Open Math Atlas
Open Math Atlas organises the 719 manuscripts of the public openai/math repository into result families, by discipline. Every result can be read at the depth you choose, with its verification status always in view.
Why this matters
On 6 October 2026, OpenAI quietly published a collection of mathematics written almost entirely by an internal AI. It is not “722 great problems solved”: it mixes proofs, improved bounds, reformulations, computations and variants of known problems. The real question is no longer “can AI do mathematics?” but “who can check, at this scale, what it produces?”
AI hardware
Matrix multiplication is the basic operation of neural networks. Shaving even a few operations off it can lower computing costs — one of the most direct links between pure research and everyday technology. The real gain still depends on actual hardware.
Number theory
Posed in 1859 and worth a $1M prize, it is not solved by this release. The work concerns a partial result — a computational “wall” — that has been checked independently on GitHub. A verified step is not a proof of the whole hypothesis.
Logic
In 1900 Hilbert asked for a general method to solve integer equations; in 1970 it was shown that none can exist. The AI explores variants of this theorem — symbolically striking, since a machine is working on a result about the limits of machines.
Combinatorics
How can you colour an infinite plane or grid while respecting simple rules? Easy to state, notoriously hard to settle. The contributions are genuinely interesting but must be checked by specialists before being accepted.
Analysis
Several texts sharpen technical properties of π. They do not change what we know about its nature: π has been known to be irrational and transcendental since the 19th century.
Reliability
On 8 October, two days after release, three texts were withdrawn over a sign error. It does not invalidate the whole collection — but it shows why no result should be accepted without careful review.
The community is divided between enthusiasm and concern: Terence Tao worries about the field's ability to absorb and check so many results; Kevin Buzzard asks how the mathematician's job will change; a group of Fields medallists (AGMAI) issued a statement.
Verification
Lean, a proof checker that guesses nothing, controls every logical step of part of the collection — provided the formal statement matches the announced problem, which still needs a human check.
Understanding
When answers become cheap, explaining why a result is true, and what it connects to, becomes the scarce skill.
Asking good questions
The AI worked on problems humans chose. Deciding what is worth solving remains a human task.
In short: AI has become a serious tool for producing mathematics, but it replaces neither verification nor expert judgement. This release is an important milestone — neither a finished revolution nor mere hype.
Sources:OpenAI announcementScientific AmericanTerence Tao (Mathstodon)Independent Riemann “wall” checkFull analysis — Jura Web Excellence
Families accompanied by a reasoning summary published by OpenAI.
Proves the ordinary two-point Chowla conjecture, with a bound for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times for .
Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational satisfies . This also proves convergence of the Flint–Hills series , with angles in radians.
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product in dimension has Gromov width 4.
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of with no nonconstant k-term progression has size at most , with positive constants depending only on k.
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.
Proves Bloch's T3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ3. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.
Resolves the free group factor isomorphism problem: , and hence all interpolated free group factors, including , are isomorphic. Their common factor has fundamental group .
Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.
In “Algebraicity of Weil classes on split abelian eightfolds” a sign error invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers. As a result, we have withdrawn the following three manuscripts:
• Algebraicity of Weil classes on split abelian eightfolds • Algebraicity of Kuga–Satake Correspondences for K3 Surfaces • The rational Hodge conjecture for products of K3 surfaces
The withdrawn papers now carry notices explaining the gap and linking to the archived manuscripts.
We have revised 14 other manuscripts with proof repairs, corrected statements, clearer hypotheses and dependencies, and one correction to an obsolete citation. The changes include: