ResearchPapers, evidence & method
AI systems from OpenAI and Anthropic produce Navier–Stokes blowup, Lean proof of Fermat's Last Theorem and faster 3SUM algorithm as mathematicians weigh the fallout
In about five weeks, AI systems from OpenAI and Anthropic produced a Lean-checked blowup for forced Navier–Stokes, a machine-checked Fermat proof and a subquadratic 3SUM algorithm. Three mathematicians call this real progress that also unsettles careers, openness and human understanding.
From the Lex Fridman Podcast podcast episode of October 26, 2026 and cited sources: OpenAI agents found a finite-time singularity in the smoothly forced version of the Navier–Stokes problem. [2] [3] [4] [6]
SOURCES[2] Transcript: #503 – AI Crisis in Math: Navier-Stokes… lexfridman.com[3] Anthropic: Formalizing Fermat's Last Theorem anthropic.com[4] Quanta: AI Has Solved One of Math's Millennium Prize… quantamagazine.org
AT Pulse research deskPublished · Updated · 5 min read · AI-assisted analysisAI-assisted, not human-edited. Methodology Corrections Who owns this site
- What happened
- OpenAI agents found a finite-time singularity in the smoothly forced version of the Navier–Stokes problem. [2] [3] [4] [6]
- Original source
- Lex Fridman Podcast podcast episode, independent perspective
- Brief by
- AT Pulse research desk AI-assisted; publisher and contact on the desk page
- When
- October 26, 2026 podcast episode published
- Covered
- Oct 27 edition
- Evidence
- 11 sources across 9 sites; transcript sourced
- 1#503 – AI Crisis in Math: Navier-Stokes, Riemann…lexfridman.com
- 2Transcript: #503 – AI Crisis in Math: Navier-Stokes…lexfridman.com
- 3Anthropic: Formalizing Fermat's Last Theoremanthropic.com
- 4Quanta: AI Has Solved One of Math's Millennium Prize…quantamagazine.org
- 5Quanta: As AI Closed In on 'Unique Games' Proof…quantamagazine.org
- 6Alman & Vassilevska Williams, Truly Subquadratic 3SUM…alphaxiv.org
RisingHow it’s measured
YouTube video · 349.5K views · 3.4K likes
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OpenAI agents found a finite-time singularity in the smoothly forced version of the Navier–Stokes problem. Lean checked the proof, but humans still had to confirm that the formal statement matches the actual problem. Anthropic reports that its agents formalized Fermat's Last Theorem in Lean in 11 days, in about 13 million lines. A separate Anthropic model supplied the core of a 3SUM algorithm that runs slightly faster than quadratic time. [2] [3] [4] [6]
The results have clear limits. The Navier–Stokes blowup happens in a non-physical regime and leaves practical fluid simulation unchanged. The quasi-Riemann method cannot reach the critical line the full conjecture requires. Only about half of OpenAI's bulk release came with Lean proofs, and some papers were later retracted or revised. [2] [8]
The 3SUM speedup undermines a hypothesis that many conditional hardness results depend on, so those problems may now need to be studied one by one. Panelists and Terence Tao also worry that researchers will share early ideas less openly if AI can quickly finish published partial work. [2] [6] [11]
For practitioners, formal math is the best case for autonomous agents: outputs can be checked mechanically and mistakes are cheap. The runs were still expensive. The Navier–Stokes run cost an estimated several million dollars, and the Fermat run used about 6B tokens. Lasting human value moves toward specifying problems, checking that formal statements match intent, and refactoring proofs into library quality. [2] [3] [4]
Lean checks made several results machine-verifiable, though only about half of OpenAI's bulk release carried Lean proofs. Implication: human value shifts toward posing problems, confirming formal statements match intent, and making proofs readable.
Sources are checked; writing the dossier now.
Executive brief
Within roughly five weeks, AI systems from two labs were credited with a Lean-checked counterexample to the forced version of the Navier–Stokes Millennium Prize problem, a proof of the Unique Games Conjecture, a 13-million-line machine-checked proof of Fermat's Last Theorem, and a faster algorithm for 3SUM, which overturns a hypothesis that much of fine-grained complexity rests on. On Lex Fridman #503, mathematicians Steven Brunton, Ankur Moitra and Alex (a Rutgers number theorist) treat this as real progress, and also as a problem for the field. Fewer humans understand the proofs, the old career playbook no longer works, and the practical engineering side of fluid dynamics hasn't changed.
What changed and event timeline
Claude formalizes Fermat's Last Theorem
Anthropic says agents working largely on their own produced a full Lean proof in 11 days. The proof is 13M lines long and uses only Lean's three standard axioms.
OpenAI claims Navier–Stokes blowup
About 10,000 agents ran for 88 hours and found a finite-time singularity under smooth forcing. Lean checked it, but humans still had to confirm that the Lean statement matches the actual problem.
Humans race the machine
Minzer and coauthors posted a deliberately rough 95-page paper on a 2-to-1 games variant after hearing rumours of OpenAI's result.
3SUM falls below quadratic time
Alman and Vassilevska Williams posted O(n^1.9992) for 3SUM and O(n^2.9995) for APSP (all-pairs shortest paths). The core algorithm is credited to an internal Claude model.
More detail
OpenAI's bulk release
OpenAI released 722 manuscripts grouped into 372 result families. Sources disagree on the date: Wikipedia says Oct 6, Aaronson says Oct 7.
More detail
Panel calls it a "crisis moment."
Three mathematicians assess the results and what they mean for careers and openness in the field.
Capabilities and access
Read the full section
- OpenAI: an internal model that is not publicly available. It averaged about 3 hours of compute per result, on roughly 4,000 attempted problems (Wikipedia; Aaronson).
- Anthropic (Fermat): an internal research model "roughly comparable to Claude Fable 5.1," run on the Prove2Me multi-agent platform (Anthropic).
- Anthropic (3SUM): a separate internal model, not the public Claude (CellCog).
- Code: the Fermat code is on GitHub under Apache 2.0 (TNW).
Technical analysis for researchers and developers
- Fermat pipeline: a graph of theorem statements is split into small pieces that agents prove in parallel.
- 3SUM method: a modified Coppersmith multiplication that computes only the needed entries of thin matrix products.
- "Canonization" gap: the Rutgers mathematician says AI-written Lean code is patched together and works well only close to existing Mathlib material.
Read the full section
- Fermat pipeline: a graph of theorem statements is split into small pieces that agents prove in parallel. The run produced 30,300 theorems, about 6B output tokens, and roughly five times the size of Mathlib. A separate "comparator" check confirmed the final statement matches Mathlib's (Anthropic).
- 3SUM method: a modified Coppersmith multiplication that computes only the needed entries of thin matrix products. The authors call it algebraic, possibly impractical, with enormous hidden constants (alphaXiv). The Lean check covers the deterministic bounds but not the randomized ones (CellCog).
- "Canonization" gap: the Rutgers mathematician says AI-written Lean code is patched together and works well only close to existing Mathlib material. Fermat broke that expectation (transcript).
Claims and evidence
Read the full section
- Navier–Stokes: Lean-verified. Fefferman credits the human groundwork by Córdoba and Martínez-Zoroa (Quanta).
- Quasi-RH: a zero-free region for Re(s) > 7/8 (Wikipedia). The panel notes this method fundamentally can't reach 1/2, the line the Riemann Hypothesis itself requires (transcript).
- Unique Games: has a Lean proof, but OpenAI's result had no human editing or independent expert review (Quanta).
- Bulk release overall: about half the results came with Lean proofs, and OpenAI disclaimed the validity of the rest. Three papers were retracted and 14 revised (Wikipedia).
Context and prior work
- In May 2026 an OpenAI model disproved the Erdős unit-distance conjecture, a result Gowers verified (Decrypt).
- Kevin Buzzard's multi-year, human-led Fermat formalization project was still underway when Claude finished its version (TNW).
- The 3SUM hypothesis underpinned a whole web of conditional lower bounds. Moitra says the field may now have to study those problems separately (transcript).
Read the full section
- In May 2026 an OpenAI model disproved the Erdős unit-distance conjecture, a result Gowers verified (Decrypt).
- Kevin Buzzard's multi-year, human-led Fermat formalization project was still underway when Claude finished its version (TNW).
- The 3SUM hypothesis underpinned a whole web of conditional lower bounds. Moitra says the field may now have to study those problems separately (transcript).
Limitations, safety and contested findings
- Readability: the proofs are hard to read. Aaronson relays one description of them as written "on psychedelics" (Aaronson).
- Publication bias: OpenAI published only its successes, about a 10% hit rate, so the line between what AI can and can't solve is unclear (transcript).
- Open science: Tao warns that AI "flattens" problems as soon as rumours spread, which threatens open sharing of ideas.
Read the full section
- Readability: the proofs are hard to read. Aaronson relays one description of them as written "on psychedelics" (Aaronson).
- Publication bias: OpenAI published only its successes, about a 10% hit rate, so the line between what AI can and can't solve is unclear (transcript).
- Open science: Tao warns that AI "flattens" problems as soon as rumours spread, which threatens open sharing of ideas. He proposes labelling some problems "analysis-required" (Decrypt). Moitra expects researchers to become more secretive (transcript).
- No engineering impact yet: Brunton says the blowup happens in a non-physical regime and doesn't change practical fluid simulation (transcript).
- Proofs aren't small: the Fermat formalization is probably much longer than it needs to be (Anthropic).
Business and practitioner implications
- Brunton's framework ranks uses by how checkable the output is and how costly an error would be.
- Agent swarms are expensive. The Navier–Stokes run cost an estimated several million dollars (Quanta), and the Fermat run used about 6B tokens (Anthropic).
- Durable value is shifting. It moves toward specifying problems, checking that formal statements match intent, refactoring proofs into library quality, and explaining results (transcript).
Read the full section
- Pick verifiable, low-stakes work first. Brunton's framework ranks uses by how checkable the output is and how costly an error would be. Formal math scores best on both, which is why it suits autonomous agents (transcript).
- Agent swarms are expensive. The Navier–Stokes run cost an estimated several million dollars (Quanta), and the Fermat run used about 6B tokens (Anthropic).
- Durable value is shifting. It moves toward specifying problems, checking that formal statements match intent, refactoring proofs into library quality, and explaining results (transcript).
The source trail.
Sources (11)
#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory
Transcript retrieved via published_transcript; language en. Timestamped text, not direct audiovisual review. Source: https://lexfridman.com/math-ai-crisis-transcript. Automatic captions/transcription may contain errors.
lexfridman.comTranscript: #503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory
published_transcript
lexfridman.com