OpenAI Drops 722 Math Papers Overnight: Riemann, BSD, Hodge Conjectures Claimed

OpenAI has published 722 mathematical manuscripts generated by a single unreleased internal model, covering 372 distinct problem groups across 17 mathematical fields. The model attempted roughly 4,000 research problems, with each result consuming compute equivalent to about three hours of ChatGPT Pro reasoning time. All papers, source code, partial Lean formalizations, and ten model inference summaries are available in the GitHub repository at https://github.com/openai/math.

Key Mathematical Claims

Quasi‑Riemann Hypothesis (Group 003): The model asserts that the Riemann zeta function and all Dirichlet L‑functions have no zeros in the region Re(s) > 7/8. A second independent proof in the same group gives a zero‑free region Re(s) > 11/12 and additionally excludes the existence of Siegel zeros — a century‑old open question in analytic number theory.

Hilbert’s Tenth Problem over ℚ (Group 004): The model claims there is no general algorithm to decide whether a polynomial with integer coefficients has a rational solution. The integer version was solved by Matiyasevich in 1970; the rational version had remained open for over half a century.

Birch and Swinnerton‑Dyer Conjecture: For elliptic curves over ℚ satisfying Selmer corank 0 or 1, the model provides a complete formula. Combined with another result using density statistics, it claims to cover the vast majority of quadratic twists of every rational elliptic curve.

Hodge Conjecture: The manuscripts address complex‑multiplication abelian varieties and products of K3 surfaces, claiming coverage of all dimensions and codimensions for these specific classes — though the full conjecture encompasses a much broader range of geometric spaces.

Irrationality measure of π: The model proves the irrationality exponent of π is exactly 2, improving the previous best bound of ~7.1 to the theoretical optimum.

Catalan’s constant: A proof that Catalan’s constant is irrational, a long‑standing open problem.

Unique Games Conjecture (Group 102): The model claims a full proof of Khot’s 2002 conjecture, along with derived hardness‑of‑approximation results for Max‑Cut, Vertex Cover, and other problems.

Other areas: Sparse spin‑glass Mézard‑Parisi formula, spontaneous magnetization in the quantum Heisenberg ferromagnet, isomorphism of free group factors, and global smooth solutions to the 3D relativistic Vlasov‑Maxwell system.

Process and Exceptions

The repository states that the vast majority of results were produced by the same fixed pipeline. Two exceptions are noted:

The Riemann zero‑free region proofs (especially the Re > 11/12 version) received human editing for readability.

The Hodge conjecture work on CM abelian varieties did not follow the standard pipeline entirely.

Verification status varies; many results lack complete Lean formalizations.

Mathematical Community Reaction

An independent advisory panel of nine scholars — including Fields medalists Timothy Gowers and Martin Hairer and physicist Edward Witten — issued a statement acknowledging the event’s significance but explicitly declaring that their participation does not constitute endorsement of the results or of OpenAI’s production process . Verification is left to domain experts.

Mathematician Francesco Maggi (UT Austin) questioned the feasibility of digesting hundreds of papers: the September Navier‑Stokes proof is still under scrutiny, and now hundreds more appear. He warns that AI output speed far outpaces human comprehension.

Terence Tao described AI’s pace as “too crazy” and “like the end of the world for mathematics.” Cédric Villani (Fields medalist) said he was “stunned” and compared the situation to an unprecedented catastrophe.

OpenAI plans to fund workshops to help mathematicians study the corpus. The advisory panel’s response: materials are public; the community must now check, explain, and decide which results enter the mathematical canon.

OpenAI’s 28‑Day Challenge – Day 2 Updates

Concurrently, OpenAI announced four Codex/Work improvements:

Auto‑review: A second AI agent checks high‑risk actions during long tasks; free for ChatGPT account users.

API pricing simplified from five tiers to three.

Meetings plugin now generates minutes and action items.

Decisions API opened for public beta.

References: OpenAI blog https://openai.com/index/sharing-ai-progress-in-mathematics/, GitHub repo https://github.com/openai/math, AGMAI https://agmai.org/, X post

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