{"schema_version":"2026-09-05.topic-graph-v1","canonical_url":"https://www.manufacturing.ai/topics/ai-in-mathematics","topic":{"slug":"ai-in-mathematics","name":"AI in Mathematics: A Dispute in Real Time","description":"A sourced timeline of the September 2026 Navier–Stokes announcement and the attribution dispute around it — built entirely from what the parties published themselves, with the prize's own rules for what counts as solved, and nothing adjudicated.","coverage_notes":"Twenty-eighth packet through the generic ingestion pipeline, researched 12 September 2026, four days after the events at its centre. It is deliberately built from party-authored documents: OpenAI's own post, Buckmaster's own signed statement, the declaration on the page its signatories published it on, and the Clay Institute's own rules. No assertion rests on a news report of what someone said, though news coverage was used to locate those documents. The central dispute is recorded and not adjudicated: where OpenAI and Buckmaster describe the same calls differently, both accounts appear with the speaker named, and this collection states plainly that it has read two documents rather than the proofs, prompts, logs or calls. Three known gaps. Fefferman's official problem statement, which defines the 'C' and 'D' labels both parties use, was not retrieved, so those labels are reported only as the parties describe them. Neither the OpenAI proof nor the Alpöge–Buckmaster papers were read, and no assertion here characterises whether any proof is correct. And no compute cost appears: third-party estimates circulated widely but OpenAI publishes agent, message and token counts without a cost, and no retrievable source establishes one. The Andreas Thom guest post is cited at the blog front page on which it was read rather than at its permanent URL, which will drift. This subject also sits outside the index's three lanes and was built at the founder's request as a test of the format on a live, contested story. Not yet editor-reviewed; every assertion reads as reported, and every quotation is attributed to the document that contains it.","primer":"In September 2026 an AI company said it had resolved one of the seven Millennium Prize Problems, and within three days twenty-five Fields Medallists had signed a declaration saying the way such results are being produced and announced is damaging mathematics. This collection is a timeline of that week, assembled from the documents the participants published under their own names.\n\nWhat OpenAI states it did: trained a new internal model from 28 August; heard rumours on 1 September that two Millennium Prize problems had been resolved; launched coordinating groups of agents against all the open problems; resolved the Euler regularity question with nearly 100 agents over about 50 hours; then reached a Navier–Stokes result on 5 September with a group on the order of 10,000 concurrent agents, about 88 hours after launch, followed by 17 hours of Lean formalisation. Across everything attempted, 4.9 million messages and about 300 billion output tokens. The company states it does not intend to claim the prize.\n\nWhat Tristan Buckmaster states, in a signed statement published about twelve hours earlier: that he and Levent Alpöge had spent most of a year on a program that was not theirs and was not proposed by a model — the credit for the idea belongs to Diego Córdoba and Luis Martínez-Zoroa — obtaining blowup under smooth forcing on 15 August and verifying it in Lean on 22 August. He describes calls on 6 September in which, he says, he was first told the model had simply been given the problem statement, and it then emerged that a team had been working on it and that even the prompt he was shown had been written by prompting Codex. He states he was told 'Why would you ruin your career?' when he said he would go public. He is also explicit about what he is not claiming: he has not seen the proof, does not know what the model did, and is not accusing anyone of anything.\n\nThe two accounts are both on this page and neither is resolved here. This index has read two documents; it has not read the proofs, the prompts, the chat logs or the calls, and it has no way to determine which account of a private conversation is accurate. Recording the conflict precisely, with each side's words attributed to the document that contains them, is the most this format can honestly do — and it is more than a summary that picks a side would do.\n\nOn whether the problem is 'solved': the Clay Mathematics Institute's own rules, adopted in 2018, require publication in a qualifying outlet, at least two years elapsed since publication, and general acceptance in the global mathematics community, before the Institute will even consider a proposed solution. By that standard nothing announced in September 2026 can be settled before September 2028, whatever the mathematics turns out to say. That is a procedural fact, not an opinion about the proof.\n\nOne more thing belongs here, five weeks earlier and involving different people: a group theorist, Andreas Thom, describing finding his own joint work inside a crucial proposition of an OpenAI paper and the public announcement describing a decade of 'no progress'. Two episodes do not make a pattern, but they make the declaration's complaint about citation legible as something other than an abstraction.","primer_note":"Manufacturing.ai's own editorial synthesis, not evidence. Every factual statement in it is separately asserted and sourced in this response's concepts and events.","reviewed_through_date":null,"source_count":6,"inspected_source_count":6,"consulted_reference_count":1,"blocked_reference_count":0,"set_aside_reference_count":1},"figures":[{"kind":"time-series","title":"Eleven days","caption":"Each date is stated by one of the two parties in its own published account. The two accounts agree on this calendar; they disagree about what was said on 6 September.","sourceNote":"Dates from OpenAI's post of 8 September 2026 and from Tristan Buckmaster's signed statement of the same day. Positions on the axis are the day of the month in September 2026, with the two August dates omitted from the plot for scale and given in the text.","unit":"Day of September 2026","points":[{"label":"OpenAI hears rumours, launches agents","value":1,"display":"1 Sep"},{"label":"Agents reach a Navier–Stokes resolution","value":5,"display":"5 Sep"},{"label":"Lean verification; OpenAI makes contact","value":6,"display":"6 Sep"},{"label":"Buckmaster statement, then OpenAI's post","value":8,"display":"8 Sep"},{"label":"OpenAI publishes investigation findings","value":10,"display":"10 Sep"},{"label":"25 Fields Medallists sign a declaration","value":11,"display":"11 Sep"}]}],"blocked_references":[],"set_aside_references":[{"title":"Navier–Stokes Equation — official problem statement","publisher":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/navier-stokes-equation/","source_type":"other","retrieval_status":"search_result_only","content_inspected":false}],"concepts":[{"slug":"agent-swarm-as-a-research-method","name":"Running a Proof Attempt as a Compute Deployment","category":"process","canonical_url":"https://www.manufacturing.ai/topics/ai-in-mathematics#agent-swarm-as-a-research-method","assertions":[{"id":"63ec5b7d-3357-4808-8a26-ef9e561db7fb","predicate":"concept.description","statement":"OpenAI states it ran coordinating agent groups — on the order of 10,000 concurrent agents for the group that produced the Navier–Stokes result, nearly 100 over approximately 50 hours for the Euler disproof — reaching a resolution on 5 September about 88 hours after launch plus 17 hours of Lean formalisation, using 4.9 million messages and about 300 billion output tokens across all problems and 2.7 million messages and approximately 130 billion output tokens on Navier–Stokes.","assessment":"reported","editor_reviewed":false,"reference_only_source_count":0,"evidence":[{"id":"a9f49663-25b8-457f-803c-f13515bc0579","stance":"supports","is_primary_for_assertion":true,"origin_independence":"unknown","note":null,"quote":null,"source":{"title":"On the Navier–Stokes Millennium Prize Problem","publisher":"OpenAI","url":"https://openai.com/index/navier-stokes-solution/","source_type":"company_website","retrieval_status":"fetched","content_inspected":true,"published_at":"2026-09-08","documents":[]}},{"id":"35b68bd7-5eea-4344-a6c7-7fa6aae0be87","stance":"qualifies","is_primary_for_assertion":false,"origin_independence":"unknown","note":null,"quote":null,"source":{"title":"On the Navier–Stokes Millennium Prize Problem","publisher":"OpenAI","url":"https://openai.com/index/navier-stokes-solution/","source_type":"company_website","retrieval_status":"fetched","content_inspected":true,"published_at":"2026-09-08","documents":[]}}]}]},{"slug":"attribution-under-machine-speed","name":"The Attribution Dispute","category":"other","canonical_url":"https://www.manufacturing.ai/topics/ai-in-mathematics#attribution-under-machine-speed","assertions":[{"id":"0de6867a-016b-4559-98e7-676a8c2e642d","predicate":"concept.description","statement":"Buckmaster states the program was not started by him and Alpöge and was not proposed by a language model: credit for the basic idea belongs to Diego Córdoba and Luis Martínez-Zoroa, who spent several years constructing forced blowups, and he states that Martínez-Zoroa 'deserves a Fields Medal' and that the smooth-force route 'is not the direction one arrives at in a few days by giving a model the problem statement'.","assessment":"reported","editor_reviewed":false,"reference_only_source_count":0,"evidence":[{"id":"16e6074a-71a8-47f1-b959-66967052b0db","stance":"supports","is_primary_for_assertion":false,"origin_independence":"unknown","note":null,"quote":null,"source":{"title":"Today, Levent Alpöge and I have made public three results (signed statement)","publisher":"Tristan Buckmaster, Courant Institute of Mathematical Sciences, New York University","url":"https://cims.nyu.edu/~tristanb/statement.pdf","source_type":"academic","retrieval_status":"fetched","content_inspected":true,"published_at":"2026-09-08","documents":[{"url":"https://cims.nyu.edu/~tristanb/statement.pdf","label":"Open cited document","format":"PDF","relationship":"cited_document","access":"not_rechecked","link_checked_at":null,"locator":null,"edition":null,"snapshot_sha256":"8d7723941bcda2fa55c1e74faa6298e04c706d17ff8abd2ad01878039c621f9d","note":"Identified as a PDF in an earlier retrieval. 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