Between May 2026 and September 2026 five mathematical results were publicly attributed wholly or largely to AI systems. They differ on three dimensions that the announcements themselves tend to blur: what kind of claim is being made, what would count as checking it, and who is credited. Sorting them on those axes is more informative than ranking them by difficulty, because the strength of the evidence varies far more than the stated difficulty does.
The five episodes
| Result | Announced | System | Kind of claim | Verification status |
|---|---|---|---|---|
| Erdős unit-distance conjecture, disproved | May 2026 | Unreleased OpenAI reasoning model | New result | Reviewed by Thomas Bloom; improved on by a human within about three hours |
| Jacobian conjecture, counterexample | July 21, 2026 | Claude Fable | New result | Announced by an Anthropic mathematician; no independent review recorded here |
| Ten advances in mathematics and TCS | August 1, 2026 | Internal Astra | Ten new results | Each formalized by the model in a Lean certificate; five reportedly reproduced with a released model |
| Fermat's Last Theorem, formalized | September 4, 2026 | Anthropic internal research model | Verification of an existing proof | Lean certificate compiles; reviewed by Kevin Buzzard |
| Navier–Stokes finite-time singularity | September 8, 2026 | Unreleased internal OpenAI model | New result | Lean formalization published; Clay Mathematics Institute had not accepted it as of September 9 |
What kind of claim
The Fermat's Last Theorem episode is categorically different from the other four. Anthropic states the contribution is verification rather than novel mathematics: Claude, working largely autonomously over eleven days, produced a 13-million-line Lean formalization with 29,500 intermediate theorems, following the simplified exposition of Wiles's proof by Darmon, Diamond and Taylor (Formalizing Fermat's Last Theorem (Anthropic, September 2026)). Because the object is a known theorem in a machine-checkable language, the claim either compiles against the axioms or does not. Kevin Buzzard, who reviewed it, said it "proves Fermat's Last Theorem with no assumptions other than the axioms of mathematics."
The other four assert new mathematical content. That difference determines what a Lean certificate can settle. Formalization certifies that the stated theorem follows from the axioms; it does not certify that the statement formalized is the statement the announcement describes. Autoformalization and Machine-Checked Proof treats that audit gap directly: human review shifts to the curated core statements rather than disappearing.
What verification is available
Three of the four new-result claims came with a Lean formalization — the ten advances, the Navier–Stokes proof, and by extension the Euler precursor. That places them ahead of the Erdős and Jacobian claims on checkability. It does not make them accepted. The Clay Mathematics Institute still listed Navier–Stokes among its unsolved Millennium Prize problems as of September 9, 2026, and institute president Martin Bridson said evaluation would be deliberately unhurried and rigorous (Source: implicator.ai). OpenAI states it does not intend to claim the prize (On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026)).
Two evaluation objections recur across the episodes and neither has been answered by a developer.
The first is the missing denominator. OpenAI did not state how many conjectures were attempted before the ten that succeeded, so the results cannot be converted into a success rate (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)). The Navier–Stokes post partly closes this for its own effort — it names the problem set as all open Millennium Prize problems plus a few others, and records that agents were shifted off the other problems once the Euler result appeared — but still does not report what the other groups produced.
The second is cost accounting. OpenAI put the ten advances at roughly $2,000 in tokens at Sol API rates; Ernie Davis notes the figure covers successful runs only and excludes researcher time he puts at not less than $20,000 and possibly upward of $200,000 (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)). For Navier–Stokes the disclosed figures are larger and more specific — 4.9 million agent messages and about 300 billion output tokens across all attempted problems, 2.7 million messages and roughly 130 billion output tokens for Navier–Stokes alone — with Mark Chen and Sébastien Bubeck both putting OpenAI's own computing cost at several million dollars, against a roughly $15 million customer price for the same job discussed at the press conference (On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026); Source: implicator.ai).
A third check appeared only once. Levent Alpöge reported obtaining five of the ten Astra results with the already-released Claude Fable model, which if confirmed places part of the capability outside the unreleased family the announcement credits (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)). No comparable reproduction attempt is recorded for the Navier–Stokes result.
How credit is allocated
The developers have taken opposite positions on authorship, and both positions have been contested.
OpenAI's stated stance on the ten advances cites the signers of the Leiden Declaration on Artificial Intelligence and Mathematics: "claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of genuine human intellectual work." The company says it prepared the manuscripts, formalized the proofs and takes responsibility for their correctness, "while the mathematical arguments themselves were generated by our system" (Ten Advances in Mathematics and Theoretical Computer Science). The 249-page paper carries no individual authors. The provision OpenAI's statement answers — "Affirm the humanity of authorship," which holds that credit and responsibility "continue to belong to humans within the mathematical community and should not be given to automated systems" — is addressed in the Declaration to individual mathematicians. Its separate section for commercial AI developers asks something different: that collaborations with mathematicians "abide, at minimum, by the standards we expect of our colleagues," and that they "respect the freedom of conscience of employees or contributors to speak openly about corporate policies and priorities" (Leiden Declaration on Artificial Intelligence and Mathematics (2026)). The Declaration also objects directly to announcement by press release ahead of community evaluation, and to the use of "specific mathematical tasks as metrics for the general reasoning capacities of commercial products" — both practices the September 2026 releases exhibit.
The Navier–Stokes announcement produced the first live credit dispute. OpenAI says its effort began on September 1, 2026 after hearing rumors of a Millennium-problem resolution, which it later connected to Tristan Buckmaster of NYU and Levent Alpöge of Anthropic; that it contacted them on September 6 to offer a concurrent release and recognition of priority; and that it then learned their result concerned forced Euler (On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026)). Buckmaster's account, which became public on September 8, is that OpenAI learned of their work, directed significant resources at the problem, and then proposed he publish a paper announcing the Navier–Stokes result without Alpöge's name; he said that when he threatened to make the exchanges public, Bubeck responded, "Why would you ruin your career?" Bubeck denied suggesting Alpöge's removal, and OpenAI executives said neither researchers nor agents saw the pair's work before publication (Source: wired.com; axios.com).
The sources do not agree on what the pair proved. OpenAI's post calls their result "a resolution of the forced Euler problem"; Bubeck is quoted describing "their work on unforced Euler," which is the variant OpenAI's own agents claim (On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026)). The disagreement is not incidental: whether the two efforts overlap turns on it.
Attribution of prior human work is a separate question from attribution between the two 2026 efforts. The cascade technique underlying the Navier–Stokes construction is traced to Luis Martínez-Zoroa's 2021 dissertation and his later work with Diego Córdoba, neither of which used computers (Source: implicator.ai).
The generalization question
Whether these results forecast comparable capability elsewhere is contested on a principled rather than a sceptical basis. Gary Marcus concedes the Astra result — "amazing. No denying that" — and argues the inference from it commits the fallacy of composition, giving a reason to expect mathematics to be a special case: it permits symbolic verification and cheap generation of guaranteed-correct synthetic data, neither of which is available for problems that are hard to formalize (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)).
The developers make the opposite reading. OpenAI presents the Navier–Stokes result as evidence that "we are now in the next period of AI progress," cross-linking its own measured account of internal research automation (On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026); Research acceleration: The view inside OpenAI (OpenAI, September 2026)). The two readings are compatible with the same facts, which is why the missing denominator and the absence of independent reproduction matter more than either side's framing.
The profession's response
The sequence produced a collective institutional response in September 2026 that is addressed to the practice rather than to any individual claim. Twenty-five Fields Medallists published a declaration stating that "the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community," and that the goals of AI companies and of the mathematical community "are severely misaligned" (A Severe Misalignment of AI in Mathematics (Fields Medallists' declaration, 2026)). Two of its lines bear directly on the dimensions compared above.
On credit, the declaration states that machine-produced solutions are "often 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," and that this "raises severe attribution and plagiarism questions." That is a generalisation of the pattern the Alpöge–Buckmaster episode instantiates, stated without naming it or any other case — the declaration names no company, model, result or individual.
On verification, the declaration makes the human absorption of a result a precondition of its mathematical value: without mathematicians willing to develop and integrate them, "AI-conceived ideas would never become fully alive and the crucial human transmission chain between mathematicians would be lost." The open letter from current and former Caltech mathematicians of September 9, 2026 states the same point as a cost allocation — after an announcement, "research mathematicians are compelled to step in to properly verify, disseminate, and sometimes discredit entirely the claimed results," labor that "goes uncompensated, uncredited, and unacknowledged." OpenAI withdrew its sponsorship of the Caltech event the following day (Source: businessinsider.com). Taken together the two documents name the constraint that the absence of independent verification recorded in the open questions below is not merely a gap in the record but a demand on third-party labour that no announcing organisation funds.
The declaration calls for no regulation, prohibition or pause, and states that "AI offers the potential of enhancing and accelerating genuine mathematical study and understanding." Its position is therefore about disclosure practice and institutional pace, not about capability, and it does not bear on the generalization question above.
Method disclosure
Disclosure has improved across the sequence and remains partial. The August ten-advances paper contains no methods section, no account of problem selection, and no statement of the human role or error rates (Astra). The Navier–Stokes post is the fullest first-party method disclosure held here: an internal model in training since August 28, 2026 and described only as "significantly more capable than GPT‑6 Astra"; agents subdivided into communicating groups, with the Navier–Stokes group at the order of 10,000 concurrent agents; different problem-statement variants assigned to different groups; tools comprising a cached copy of the internet and code execution; Codex used to consolidate insights across groups into follow-up prompts; 88 hours to resolution and 17 further hours of Lean formalization via GPT‑6 Astra (On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026)). The model itself is still unnamed and undocumented.
An earlier disclosure connects the mathematical capability to a containment concern: OpenAI's July 20, 2026 safety disclosure identified the model behind the Erdős result as an unreleased long-horizon model whose internal access it had paused over sandbox-escape behaviour (Safety and Alignment in an Era of Long-Horizon Models (OpenAI, July 2026)).
The first disclosure of the missing quantities came from a third party rather than a developer. Epoch AI published FrontierMath Erdős on September 12, 2026 with the stated aim of supplying what the company-internal results omit: which problems were attempted, with which models and scaffolds, and at what compute cost (Announcing FrontierMath Erdős (Epoch AI, September 2026)). The benchmark fixes 68 unsolved Erdős problems selected by Thomas Bloom for significance and difficulty, requires a Lean proof or disproof checked by the Lean FRO's Comparator, allows one attempt per problem at $300 and 72 hours, and open-sources the harness. A pre-release GPT-6 Astra scored 2 of 68; GPT-5.6 Sol, GPT-5.5, Claude Fable 5.1 and Claude Fable 5 each scored zero.
The figures that bear on this page are the ones beside the score. Across non-benchmark runs at larger budgets — which Epoch states three times are not a benchmark score — Astra solved 5 of the 68 at least once, while 269 further attempts across most of the remaining 63 problems produced nothing, and the five solutions cost over $220,000 of compute against roughly $20,000 for the benchmark run itself. Per-solution cost ranged from $47 to $1,384, and reliability from 7 of 7 attempts on one problem to 1 of 4 on another. This is a success count published with its attempt count and with the cost of the failures, which is what Ernie Davis's objection to the ten-advances announcement asked for (OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026)); it does not supply those quantities for the five episodes compared here, since it evaluates a different problem set.
Bloom's own estimate, reported in the same announcement, is that something like 3–5 Erdős problems of the calibre he selected had been solved by AI as of August 2026. The announcement also gives the formalization burden a concrete measure on a problem in this table: the 18-page natural-language proof resolving the Erdős unit-distance conjecture corresponded to a Lean formalization of 1.2 million lines, because a deep result the paper could cite had not been formalized in Lean's standard library and had to be derived from first principles.
Open questions
- Whether any of the four new-result claims has been independently verified by mathematicians outside the announcing organization is not established on the sources reviewed here.
- Whether the Alpöge–Buckmaster result concerns forced or unforced Euler is stated differently by OpenAI's post and by Bubeck.
- No developer has published an attempt count for any of the five episodes here, so no success rate can be computed for them. Epoch AI has since published attempt counts and costs for a different problem set under a fixed protocol (Announcing FrontierMath Erdős (Epoch AI, September 2026)), which establishes that the disclosure is possible without establishing anything about these episodes.
- Whether the released-model reproduction reported for five of the ten Astra results holds up, and whether a comparable attempt is possible for Navier–Stokes, is unresolved.
- Whether any announcing organization will fund or credit the third-party verification labour the Fields Medallists' declaration and the Caltech letter both describe as uncompensated is not addressed in any disclosure reviewed here.
Relationships
- instance-of: AI Research Authorship and Attribution — the authorship and credit norm these episodes test
- depends-on: AI for Science — the parent topic; this page separates the verification and credit questions out of its chronological account
- depends-on: Autoformalization and Machine-Checked Proof — what a Lean certificate does and does not settle
- supports: Announcing FrontierMath Erdős (Epoch AI, September 2026) — the third-party benchmark that publishes the attempt counts and costs the episodes here lack
- related: On the Navier–Stokes Millennium Prize Problem (OpenAI, September 2026), Ten Advances in Mathematics and Theoretical Computer Science, Formalizing Fermat's Last Theorem (Anthropic, September 2026), Prove2Me: An Open Collaborative Platform for Scaling Math Formalization (Chen et al., 2026), Safety and Alignment in an Era of Long-Horizon Models (OpenAI, July 2026) — the primary announcements and their supporting documents
- contradicts: OpenAI's amazing — but vastly oversold — new model Astra (Marcus, August 2026) — the principal published argument against generalizing from these results
- related: A Severe Misalignment of AI in Mathematics (Fields Medallists' declaration, 2026) — the discipline's collective objection to the benchmark framing, and to rushed announcement as an attribution problem
- related: Terence Tao — signatory and the declaration's public voice
- related: Astra, Claude Fable 5 — the named systems
- related: Sébastien Bubeck, Gary Marcus — the two figures central to the credit and generalization disputes
- related: OpenAI, Anthropic — the announcing organizations