10-07 · open in the interactive timeline