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.
9 moments · one accelerating frontier
Follow the record across 2026.
Spacing follows publication or announcement time. Clustered nodes show activity accelerating—not stronger evidence. 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.
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.