Mathlib

An OpenAI repository containing Lean 4 formalizations of ten results presented in its paper on advances in mathematics and theoretical computer science. The formalizations cover high-dimensional sphere packing; binary and spherical codes; non-sofic groups; Connes’s rigidity conjecture; arithmetic circuit complexity; quantum parallel repetition; the closest vector problem; Ehrhart’s volume conjecture; multicolor Ramsey numbers; and compactness and degeneracy conjectures in extremal graph theory. Each result is implemented as a separate Lean module, including `SpherePacking.lean`, `MetricCodes.lean`, `NonSoficGroup.lean`, `ConnesRigidity.lean`, `Permanent.lean`, `QuantumParallelRepetition.lean`, `GapCVP.lean`, `EhrhartVolumeInequality.lean`, `MulticolorTriangleRamsey.lean`, and `CompactnessAndDegeneracy.lean`. The project uses Lean 4.32.0, mathlib, and Lake. With the mathlib cache installed, all formalizations can be built with `lake build All`, or individual modules can be built by name; the repository also provides reasoning walkthroughs and documents independent checking with Comparator.

Lean 4 is a programming language and theorem prover used to formalize mathematical proofs, including proofs generated by AI systems. Its project repository provides the language implementation along with theorem-proving and functional-programming tutorials, reference documentation, examples, installation guidance, and instructions for building from source.

A research artifact containing a complete machine-checked Lean 4 proof of Fermat's Last Theorem, following the Frey–Serre–Ribet–Wiles and Taylor–Wiles argument and built on Mathlib. The theorem states that for natural numbers n, a, b, and c with n ≥ 3 and a, b, and c positive, a^n + b^n ≠ c^n. Lean's kernel checks the proof, which uses only Lean's three standard axioms—propext, Classical.choice, and Quot.sound—and the repository excludes sorry statements, added axioms, native_decide, unsafe code, and related escape mechanisms. A comparator independently replays the environment against a Mathlib formulation, while Nanoda, an independent Lean kernel implemented in Rust, checks an exported version of the environment. The repository also includes PROOF-PATH.md and an offline HTML presentation of the proof's theorems, definitions, dependencies, and generated documentation. It is released under the Apache License 2.0 and is described as not maintained and not accepting contributions.

NavierStokesAndEuler is an open-source Lean 4 formalization of results concerning finite-time breakdown in the Navier–Stokes and incompressible Euler equations, developed by OpenAI. For Navier–Stokes with positive viscosity, it formalizes constructions on both three-dimensional Euclidean space and the periodic torus using smooth initial data and external forcing; the results rule out global smooth solutions under the stated kinetic-energy or existence conditions. For the unforced Euler equations, it formalizes a smooth, compactly supported, divergence-free initial velocity whose solution develops a finite-time singularity, with the velocity’s C1 norm becoming unbounded and the time integral of the vorticity’s L-infinity norm diverging. The repository uses Lean 4, Mathlib, and Lake. Its formalizations can be built after fetching the Mathlib cache, and the repository provides separate instructions for independently checking them with Comparator.

A formally verified 3D constructive solid geometry (CSG) kernel for intersecting triangle meshes. Implemented in Lean 4, it is checked against a concise formal specification that defines the resulting surface and guarantees stated well-formedness properties for the output triangulation. The project includes a browser demo that imports STL files and runs the compiled Lean code locally without sending mesh data to a server. The kernel is formally verified, but the demo's UI and glue code are not; the implementation prioritizes correctness review over performance and may generate meshes that are finer than necessary.