Executable proof checked by a proof kernel
AI × MATHEMATICS · PUBLIC RECORD
From Erdős problems to a formalization frontier.
A sourced timeline of AI contributions to open mathematics—and the evidence behind each claim. This record separates discovery, human judgment, formal checking, and public announcement.
How to read this record
“AI solved it” is not one evidence state.
Paper or construction checked by mathematicians
A bounded algebraic claim can be recomputed
Public claim whose scholarly record is still forming
Lean replay desk
Claim → source → executable check.
These are the shortest honest paths from the public record to a kernel-checked artifact. External repositories are linked directly; Proofweave workspaces are entry points for a bounded local replay and do not imply that a result has already been verified here.
Ten advances
Ten Lean 4 formalizations, with a documented all-project build and individual modules.
Zeta23
A pinned Lean formalization for the more-than-two-thirds lower-bound result; it does not prove RH.
Owner-approved replay
Select a pinned target, connect a local Agent, run Lean locally, and publish only the checkpoint you approve.
Latest source review: 19 August–10 September 2026. Includes the S² × S² human research milestone for context. Selected X announcements are cross-checked against authors’ papers and repositories. Dates distinguish public releases from earlier proof work; X timestamps use UTC. External verification claims are attributed to their authors; this update includes no new Proofweave Lean replay or receipt.
19 moments · one accelerating frontier
Follow the record across 2026.
Zoom in to separate busy days. Drag the timeline or swipe sideways to explore. Records from the same day stay stacked; month-only records sit at mid-month.
Aristotle
Erdős Problem #728
A Lean-checked result became the first recognized Erdős problem resolved autonomously by an AI system.
- AI contribution
- Generated the proof and its Lean formalization.
- Human contribution
- Selected and presented the result, and searched the literature for prior work.
- Why it mattered
- The headline was bounded by an executable artifact: the mathematical proof could be replayed independently of the model narrative.
ChatGPT + Aristotle
Erdős Problem #650
A model-proposed strategy was made rigorous and formally verified, showing a distinctly collaborative path to a result.
- AI contribution
- ChatGPT proposed the strategy; Aristotle produced a detailed Lean-verified argument.
- Human contribution
- Checked the reasoning, supplied context, and wrote the final exposition.
- Why it mattered
- It made clear that “AI solved” can describe a chain of different systems and human decisions, not one isolated act.
GPT-5.4 Pro + mathematicians
Erdős Problem #1196
A new Markov-chain method with von Mangoldt weights emerged from model output and was refined into a mathematical paper.
- AI contribution
- Suggested the central method and an initial route through the argument.
- Human contribution
- Reworked gaps, established the rigorous result, and authored the paper.
- Why it mattered
- The valuable contribution was an idea that changed the proof route—not a claim that the first generated answer was already a proof.
OpenAI reasoning model
A planar unit-distance conjecture falls
A general-purpose reasoning model produced a counterexample to a longstanding Erdős unit-distance conjecture.
- AI contribution
- Found the construction and the core disproof.
- Human contribution
- External mathematicians checked it, simplified the construction, and situated the result in the literature.
- Why it mattered
- Counterexamples became a visible AI research mode: one exact construction can decisively close the wrong branch.
Google DeepMind · AlphaProof Nexus
AlphaProof Nexus scales the search
A formal proof-search framework coordinated LLM prover subagents, Lean compiler feedback, optional AlphaProof calls, and evolutionary selection across open-problem benchmarks.
- AI contribution
- Generated and revised Lean proof sketches, using compiler feedback until a complete, sorry-free proof passed validation.
- Human contribution
- Formalized targets and evaluated statement fidelity, novelty, and mathematical relevance.
- Why it mattered
- Its reported 9/353 Erdős and 44/492 OEIS results shifted the frontier from one celebrated example to a repeatable, measurable workflow.
GPT-5.6 Sol Ultra + Codex
Cycle Double Cover Conjecture
OpenAI released a proof that every finite bridgeless loopless multigraph admits a family of cycles covering each edge exactly twice.
- AI contribution
- GPT-5.6 Sol Ultra developed the proof; Codex helped produce the write-up and the accompanying Lean formalization.
- Human contribution
- Specified the exact target and adversarial audit requirements, published the complete prompt, and opened the artifacts to independent review.
- Why it mattered
- A concise flow-and-linear-algebra argument arrived with a pinned Lean project that checks the full unconditional theorem—not only a finite computation or special graph class.
Claude-Fable · reported by Levent Alpöge
A three-dimensional Jacobian counterexample is announced.
An explicit polynomial map over ℂ was reported with constant nonzero Jacobian determinant and three distinct inputs sharing one output—the exact shape required to refute the classical Jacobian Conjecture in dimension three.
CONSTANT JACOBIANdet JF = −2
NON-INJECTIVITY3 inputs → (−¼, 0, 0)
Inspect the exact polynomial map
F₁ = (1 + xy)³z + y²(1 + xy)(4 + 3xy)F₂ = y + 3x(1 + xy)²z + 3xy²(4 + 3xy)F₃ = 2x − 3x²y − x³zCollision: (0, 0, −¼), (1, −3/2, 13/2), and (−1, 3/2, 13/2) all map to (−¼, 0, 0).
- AI contribution
- Claude-Fable produced the explicit map during an open-ended mathematical exploration.
- Human contribution
- Levent Alpöge directed the investigation, checked the construction, and announced the result with exact coordinates.
- Evidence today
- The determinant and collision are directly reproducible by symbolic or exact arithmetic. Journal review and a formal Proofweave receipt are not yet recorded.
What this does—and does not—establish
The displayed algebraic claims can be checked now. The scholarly publication and attribution record are still forming, and the two-dimensional Jacobian Conjecture remains open.
OpenAI · internal Astra
OpenAI publishes ten advances
OpenAI published ten results spanning geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics.
PORTFOLIO10 results
PUBLIC ARTIFACTLean 4 modules
Show the ten result areas
- Sphere packing
- Binary and spherical codes
- Non-sofic groups
- Connes's rigidity
- Arithmetic circuit complexity
- Quantum parallel repetition
- Closest vector problem
- Ehrhart's volume conjecture
- Multicolor Ramsey numbers
- Extremal number conjectures
Lean replay pathOpen the ten Lean certificates
The public repository lists one Lean 4 module for each result and documents `lake build All`.
lake exe cache get · lake build All- AI contribution
- The internal Astra system generated the mathematical arguments and then formalized each one in a Lean certificate.
- Human contribution
- OpenAI researchers prepared the manuscripts, selected the public record, and took responsibility for the formalized proofs.
- Why it mattered
- The unit of evidence changed from one headline result to a portfolio. The public repository makes all ten certificates available for independent Lake builds; Proofweave still treats them as external artifacts until a local replay receipt is submitted.
Claude · unreleased research version
Claude improves a Riemann-zeta lower bound
Claude did not solve the Riemann Hypothesis, but Anthropic reports a new lower bound for the fraction of zeta zeros on the critical line: 41.6% to 67.2%.
KNOWN LOWER BOUND41.6% → 67.2%
RH ITSELFNot proved
Lean replay pathOpen the Zeta23 Lean formalization
The artifact documents a pinned Lean toolchain and headline theorems for the more-than-two-thirds result.
lake build · lake build Solution Solution.Multiplicity Solution.XiPrime- AI contribution
- Claude searched, tested, criticized, and formalized the argument across coordinated research sessions.
- Human contribution
- Anthropic mathematicians examined the paper, related it to prior work, and released the supporting note and formal artifact.
- Evidence today
- This is a substantial result about a related problem, not a proof of the Riemann Hypothesis. The exact formalization is now public and can be replayed independently; a Proofweave receipt is still not recorded.
What this does—and does not—establish
This improves a lower bound for a related zeta-zero problem. It does not prove or disprove the Riemann Hypothesis; the formal artifact is public, while a Proofweave replay receipt is still pending.
Simon Brendle + Pei-Ken Hung · human research
Positive sectional curvature on S² × S²
Brendle and Hung posted a construction of a positively curved metric on S² × S², addressing the Hopf product conjecture. Their 24 August revision corrects manuscript and notebook typos.
- AI contribution
- No generative-AI contribution is disclosed in the cited paper. Mathematica supports calculations.
- Human contribution
- The authors constructed a third-order perturbation of a Cheeger–Müter metric and supplied the argument and Mathematica notebook.
- Evidence today
- A human research milestone included for context alongside AI-assisted work; computer algebra alone does not establish AI authorship.
What this does—and does not—establish
This is the Hopf product problem for positive sectional curvature on S² × S², distinct from the complex-structure problem for S⁶. No Proofweave verification receipt is recorded.
Jihao Liu + GPT-5.6-sol, Fable 5 and Danus
A counterexample to the cscK Yau–Tian–Donaldson conjecture
Liu’s preprint reports a polarized smooth projective fivefold that is K-polystable but admits no constant scalar curvature Kähler metric. arXiv records submission on 19 August; the manuscript is dated 24 August.
- AI contribution
- The paper credits GPT-5.6-sol, Fable 5 and Danus with the counterexample and proof, and documents a separate Danus run.
- Human contribution
- Liu directed the investigation, checked and edited the argument, and takes responsibility for its correctness; the appendix documents contributions with Bin Dong and Guoxiong Gao.
- Evidence today
- The result targets K-polystability as a criterion for constant scalar curvature, where the distinction from uniform stability matters.
What this does—and does not—establish
The claim concerns the cscK formulation, not a reversal of the established Fano Kähler–Einstein theorem. No Lean certificate or Proofweave replay is asserted here.
Levent Alpöge + Claude · formalization by Boris Alexeev
A complex structure on the six-sphere is announced
Alpöge announced a proposed complex structure on S⁶, built from a family of complex two-tori. His X post is dated 23 August UTC, or 24 August in China. A subsequent public Lean project formalizes the existence statement.
Lean replay pathInspect the HopfProblem certificate
The repository supplies a Comparator configuration for the formalized statement. This update has not run it locally.
lake exe cache get · lake build lean4export · lake exe comparator comparator/config.json- AI contribution
- Alpöge credits Claude in the original announcement; the released formalization provides a separate executable artifact.
- Human contribution
- Alpöge presented the construction and manuscript. Boris Alexeev released the HopfProblem formalization with a Comparator setup for its statement.
- Evidence today
- The target is an integrable complex-manifold structure compatible with the sphere’s standard topology, a stronger requirement than an almost-complex structure.
What this does—and does not—establish
This records a manuscript and an external formalization, without claiming a fresh Proofweave kernel replay, statement audit, or receipt. It is distinct from the S² × S² curvature problem.
Youness Lamzouri + AxiomProver
A simpler zeta-zero proof receives an AI formalization
Lamzouri gave a simpler proof that more than 67.25% of zeta zeros are simple and on the critical line. Axiom announced its Lean formalization on 3 September.
- AI contribution
- AxiomProver formalized the argument in the ZetaZeros project.
- Human contribution
- Lamzouri replaced the earlier matrix argument with a Hilbert-space inequality; his 8 September revision adds further estimates.
- Why it mattered
- The public Lean statement assumes the Riemann–von Mangoldt and pair-correlation formulas. A formal certificate of the remaining argument is useful evidence, with that dependency boundary preserved.
Axiom Math · mathematicians + AxiomProver
Axiom announces prime gaps at most 212
Axiom released a draft proving that infinitely many consecutive prime gaps are at most 212, improving on Julia Stadlmann’s 240 bound.
- AI contribution
- AxiomProver joined the research effort and extensive numerical experiments on a distributed cluster.
- Human contribution
- The authors developed and checked the optimization, explicitly crediting Stadlmann’s analytic framework and earlier sieve work.
- Why it mattered
- The announcement linked a preliminary paper and said a final paper and Lean formalization would follow. The separate PrimeGapsLib project for 246 does not certify this new 212 bound.
GPT-6 Astra · OpenAI
OpenAI releases a prime-gap bound of 186
OpenAI’s short-gap paper, dated 30 August and released with Astra, claims infinitely many consecutive prime gaps of at most 186 via DHL[40, 2].
- AI contribution
- Astra developed the argument and associated numerical certificate.
- Human contribution
- OpenAI published the proof and verification materials, building on Polymath and Stadlmann’s distribution estimates.
- Evidence today
- PrimeGaps186 explicitly retains three input axioms covering analytic estimates and numerical bounds. Its Lean theorem and Python-FLINT certificate have different verification scopes.
What this does—and does not—establish
The paper claims an unconditional mathematical bound. The released Lean proof is conditional on stated inputs; it is not an end-to-end formalization of every input. A bound of 186 does not establish the twin prime conjecture, which asks for a gap of 2.
GPT-6 Astra · OpenAI
A stronger lower bound for long prime gaps
A companion paper gives a stronger asymptotic lower bound for the largest gap between consecutive primes below X, with an additional iterated-logarithm gain.
Lean replay pathOpen the long-gap formalization
Use the repository’s pinned toolchain and comparator instructions to check the exact formal statement.
lake exe cache get · lake build- AI contribution
- Astra produced the proof presented in the paper.
- Human contribution
- OpenAI released the manuscript and Lean project; the argument builds on the classical long-gap literature.
- Why it mattered
- This concerns unusually large gaps, a different question from small-gap bounds such as 186. The repository supplies a Lean 4.33.0 build and comparator instructions; no replay was performed for this record update.
Claude + Prove2Me · Tianyi Peng
Fermat’s Last Theorem is formalized at scale
Anthropic announced a complete Lean formalization of Fermat’s Last Theorem, produced over 11 days. The reported proof completion was 18 August; 4 September is the public release date.
- AI contribution
- Claude agents formalized the proof through the shared Prove2Me theorem graph.
- Human contribution
- Tianyi Peng directed the project; the proof follows the Wiles tradition and community formalization work. Kevin Buzzard reviewed the released result.
- Why it mattered
- The contribution is formal verification of an established theorem. Anthropic reports only Lean’s three standard axioms and a comparator check against Mathlib’s FLT statement.
Levent Alpöge + Tristan Buckmaster · Claude and Codex
Smoothly forced fluid blowup results become public
Alpöge and Buckmaster released results on finite-time blowup with smooth forcing for incompressible porous media, Boussinesq, and three-dimensional incompressible Euler.
- AI contribution
- Claude and Codex assisted with proof development, exposition, and auditing.
- Human contribution
- The mathematicians chose and advanced Córdoba and Martínez-Zoroa’s program. They describe this as a personal collaboration, independent of their employers.
- Evidence today
- The authors report Lean verification on 22 August, followed by work to understand and present the proof. Public discussion on X followed the September release.
What this does—and does not—establish
These results include a smooth external force. They are distinct from unforced Euler and from the full-viscosity Navier–Stokes problem. Proofweave has reviewed the linked sources, not rerun the formal proofs.
OpenAI · unreleased research model
OpenAI announces Navier–Stokes and Euler blowup proofs
OpenAI released papers and a Lean repository claiming forced Navier–Stokes breakdown in ℝ³ and the periodic setting, together with a separate unforced Euler blowup result.
Lean replay pathInspect the Navier–Stokes and Euler artifacts
Start with the theorem statements, assumptions, and pinned build instructions in the public repository.
- AI contribution
- A group of agents using an internal model developed and formalized the arguments.
- Human contribution
- OpenAI researchers organized the effort and released the materials for scrutiny. The preceding Alpöge–Buckmaster work has its own attribution and scope above.
- Evidence today
- The Navier–Stokes claim targets the smooth-forcing alternatives C and D of the Millennium formulation. The public repository exposes the claimed theorems and reproduction materials.
What this does—and does not—establish
The announced Navier–Stokes result uses smooth forcing; it does not settle unforced Navier–Stokes. The unforced Euler claim is separate. Publication and external Lean artifacts do not by themselves establish prize recognition, scholarly acceptance, or a Proofweave replay receipt.
Status check · 10 September 2026
Hodge conjecture: no verified resolution recorded.
Reports of a possible result under review are circulating. This source review found no public proof or official result announcement that establishes a resolution. The Clay Mathematics Institute still lists it as unsolved. It is therefore not counted as a solved milestone on this timeline.
A record, not a leaderboard
The unit of history should be a verifiable contribution.
Problem selection→AI exploration→Human judgment→Exact evidence→Independent record
Proof search, problem selection, literature review, formal verification, and publication are different contributions. This archive preserves those boundaries instead of awarding a whole result to the most dramatic headline.
The next entry
Make the next mathematical advance reproducible from the start.
Proofweave turns local Agent work into signed evidence, independent replay, and contribution records attributed to the person who delegated it.