AI for Mathematics

Definition

The use of language models to produce new mathematical results — not to tutor, not to retrieve known theorems, but to settle questions the literature had left open — and the separate machinery used to check that what they produced is correct.

The field splits along that seam. Generation is a frontier-model capability, run at inference time with long chain-of-thought. Verification is a distinct artefact: a proof assistant such as Lean 4 re-checks the argument line by line, mechanically, without trusting the model that wrote it. The two halves have different vendors, different licences and different failure modes, and conflating them is the main way claims in this area get overstated.

Why It Matters

• It is the cleanest available test of whether a model produced something new. Most benchmark disputes reduce to harness configuration — see Eval Harness Configuration. A previously open conjecture has no harness to configure. Either the proof stands or it does not.
• Verification is mechanical in a way almost nothing else in evaluation is. A Lean certificate can be checked by anyone with a laptop and no access to the model, the weights, or the lab. That is a stronger form of reproducibility than any leaderboard offers.
• It is a pacing signal. Labs have begun quoting research output as evidence of capability instead of benchmark scores — see Frontier Pacing. When the claim moves from "scores 51 on an index" to "settled a question open since 1978", the thing being measured has changed, and so has who is qualified to check it.

State of the Art (2026-10-04)

The largest result on this page was published 30 days ago and read today. Formalizing Fermat's Last Theorem (Anthropic Research, 2026-09-04, primary author Tianyi Peng, with Kevin Buzzard consulted) reports what it calls

the first end-to-end, computer-checked proof of FLT.

(source)

Quantity	Value
Theorems proved	30,300
Theorems in the final proof	29,500
Lean output	13 million lines
Wall-clock duration	11 days
Output tokens	about six billion
Model	"a general-purpose internal research model roughly comparable to Claude Fable 5.1"
The formalization "follows a simplified version of Wiles's proof from Darmon, Diamond, and Taylor", and human mathematical input stayed "limited to occasional high-level instructions". The infrastructure is named and is not Anthropic's:
Prove2Me, "an open collaborative platform for formalizing mathematics designed by Tianyi Peng and his collaborators at Columbia University", which held **a
directed acyclic graph (DAG) of theorem statements** so that **multiple agents could
work in parallel**.

This is the cleanest instance of the generation/verification seam this page's ## Definition is built around, and it falls entirely on the verification side. FLT was proved by Wiles in 1994. Nothing here is a new mathematical result; what is new is that a 13-million-line Lean artefact exists and can be re-checked by anyone with a laptop and no access to the model. Measured against the nine-loop result below — where the novelty claim is contested and the check came from a rival lab's model — this one has no novelty claim to contest and a mechanical check. Those are opposite trade-offs, and the page carries both rather than ranking them.

The disclosed limitations are the part that generalises, and they match the failure mode Claude-shaped science names in the section below:

likely much longer than it needs to be

relative to concise peer-reviewed alternatives — and early agent attempts

quickly lost track of the project's state and stopped collaborating effectively

contributing roughly 7% of non-boilerplate lines through failed iterations. A 13-million-line proof of a theorem whose human proof runs to a few hundred pages is the "grinds through calculations rather than building tools" tendency at scale, in a medium where it is countable.

Not stated, and so not on this page: any dollar cost, any named public model (the research model is characterised only by comparison to Fable 5.1), any compute or GPU figure, and any statement that the Lean proof has been reviewed or accepted by the Lean mathlib maintainers or by a journal. "Computer-checked" is a claim about the proof assistant, not about peer review.

Capture note: +30 days. The post sits under anthropic.com/research, a path this pipeline did not poll until 2026-10-02 and which that run recorded as a title only. www.anthropic.com has answered first-party on every run since, so this was never a reachability gap — it was a path nobody asked for.

State of the Art (2026-10-02)

Two Anthropic research posts, and between them they give this page its first independently cross-checked result and its sharpest statement of what the model is not.

Nine loops, verified by a rival model rather than by a reviewer

Fable 5.1, via the Claude Science platform, computed a nine-loop amplitude in N=4 super-Yang-Mills theory, past the eight-loop record Lance Dixon set in 2023 — and computed it two different ways, by the original bootstrap and by the indirect form-factor approach (source).

The verification is what makes this different from every other result on this page. Song He's group reached the same answer concurrently using GPT-6 assistance. Two groups, two frontier models from two labs, one answer — which is a stronger check than any of the self-reported benchmark figures above, and it is a check on the arithmetic, not on the novelty.

Cost is unusually disclosed: "one or two thousand dollars" in total, with the bootstrap alone taking "$100 of the budget, corresponding to running 96 CPUs for a week". A frontier physics record for four figures is the datum, not the loop count.

The guest author deflates his own headline, and the page records that rather than the headline:

Instead, it did something it turned out humans were also able to do. Claude used known methods, with a bit more compute than people had tried to use before.

Dixon's addendum supplies the counterweight — "It's quite a triumph, in my opinion, for a large language model to execute all of the steps in the complicated recipe" — so the disagreement on this page is about whether executing a known recipe further than anyone bothered to is a mathematical result or a compute result. Not resolved here.

"Claude-shaped science" — select the problem, not the collaborator

Matthew Schwartz, guest post, 2026-10-01 (source). The thesis is about problem selection: models do well where broad knowledge, coding and mathematics are what the task needs, and the productive move is finding those problems rather than casting the model as a researcher.

Claude and GPT are good at science, but they are not scientists: yes, they are smart, but it can take a lot of hand-holding to get them to produce anything of scientific value.

Reported scale, with Claude Opus 4.5 and Claude Fable 5 named: 30 elliptic Feynman integrals — 15 reproductions and 15 novel, 36 manuscripts across 18 fields with 19 coauthors over three months, 4,452 economics papers' replication packages converted to open source, 30,000 routines ported off commercial tools, 6,072 languages in the AccStack word-stress database against 160,000 phonology works in its bibliography, and 5.7 billion pairs of mutations from the 1000 Genomes Project.

The stated limitations are the usable part, and two of them are mechanical rather than philosophical: "Claude has no sense of time", and a tendency to grind through calculations rather than build efficient tools — which is a specific, checkable failure mode, and the reason the nine-loop result reads as compute applied to a known recipe. The others: context loss over long projects, and that the model's judgement about what is scientifically interesting cannot be trusted without an expert. Set against "15 novel" Feynman integrals in the same post, the two together say the model can produce new results inside a problem a human chose, and cannot choose the problem.

State of the Art (2026-09-28)

Every result on this page is scored against a problem somebody set. This one optimises the choice of problem, and reports the number that says it worked. Learning to Discover Interesting Mathematics — FAIR @ Meta with CERMICS/ENPC and NYU (Niket Patel, Ahmad Rammal, Amaury Hayat, Rémi Munos, Julia Kempe; submitted 2026-09-23, author list from 2 search passes, arxiv.org blocked from this run) — defines the intrinsic interestingness of a theorem as proof length ÷ statement length, shows it correlates strongly with an extrinsic measure of downstream utility, and reduces both to one computable primitive: the difficulty of a proof conditioned on a premise set. A 27B model trained to predict that difficulty is reported more accurate than frontier general-purpose models (source).

The measurement worth keeping is the overlap figure. Optimising for the metric drops substantial-or-full overlap with Mathlib from 91.9% to 30.6% — meaning a generator without the metric spends most of its output re-deriving the library it was handed. Both sides of that figure are machine-verified, so it measures novelty relative to an existing library, not correctness and not value.

Why it sits at the top of this page rather than in its capability list. The 25 Fields Medallists' declaration below objects that "mathematical problem-solving used as an AI benchmark is misaligned with mathematics' own needs" — that the field is being measured on solving, when what it wants is judgement about what is worth proving. This is the first entry on this page that attacks that layer with a number instead of a preference, and the first whose loop runs without human-supplied targets: generate candidates → select the most interesting → extend a machine-verified library → build on the extension.

Three limits, and the first is the whole argument. The correlation between the intrinsic ratio and downstream utility is the only thing separating "interesting" from "expensive to verify" — a proof-to-statement ratio rewards a deep theorem and an obfuscated one identically — and the snapshot states the correlation is strong without giving a coefficient. Second, no extrinsic score is reported for the generated theorems, only for the correlation study, so whether the 69.4% that is not in Mathlib is mathematics anyone wants is unanswered on the paper's own terms. Third, the whole method presupposes a decidable truth and an enumerable premise set: it is demonstrated inside a formal library and nothing here transfers to a conjecture that cannot be machine-checked. The paper does not claim otherwise.

State of the Art (2026-09-11)

Twenty-five Fields Medallists signed a declaration saying the thing AI labs measure is not the thing mathematics needs — and the objection is to the benchmark, not to the machine. A Severe Misalignment of AI in Mathematics was published on 2026-09-11 on Terence Tao's blog and at mathandai.org, with 25 signatories, all Fields Medallists, spanning medal classes reported as 1978 through 2026 (4 passes on the count; one pass enumerated the names). The page is reported to remain open for further signatures, as the Leiden declaration was (source).

The word in the title is doing precise work. The claim is that using mathematical problem-solving as a benchmark is "severely misaligned with the needs of mathematics itself" (2 passes) — a statement about incentives, not about capability, and not a claim that the results are wrong. The named mechanism is that results are, verbatim, "announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others" (1 pass), and that without the writeup step it becomes unclear whose ideas a proof rests on (source).

This page's own Open Problems predicted both halves of that. "Attribution is unresolved" and "peer review has no defined role yet" have been standing here since before this declaration existed, recorded as the second Leiden risk. What is new is not the diagnosis but who is making it and how many of them — the Leiden declaration was an institutional statement endorsed by the IMU; this is twenty-five named individuals, and the field's most decorated ones.

Two incidents this wiki already holds are reported as the context. Write-ups name the Jacobian Conjecture counterexample — Anthropic researcher Levent Alpöge using Claude Fable 5, announced on social media with no peer review — and the 2026-09-08 Navier–Stokes announcement and the plagiarism question Tristan Buckmaster raised about it, recorded in this page's ## Conflicting Reports. Whether the declaration itself names them is carried by two passes and no pass quotes the sentence that does, so the link is the write-ups' and is recorded as theirs (source).

What the declaration asks for is not recorded here, because it was not read. terrytao.wordpress.com and mathandai.org both answer EGRESS_BLOCKED from this run, so the full text was never opened; every source read describes the concern and none states a demand, request or recommended practice. That gap is left as a gap (source).

And the same week supplied the counter-example to the "rush" charge. An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics, surfaced 2026-09-13, reaches the IMO 2026 gold threshold at 30/42 and releases the checkpoints, the training data, the training and inference code, the submitted solutions and a new benchmark. An IMO score is the purest instance of the genre the declaration objects to — a competition number containing no new mathematics — and it is also the most completely documented mathematical AI result on this page. The two are recorded side by side rather than reconciled (source).

State of the Art (2026-09-08)

Two groups reach finite-time blowup for forced fluid equations within a day of each other, and the interesting part is that neither the mathematics nor the verification is what is in dispute. On 2026-09-08 OpenAI published On the Navier–Stokes Millennium Prize Problem, reporting a proof of finite-time blowup for the forced 3D Navier–Stokes equations produced by up to 10,000 agents running in parallel for about 88 hours, reaching resolution on 2026-09-05, followed by a further ~17 hours of Lean formalization and verification attributed to GPT-6 Astra (3, 3 and 2 passes respectively). The generating model is described only as internal and unreleased; compute cost is reported as "into the millions of dollars" (1 pass). OpenAI states the effort began on 2026-09-01 after it heard a rumour that another group was close (source).

That other group published the same day: Levent Alpöge (Anthropic) and Tristan Buckmaster (NYU) posted three papers proving finite-time blowup with a smooth forcing term for the incompressible porous medium equation, the 2D Boussinesq equations and the 3D incompressible Euler equations, building on Córdoba and Martínez-Zoroa, all three verified in Lean with the formalizations public. Their own account of the AI's role is unusually specific: Claude to identify key elements of the prior proof and reproduce its argument, Claude and Codex to write the main body of the text (source).

The verification half of this page's Definition held perfectly and the generation half did not. OpenAI's Lean development is public at github.com/openai/NavierStokesAndEuler (Lean 4.34.0-rc2, Apache-2.0), and it was read directly by this run — the only first-party artefact available, since openai.com and cdn.openai.com are both blocked here. It formalizes blowup for Navier–Stokes with positive viscosity and forcing on both ℝ³ and the periodic torus ℝ³/ℤ³, plus an Euler result from smooth, compactly supported, divergence-free initial velocity on ℝ³. The README states these correspond to alternatives (C) and (D) of the Clay Institute's official problem description. It makes no statement about whether the development is sorry-free, and this run did not build it (source).

What is disputed is provenance, and it is the first time on this page that the question has been asked about a specific person's private data. Buckmaster alleges that Sébastien Bubeck told him on a Sunday call that an internal OpenAI model had already produced a roughly 100-page proof, and proposed publication arrangements that would drop Alpöge from authorship or give OpenAI top billing (2 passes); he has said publicly that he wonders whether the private drafts he had been feeding into OpenAI's tools were accessible to OpenAI (3 passes). OpenAI's reply denies access outright and declines to deny influence, in two sentences that do different work. Both are quoted in full on Safety Monitoring and Data Retention, which is the page that owns retention and access, and they are not restated here; this page keeps the attribution half (source).

"Attribution is unresolved" was written on this page as a convention gap and came back as a conflict. The Open Problem below says that when a model generates the idea, a mathematician writes the paper and a prover checks it, no convention says how the result is credited. Every prior entry on this page had a single lab and no rival claimant, so the gap was theoretical. Here two groups, one of them containing an employee of a competing lab, arrive at neighbouring results in the same week, and the absence of a convention is what the argument is made of.

Terence Tao calls the Alpöge–Buckmaster work a "remarkable achievement" and says he sees no fundamental obstacle to extending their approach to the full Navier–Stokes problem (1 pass each), while lamenting that AI companies use longstanding mathematical problems as marketing proof points (2 passes). His blog and his Mastodon account are both blocked from this sandbox, so both readings come through extracts and neither is a quotation this wiki verified (source).

No independent mathematical review of the OpenAI proof appears in anything read. Every extract discussing reception describes mathematicians as not yet having seen it — the manuscript's host is blocked here, and the Lean certificate, which anyone can in principle check, is the only part of the claim currently open to inspection. That is the shape this page's Definition predicts, arriving intact.

State of the Art (2026-08-31)

Novel results on five open problems from a population of agents with no coordinator — Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment, arXiv 2026-08-24. Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment (Stephen Chung, Wenyu Du, William J. Wesley) reports agents drawn from different model families sharing one research goal in an environment called the Station, with no central agent and no scripted pipeline: they pick their own directions, run experiments, collaborate, and write a shared literature that later agents read. Reported outcome: results novel to the mathematical literature on five open problems, with primary discovery attributed to a Claude agent for 18 (64.3%), a GPT agent for 9 (32.1%) and a Gemini agent for 1 (3.6%) (source).

Every other result on this page pairs one lab's model with separate checking machinery, and this one does not. OpenAI's ten advances and the Claude zeta bound each ran a closed frontier model into a human editing step; the Lean 4 line runs generation into an external prover. Here the population is mixed-family and the verification artifacts are reported as produced inside the same environment. Whether that is the same guarantee is the question, and it could not be read — arxiv.org is blocked from this run's sandbox and this page's entry rests on two agreeing search passes, not on the paper.

The 64.3% share is the number most likely to be misquoted, including by this wiki. Nothing read states how many agents of each family were present, so a larger population, more turns or more budget would produce more primary discoveries at equal capability. It is not a capability comparison between Claude Opus 5 and anything else, no version strings were published, and it must not be carried onto a model page.

And it lands directly on the last Open Problem below. "No shared measure exists — only incommensurable lists of results" was written about labs. This paper puts three families in one environment on one goal, which is the first arrangement read here where their mathematical output is even in principle commensurable — and then reports no baseline, no ablation and no single-agent or coordinated comparison, so the arrangement is demonstrated rather than tested.

State of the Art (2026-08-11)

A record moves by 0.000162, and the AI is refining an optimizer — AlphaEvolve, 2026-08-19. Improving the matrix multiplication exponent with modern optimization and AlphaEvolve (arXiv:2608.16884) improves the best known upper bound on the matrix multiplication exponent ω from 2.371339 to 2.371177, in three steps: reformulating the optimization problem at the core of combination loss analysis (the laser-method refinement credited to Duan et al. 2022, Williams et al. 2024, Alman et al. 2025) so it can be solved in a larger setting, designing a new optimization algorithm for it, then refining that algorithm with AlphaEvolve (source).

The division of labour is the point, and it is the opposite of the usual framing. AlphaEvolve proves nothing here. It refines a search procedure that humans reformulated and designed, over a space humans defined, inside a method three prior papers built. That is a narrower and more legible claim than the entries below it — and unlike them it needs no certificate machinery: the previous record is a published number, the new one is a published number.

The magnitude belongs in the same breath as the result. The bound moves by 0.000162, records on ω have moved in increments of that size for years, and the paper calls itself a note. What it demonstrates is a tool contributing at the frontier of a well-worked problem, not a problem that moved.

Caveat carried from the paper page: no author or affiliation was read, so whether this is a Google DeepMind paper or third-party use of AlphaEvolve is not established here.

An unreleased Claude — Anthropic, 2026-08-10. The proven lower bound on the fraction of nontrivial zeros of the Riemann zeta function lying on the critical line moved from 41.6% to 67.2%, reported as the largest single improvement in the problem's history, displacing a figure that took decades of human work. The result is formally verified in Lean with the proof public, and reviewed by named external number theorists (Brian Conrey, Dan Goldston). Anthropic states it is not a path to a full proof and that the bound was an unintended byproduct of attempting the hypothesis itself (source).

This is the first entry on this page where the verification is public, machine-checkable and independent of the lab's own account. The generation/verification split described below is the reason that matters: it is the first result here a sceptical reader can check without being granted access to anything. The mechanism is the other half — 650 ideas, ~60 subagents, 2,400 shell commands, 31M output tokens in an agent harness — which makes this search at a scale a person cannot run rather than a model emitting a proof. Full detail on More than two thirds of the zeros of the Riemann zeta function lie on the critical line.

Astra — OpenAI, 2026-08-01. Ten results in mathematics and theoretical computer science, each shipped with a Lean 4 certificate and a chain-of-thought walkthrough, plus a 249-page manuscript. Named results include the first explicit non-sofic group, a disproof of Connes' Rigidity Conjecture, a quantum parallel repetition theorem for general two-player entangled games, Ehrhart's volume conjecture, and the first improvement to the general high-dimensional sphere-packing upper bound since 1978. Problems are stated to have been open at least ten years. Total token cost approximately $2,000 at Sol API prices (source).

The human step is not incidental. Coverage of the Astra release states that humans organized the proofs into papers, which were then converted into Lean certificates (source). The same division of labour appears in the earlier result: when an internal OpenAI reasoning model disproved the planar unit distance conjecture (Erdős, 1946) on 2026-05-20, professional mathematicians read the model's reasoning transcript, extracted the key ideas and rewrote them as a succinct formal proof, and Princeton's Will Sawin refined the bound to δ = 0.014 (source). In neither case is the end-to-end pipeline reported as autonomous.

What changed between May and August is scale and checkability: one result became ten, and human-refereed prose became machine-checkable certificates.

Verification tooling is open where generation is not. Leanstral 1.5 (Mistral AI, 2026-07-01/02) is an Apache 2.0 open-weight Lean 4 theorem prover with a free API endpoint. The asymmetry is worth naming: the models that generate these arguments are closed and, in Astra's case, not released at all, while the tooling that checks them is open.

Search-based discovery is a separate lineage. AlphaEvolve (Google DeepMind) reaches mathematical and algorithmic results by evolutionary search over programs scored against an evaluator, rather than by long-form reasoning; it reached GA on the Gemini Enterprise Agent Platform on 2026-07-10. Gemini 3.1 Deep Think is the reasoning-side counterpart in that lab's line.

The community has stated its terms. The Leiden declaration on artificial intelligence and mathematics — published June 2026, endorsed by the International Mathematical Union, signed by Terence Tao, Peter Scholze, Kevin Buzzard and Scott Aaronson among others — lists five risks: unreliable results, missing citations, dependence on closed commercial systems, exaggerated claims, and loss of scientific independence. OpenAI cites it in the Astra post (source).

Open Problems

• A Lean certificate proves the theorem, not the interest. Formalization confirms that the argument establishes the formal statement. It cannot confirm that the formal statement is the one mathematicians care about — the translation from informal conjecture to Lean proposition is itself a human judgement, and an error there produces a machine-checked proof of the wrong thing.
• Peer review has no defined role yet. A Lean-checked argument is not a peer-reviewed one, and no journal process read here has stated how it treats one.
• Attribution is unresolved. When a model generates the idea, a mathematician writes the paper and a prover checks it, no convention read here says how the result is credited or cited — the second risk named in the Leiden declaration.
• The claims are not independently reproducible. Astra is unreleased, and so is the Claude research version behind the zeta bound — both labs' headline mathematical results in 2026 come from models nobody outside the lab can run. The certificates can be checked by anyone; the generation cannot be repeated by anyone, which is the third Leiden risk stated concretely. Public verification narrows this gap without closing it.
• No shared measure exists. There is no benchmark, baseline or common problem set on which two labs' mathematical output can be compared — only incommensurable lists of results.
• The community has now said the measure is the problem, and nobody has proposed a replacement. The 2026-09-11 declaration objects to mathematical problem-solving used as a benchmark; the bullet above objects that there is no benchmark. Both are recorded because they are both true and they point opposite ways, and nothing read this run reconciles them (source).
• Natural-language-only pipelines have no certificate at all. An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics reaches a gold-medal score with the model verifying its own proofs inside the search loop. Everything this page says about Lean certificates — that they can be re-checked by anyone, without the model or the lab — does not apply to it, and no independent re-reading of its natural-language proofs is reported (source).

Key Papers

• 2026-09-23 — the first entry here that optimises the choice of problem. Learning to Discover Interesting Mathematics defines interestingness as proof length ÷ statement length, trains a 27B proof-difficulty predictor that beats frontier general-purpose models on that task, and cuts Mathlib overlap 91.9% → 30.6% by optimising for it; the correlation with downstream utility is asserted as strong and no coefficient is given (source)
• 2026-09-14 — the automated half of scrutiny has a proved blind spot. Beyond Solver Verdicts: Generative Reward Models for Autoformalization names Verdict-Preserving-Unfaithfulness: an incorrect formalization that executes successfully and matches the expected verdict, so the solver certifies the encoding and never the translation into it. The paper proves that verdict-only structural heuristics are bounded to chance-level detection on such traces, and answers with Generative Verification (GenV) — an offline Z3-equivalence oracle distilled into a reference-free continuous score, 0.961 AUROC, +11.3 points downstream on agentic test-time compute allocation. Z3, not Lean 4, so it does not yet reach this page's own setting (source)
• 2026-09-13 — the first result on this page a reader can actually run. An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics scores 30/42 at IMO 2026, the stated gold threshold, from Nemotron 3 Ultra plus two post-trained specialists driving a generate → verify → refine search with a separate high-compute selection stage — entirely in natural language, with no formal prover, no external tools and no internet access. Both checkpoints, the training data, the code, the submitted solutions and Nemotron-IMO-Bench (200 novel problems) are released. It has no verification half: every other entry below pairs generation with a Lean certificate, and this one verifies with the model itself inside the loop (source)
• A Severe Misalignment of AI in Mathematics (2026-09-11) — declaration signed by 25 Fields Medallists, arguing that mathematical problem-solving used as an AI benchmark is misaligned with mathematics' own needs; text not readable from this run (source)
• 2026-09-07 — a protocol for the convention this page's Open Problems say is missing. Scores Alone Do Not Prove Discovery: The Discovery Certification Protocol for Auditing AI Research Agents turns a research agent's claimed discovery into executable recovery tests: Gate 1 validates useful improvement on a sealed evaluation; Gate 2 gives matched agents the registered starting information and the observed web content while withholding the target research history, and any valid method that reaches the numerical target supplies a recovery witness and triggers a veto. Two controlled audits produced zero recoveries in 96 episodes (upper bound 0.0468), each paired study 30 truthful against 0 neutral recoveries, and a deterministic, LLM-free verifier reproduces every decision from frozen evidence. It cannot be applied retrospectively — sealing and registration happen before the work — so it does not touch the dispute recorded above; it is a proposal for the discoveries that come next (source)
• OpenAI, Finite time blowup for Navier–Stokes and Finite time blowup for the Euler equation (2026-09-08) — Lean 4 certificates public at github.com/openai/NavierStokesAndEuler under Apache-2.0; the manuscript itself was not readable from this run (source)
• Alpöge and Buckmaster, three papers on finite-time blowup with a smooth forcing term for the incompressible porous medium, 2D Boussinesq and 3D incompressible Euler equations (2026-09-08) — Lean-verified, formalizations public, and the authors state which parts of the writing were done by Claude and Codex (source)
• Improving the matrix multiplication exponent with modern optimization and AlphaEvolve (2026-08, arXiv:2608.16884) — ω bound 2.371339 → 2.371177; AlphaEvolve refines the optimizer rather than proving the result → Improving the matrix multiplication exponent with modern optimization and AlphaEvolve (arXiv:2608.16884) (source)
• Claude / Anthropic, More than two thirds of the zeros of the Riemann zeta function lie on the critical line (2026-08-10) — bound raised 41.6% → 67.2%, Lean formalisation public at anthropics/zeta-23-lean, reviewed by Brian Conrey and Dan Goldston → More than two thirds of the zeros of the Riemann zeta function lie on the critical line (source)
• OpenAI, Ten advances in mathematics and theoretical computer science (2026-08-01) — 249-page manuscript, Lean 4 certificates on GitHub (source)
• OpenAI, An OpenAI model has disproved a central conjecture in discrete geometry (2026-05-20) — the planar unit distance problem, via Golod–Shafarevich theory and infinite class field towers (source)
• Leiden declaration on artificial intelligence and mathematics (2026-06) — IMU-endorsed statement of five risks (source)

Related Concepts

• Reasoning Models — the generation half of the pipeline
• Test-Time Compute (Inference-Time Compute Scaling) — what the long chain-of-thought runs consume
• Eval Harness Configuration — why an open conjecture is a cleaner claim than a benchmark score
• Frontier Pacing — research output used as a capability claim
• Safety Monitoring and Data Retention — the data half of the 2026-09-08 dispute: what a lab retains from a user's sessions, and what it can rule out

Conflicting Reports

Whether the 2026-09-08 Navier–Stokes result meets the Millennium Prize criteria. Two accounts are in circulation and this wiki does not choose between them (source):

Account	Claim	Carried by
Reporting	the Clay criteria are defined on the unforced equations; the result addresses the forced version; the prize therefore remains unclaimed	2 search passes
OpenAI, and its Lean repository	the work establishes alternatives (C) and (D) of Fefferman's official problem statement, which permit a breakdown example carrying a smooth external force under stated decay and periodicity conditions	the repository README, read first-party, plus 1 pass quoting the manuscript
Two further facts sit awkwardly with both readings and are recorded rather than
reconciled: OpenAI says it does not intend to pursue the $1,000,000 award (2
passes), and **Charles Fefferman — who wrote the Clay Institute's official
description of the problem — is quoted saying "I was thrilled that the problem was solved"** (1 pass). Nothing read addresses whether these are consistent.

This is not a resolvable dispute from here. It turns on the text of Fefferman's problem statement against the theorem actually formalized, and the manuscript's host is blocked from this sandbox. The Lean development is public and is the artefact that would settle it.