AxiomMath
IMO2026
AxiomProver is an autonomous multi-agent ensemble theorem prover for Lean 4 that achieved a perfect score at IMO 2026 by solving all six problems.
ten-proofs
This repository contains Lean 4 formalizations of ten major mathematical and theoretical computer science results from OpenAI's 'Ten Advances' paper, including theorems on sphere packing, non-sofic groups, Ramsey numbers, and more.
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Apache-2.0
Last updated
2026-08-01
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FR-AI / ANALYSISWhy it is worth attention
It turns a set of notable recent advances—several resolving long-standing conjectures—into machine-checkable Lean proofs, making them more accessible to the formal verification community.
Who it is for
- Mathematicians interested in formal proof verification
- Theoretical computer scientists working on complexity, coding, and quantum games
- Lean/mathlib users and formal proof developers
- Graduate students in mathematics or computer science
Use cases
- Verifying or exploring machine-checked proofs of major theorems
- Reusing the formalized lemmas and constructions in new Lean projects
- Learning how advanced mathematics and TCS results are encoded in Lean
- Benchmarking proof assistant capabilities on large, diverse formalizations
Strengths
- Covers ten distinct advanced results across mathematics and TCS in one repository
- Modular structure: each result has its own Lean file and can be built independently with Lake
- Uses Lean 4.32.0, mathlib, and a cache command, making setup reproducible
- Provides Comparator instructions for independent proof checking
Considerations
- Requires the Lean ecosystem (elan, Lake, mathlib) and some familiarity with Lean tooling to build or inspect
- The README lists the results but does not provide detailed mathematical explanations or tutorials
- Formalizations are tied to Lean 4.32.0, so future library or version changes may require adaptation
README quick start
Ten Advances in Mathematics and Theoretical Computer Science
This repository contains Lean 4 formalizations of the results presented in Ten advances in mathematics and theoretical computer science by OpenAI.
The results
- High-dimensional sphere packing: Improved asymptotic upper bounds on
sphere-packing density, reaching the Cohn–Elkies threshold.
(
SpherePacking.lean) - Binary and spherical codes: Exponentially stronger upper bounds for
binary codes at every minimum distance, together with corresponding bounds
for spherical codes. (
MetricCodes.lean) - Non-sofic groups: A construction of a non-sofic group, resolving whether
every group admits finite permutation approximations.
(
NonSoficGroup.lean) - Connes’s rigidity conjecture: A counterexample to the conjecture that
certain groups are determined by their group von Neumann algebras.
(
ConnesRigidity.lean,ConnesRigidity/) - Arithmetic circuit complexity: New lower bounds for computing the
permanent with arithmetic circuits and formulas, including an
$n^4 / \log n$ formula lower bound.
(
Permanent.lean) - Quantum parallel repetition: Exponential parallel repetition for
arbitrary finite, two-player quantum games.
(
QuantumParallelRepetition.lean) - Closest vector problem: Polynomial-factor hardness of approximation for
the closest vector problem, with related consequences for decoding and
lattice problems. (
GapCVP.lean) - Ehrhart’s volume conjecture: The sharp maximum volume in every dimension
for a convex body whose centroid is its only interior lattice point.
(
EhrhartVolumeInequality.lean) - Multicolor Ramsey numbers: A superexponential lower bound for multicolor
triangle Ramsey numbers, resolving Erdős problem 183.
(
MulticolorTriangleRamsey.lean) - Extremal number conjectures: Counterexamples to the compactness and degenerac
Description
Lean certificates accompanying proofs in mathematics and theoretical computer science
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