The OpenAI math repository: 722 manuscripts, 372 families and the Lean check

On the evening of 6 October 2026 a striking collection appeared on GitHub. OpenAI published 722 manuscripts grouped into 372 result families in a public repository. The producer is described as an unreleased internal frontier model. The scale is matched by ambition: 17 fields from number theory and geometry to theoretical computer science and mathematical physics. This article clones the repository and reads the catalogue, the papers and the Lean traces to answer what is actually there. The OpenAI announcement frames the process as transparent.

722 and 372 are not the same thing. A manuscript is a single document, while a result family bundles documents that pursue one goal: a main result, a companion argument, an application or a second proof. The catalogue peaks at number 377, with 045, 061, 070, 123 and 163 absent. So 372 is the true counted total. Paper dates spread from 10 September to 6 October; 6 October is only the catalogue release day. The inspected snapshot is commit adc7f1241. The Kingy survey establishes this distinction clearly.

What the numbers mean

The production routine is described as uniform. After its existing evaluations saturated, the model was posed about 4,000 open problems. Outputs that passed a significance filter were grouped into families and manuscripts. Each result used roughly three hours of ChatGPT Pro thinking on average. Two threads sit outside that routine: a zero-free region for the Riemann zeta function and Hodge for CM abelian varieties. The Re(s) > 11/12 write-up was human edited for readability. No aggregate dollar total, token count or accelerator hours were disclosed. The CellCog summary flags that costing gap openly.

A tour of the repository has three stops. First the overview catalogue: short descriptions and links for all 372 families. Then the CONTENTS map: from family to paper. Finally the preprints directories: each with a PDF, source text and citation data. The license is Apache 2.0. Version discipline is preserved, with corrections added as new versions. Downloading one paper is enough; cloning everything means 2.4 GB and over 130,000 files. The BigChange inventory confirms a PDF in each of the 722 directories.

Headline claims

The number-theory shelf is crowded. Family 002 asserts the full Birch–Swinnerton-Dyer formula in low Selmer corank with finiteness of Tate–Shafarevich. Family 003 asserts a zero-free half-plane Re(s) > 7/8 for Dirichlet L-functions plus a second argument for 11/12. Family 005 asserts the irrationality of Catalan's constant, while family 017 asserts that the irrationality exponent of pi equals 2. Convergence of the Flint–Hills series sits in the same bundle. Two-point Chowla and Deligne–Drinfeld are listed too. The OpenAI catalogue presents these as proved.

The geometry and physics wing is equally broad. Family 032 gives the rational Hodge claim for CM abelian varieties and K3 products, with consequences toward Tate and the Hodge standard. Symmetric and general Mahler claims sit in family 087. A counterexample to Kaplansky direct finiteness in characteristic two is family 197. Spontaneous magnetization in the quantum Heisenberg ferromagnet is 271, the 3D relativistic Vlasov–Maxwell system is 362, and the isomorphism of free group factors is 287. Each demands a long chain of reasoning that takes humans weeks to read.

Theoretical computer science is the largest block with 40 families. The Unique Games Conjecture, the claim that randomness adds no power in log-space (L = RL = BPL), a 9/4 upper bound for the matrix-multiplication exponent, and sub-n-log-n integer multiplication and Fourier transforms sit here. Non-amenability of Thompson's group F, Hilbert–Smith, Kakeya in three and four dimensions, and the uncolourability of the plane with five colours under the unit-distance rule share the shelf. By the Kingy count, combinatorics holds 37 families, geometry 36 and number theory 31. Breadth also enlarges the checking burden.

Verification and debate

The formal-proof side needs the most careful reading. By the CellCog count, the main result of 162 papers has a Lean formalization. Scope documents cover 235 families. The pile is huge: about 122,000 Lean files, 26 million lines, drawing on some 30 outside formalization projects. Checking runs through Comparator, where the challenged statement and the solution live in separate modules. The sorry marker in a challenge is not incompleteness but the placeholder for the statement to match. The sample configuration disables nanoda and allows only three standard axioms. Catalogue status notes read partial progress and unchecked. The BigChange review stresses that these notes are publication metadata, not the outcome of an executed check.

The method debate matters as much as the results. The independent AGMAI group at the IAS has nine members, including Gowers, Witten, Hairer, Vakil and Wood. On 29 September, after more than 600 responses, it published principles: no probing of advanced problems on closed models, and releases built with the community. On the evening of 6 October the group wrote that advice is not endorsement and that publication is the start of human understanding. OpenAI announced revision and citation protocols, exploration of community hosting, and support for workshops. The AGMAI line is clear: without equal access there is no free inquiry.

A practical route for readers follows. Find the family sketch in the overview catalogue, then pin the paper PDF and citation data in its preprints directory: commit hash, directory name, version. Where a scope document exists, note which statement Lean covers and what is excluded. A Lean pass is a strong signal but not acceptance by itself; check whether the formal statement matches the paper claim. The order is catalogue, paper, Lean trace, independent source. Do not cite before verification. OpenAI repeats its aim to share the model responsibly; AGMAI says the community will set the measure.