AI for Science in depth
What has AI done on open problems?
In math, a historic conjecture fell and an AI-generated proof appeared for a Millennium Prize Problem. In physics, a textbook "zero" turned out not to be. In biology, AI-designed proteins bound their targets in the lab. We map the record and the limits across three fields — and the backlash from mathematicians.Oct 1, 2026
In 2026, AI moved from being a tool for attacking open problems to being the thing that actually closes them. In May, Erdős's unit distance conjecture was disproved; in September, an AI-generated proof was published for the Navier–Stokes Millennium Prize Problem.Verified At the same time, mathematicians are pushing back hard: we have answers, they say, but not more understanding. This feature uses primary sources to map what AI has done on open problems in mathematics, physics and biology — and what it still cannot do.
- Verified Confirmed in papers, official announcements, peer-reviewed journals or formal proofs
- Vendor claim Announced by the AI company; external verification not complete
- Independent check Confirmed or assessed by outside researchers or institutions
- AIfrontier view Our interpretation or hypothesis
1. The big picture: twelve months
| When | What happened | Field |
|---|---|---|
| 2026.01 | Two people in their early 20s resolve Erdős Problem 728 with GPT-5.2 Pro. A Google DeepMind-led team screens 700 "open" problems, solves four and finds nine forgotten existing solutions | Math |
| 2026.02 | GPT-5.2 Pro conjectures a formula showing single-minus gluon tree amplitudes are nonzero; an internal model proves it | Physics |
| 2026.03 | Gemini Deep Think plus tree search finds exact solutions for the gravitational-wave spectrum of cosmic strings | Physics |
| 2026.05 | OpenAI internal model disproves Erdős's unit distance conjecture; a DeepMind agent autonomously resolves 9 of 353 problems (a few hundred dollars each); Google's Co-Scientist is published in Nature | Math, biology |
| 2026.07 | A counterexample to the Jacobian conjecture is reported with Fable 5 assistance | Math |
| 2026.08 | Internal Astra resolves or advances ten open problems (Aug 1); Claude raises the zeta-zeros lower bound (announced around Aug 10; paper revised Aug 13); Claude's protein binder designs succeed in the wet lab (Aug 18) | Math, biology |
| 2026.09 | Anthropic formalizes Fermat's Last Theorem end-to-end in Lean (Sep 4); AI-generated Navier–Stokes proof (Sep 8); Fields Medalists' declaration (Sep 11); OpenAI sets up a mathematics advisory group (Sep 21) | Math |
The overall shift is from "humans solve with AI" to "AI solves alone, humans interpret." But most results sit in fields where verification is fast (mathematics, symbolic manipulation in theoretical physics). In biology, the division of labor is still "AI designs, the wet lab decides."
2. Math I: the Erdős problems as a proving ground
Many of AI's mathematical results come from problems left by Paul Erdős. erdosproblems.com, a database British mathematician Thomas Bloom built for himself in 2023, became a collaboration hub after adding comments in August 2025 — and then an informal benchmark for AI labs. Source: Quanta Magazine
| Who | Result | Label |
|---|---|---|
| Kevin Barreto and Liam Price (early 20s) | Resolved Problem 728 with GPT-5.2 Pro (Jan 4, 2026); logic checked with Harmonic's Aristotle. Their first claimed "solve," Problem 333, turned out to have been solved by Erdős himself in 1977 | Independent check |
| Same, with Terence Tao and others | Co-authored paper resolving Problem 1196 on primitive sets (May 2026) | Verified |
| Google DeepMind (24 researchers) | Screened 700 "open" problems with Gemini; solved four, found nine forgotten solutions (January) | Verified |
| Google DeepMind (21 researchers) | Most capable agent autonomously resolved 9 of 353 formalized open problems at a few hundred dollars each (May) | Vendor claim |
Even at this stage, quality depended on literature search and human verification. Bloom warned of 100–200-page papers where AI wrote the proof, checked it and wrote the paper: "no human has read it, and no human is going to read it." Source: Quanta Magazine
3. Math II: historically significant proofs
Unit distance conjecture disproved (May 20)Verified
How many pairs among n points in the plane can be exactly distance 1 apart? Since Erdős posed it in 1946, grid-like constructions were believed essentially optimal. An OpenAI internal model disproved this. What surprised mathematicians was that a general-purpose reasoning model — not a math-specific system — brought algebraic number theory (class field towers, Golod–Shafarevich theory) to a geometry question. Princeton's Will Sawin made the improvement explicit with δ = 0.014. Source: OpenAI
"If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that." — Tim Gowers, Fields Medalist Source: Quanta Magazine
Ten results from internal Astra (Aug 1)Vendor claim
OpenAI said an internal version of Astra resolved or substantially advanced ten long-standing problems. Tokens used would cost about $2,000 at Sol API rates. The model formalized each argument in Lean, published on GitHub. Source: OpenAI
| Area | Result |
|---|---|
| High-dimensional sphere packing | New upper bounds on density |
| Coding theory | Exponentially improved bounds for binary and spherical codes |
| Group theory | Construction establishing non-sofic groups exist |
| Operator algebras | Disproof of Connes's rigidity conjecture |
| Complexity theory | Arithmetic circuit and formula lower bounds for the permanent (n⁴/log n) |
| Quantum complexity | Exponential parallel repetition for two-player quantum games |
| Lattice cryptography | Polynomial-factor hardness of approximating the closest vector problem |
| Convex geometry | Ehrhart's volume conjecture settled in every dimension |
| Ramsey theory | Superexponential lower bound for multicolor triangle Ramsey numbers (Erdős Problem 183) |
| Extremal graph theory | Compactness and degeneracy conjectures (Erdős Problems 146, 180) |
OpenAI states the arguments were generated by its system — claiming human authorship "would misrepresent both the system's contribution and the nature of genuine human intellectual work" — and that it takes responsibility for correctness.
Zeros of the Riemann zeta function (August)Independent check
An unreleased research version of Claude raised the lower bound on the fraction of zeta zeros satisfying the Riemann hypothesis from 41.6% to 67.2%. Two Anthropic mathematicians validated it, Claude produced a Lean proof (the paper was revised on Aug 13), and experts Brian Conrey and Dan Goldston examined the paper. Anthropic itself says: "We don't expect that the techniques Claude used will lead to proving the Riemann hypothesis." Source: Anthropic
Fermat's Last Theorem fully formalized (Sep 4)Independent check
An Anthropic internal model formalized a complete proof of Fermat's Last Theorem in Lean — according to Anthropic, Claude worked largely autonomously for 11 days, writing about 13 million lines of Lean and proving 29,500 intermediate theorems. Source: Anthropic Kevin Buzzard (Imperial College), funded to do the same, compiled it and ran the checker himself. He says it tells us "essentially nothing" mathematically (the proof was already trusted) but shows thousands of pages of literature can now be autoformalized end-to-end. Source: Xena Project
4. Math III: Navier–Stokes — a solution and a "loophole"
OpenAI's announcementVerified
On September 8, OpenAI said an internal system proved that a fluid that starts smooth can develop a singularity (infinite speed) in finite time under the Navier–Stokes equations, establishing statement C (and D) — breakdown with a smooth external force — in the Clay Institute's official formulation. Source: OpenAI
| Item | As announced |
|---|---|
| Model | Internal model in training since Aug 28, far more capable than GPT-6 Astra; swapped for a further-trained version mid-run |
| Setup | ~10,000 concurrent agents with a cached internet and code execution |
| Path | First, ~100 agents over ~50 hours proved blowup for the unforced Euler equations; resources then shifted to Navier–Stokes |
| Time | ~88 hours to the result (Sep 5); Lean formalization and verification took ~17 hours with GPT-6 Astra |
| Scale | 2.7M messages and ~130B output tokens for Navier–Stokes (4.9M and ~300B across all problems) |
| Output | 166-page paper and Lean formalization; prize not claimed |
The priority disputeIndependent check
About 12 hours earlier, NYU's Tristan Buckmaster announced that he and Levent Alpöge, a mathematician employed at Anthropic, had proved blowup with smooth forcing for Euler and related equations (obtained Aug 15, verified in Lean a week later). They had used OpenAI's Codex in their research. OpenAI began its effort on September 1 after rumors of their progress, and initially said it "cannot rule out" that de-identified data from their usage improved its models. It later said an investigation showed the prior two months of prompts could not have influenced the system, and on September 13 that "no user inputs past July 3rd" could have. Source: Wikipedia (aggregating coverage)
Terence Tao called the pair's work "a remarkable achievement" and criticized AI companies for announcing math by press release rather than preprints. Diego Córdoba, whose program underlies the approach, said that without his group's work, "AI would not have solved the problem."
The second dispute: was it the right problem?Independent check
Correctness is broadly accepted because the Lean code compiles. The question is meaning. OpenAI's construction uses a highly contrived external force. On September 17, Vlad Vicol and colleagues proved that solutions with the key features of OpenAI's construction are regular if the force vanishes near the singular point or is real-analytic — so this method cannot be extended to the unforced problem. Source: arXiv:2609.20803
"The Clay problem is settled, but the main problem for the Navier-Stokes equations is not." — Luis Silvestre, University of Chicago Source: Scientific American
Readability is another concern. Oxford's James Maynard: "So far it's been very difficult to really extract any human understanding from this new AI proof." Brown's Javier Gómez-Serrano: "The paper is not written for humans… as of today, the paper doesn't teach us much." Source: NPR The Clay Institute said on September 11 that the problem "has apparently been settled," but requires peer-reviewed publication and calls its process "deliberately unhurried"; its website lists the problem as "Active."
5. Physics: AI that makes complicated formulas simple
Gluon scattering amplitudes (February)Independent check
Textbooks treat the tree-level amplitude with one negative-helicity gluon and the rest positive as zero. A paper by physicists at IAS, Cambridge and Harvard with OpenAI shows it is nonzero in a special "half-collinear" regime. GPT-5.2 Pro greatly simplified expressions humans had computed by hand up to n = 6 and conjectured a formula for all n; an internal model derived and proved the same formula in about 12 hours, and the humans checked it against the Berends–Giele recursion and the soft theorem. Extensions to gravitons are under way. Source: OpenAI
"To me, finding a simple formula has always been fiddly, and also something that I have long felt might be automatable by computers." — Nima Arkani-Hamed, IAS Source: OpenAI
Gravitational waves from cosmic strings (March)Verified
Michael Brenner and colleagues combined Gemini Deep Think with tree search and numerical feedback to derive exact analytical solutions for the core integral of the gravitational-wave power spectrum from cosmic strings, for arbitrary loop geometries — where earlier AI-assisted work had produced only partial asymptotic results. The system found six different methods. Source: arXiv:2603.04735
An old statistical mechanics problem (2025)Independent check
In 2025, Brookhaven National Laboratory's Weiguo Yin used OpenAI's o3-mini-high to solve the 1D frustrated Potts model, unsolved even for three spin orientations, publishing in Physical Review B. He first tested the AI on his own unpublished paper; it derived an equivalent, more elegant form of a key equation, convincing him it was doing its own math. Source: Brookhaven National Laboratory
6. Biology and chemistry: AI designs, the wet lab decides
In biology, passing a proof checker does not end the story — experiments do. So results are reported as "AI proposed the design or hypothesis, an outside lab confirmed it."
Autonomous protein binder design by Claude (Aug 18)Independent check
Anthropic had Claude (Mythos Preview and Opus 4.8) design binders for 15 target proteins. Beyond the initial prompt and access approvals, humans were not involved; Claude orchestrated existing design and folding models. Adaptyv Bio and Twist Bioscience produced and tested the designs. Source: Anthropic
| Metric | Result |
|---|---|
| Targets | Binders against 14 of 15 (354 binders from 1,320 designs) |
| Hit rate | 22.6–35.1% depending on setup, versus 10–15% typical today |
| Hard target: TNF | Opus 4.8 succeeded, including designs cross-reactive to human, cynomolgus and mouse TNF; Mythos Preview failed |
| Failure | Maltose-binding protein (MBP): none of 90 designs confirmed |
Anthropic also reported Opus 5 processing raw NMR and LC-MS files in about 20 minutes, matching a contract lab's purity (96.4% vs 96.33%). Calling these capabilities dual-use, Anthropic keeps protein design unavailable in generally available Fable 5. In September its head of life sciences confirmed the company has set up a wet lab in the Bay Area. Source: Reuters
Google Co-Scientist (Nature, May 19)Independent check
Co-Scientist, a Gemini-based multi-agent system that generates, debates and refines hypotheses, published end-to-end experimental validation in three biomedical areas. Source: Nature
- Acute myeloid leukemia: proposed repurposed drugs showing selective cytotoxicity at clinically relevant concentrations
- Liver fibrosis: proposed new epigenetic targets with anti-fibrotic activity and hepatocyte regeneration in human hepatic organoids (including FDA-approved vorinostat)
- Antimicrobial resistance: independently proposed how phage-inducible chromosomal islands expand host range — matching researchers' unpublished results
The paper lists its own limits: reliance on open-access literature (missing paywalled work and negative results), propagation of irreproducible findings, and no guarantee that in-vitro results hold in vivo.
Around the edges
- GPT-Rosalind (OpenAI): life-sciences model updated Sep 11 with stronger medicinal chemistry and genomics; trusted-access organizations only Source: OpenAI Vendor claim
- Pharma's private data: Nature reports an AI trained on more than 20,000 protein structures from drug companies outperforms public-data-only AlphaFold-like models (Sep 14) Source: Nature
- Virtual Cell Challenge 2026 (Arc Institute): predict CRISPRi knockdown responses in six never-perturbed cell lines; final submissions Nov 5, prizes up to $100,000 Source: Arc Institute
7. Pushback and institutions
On September 11, 28 Fields Medalists (25 initially), including Terence Tao, Peter Scholze and Maryna Viazovska, published "A Severe Misalignment of AI in Mathematics." Source: mathandai.org
"Solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal."
The declaration cites rushed announcements without proper write-ups or citations, attribution and plagiarism concerns, and the loss of the human chain of transmission. On September 21, OpenAI set up an independent mathematics advisory group — including Gowers, Hairer and Witten — unpaid and free to publish its advice. But the group will not advise on how OpenAI paces its internal progress. In the same post, OpenAI said its internal model has "resolved more than 100 long-standing open problems," without publishing a list or proofs. Source: OpenAI Vendor claim
8. What we have learnedAIfrontier view
| Angle | Where things stand |
|---|---|
| Where AI is strong | Finding explicit constructions and counterexamples (unit distance, Jacobian, Navier–Stokes blowup), simplifying and generalizing formulas (gluons), large-scale formalization (Fermat) |
| Where AI is still weak | Building new theory to show something can't happen. As one researcher put it, LLMs are "especially good at constructing things that are very explicit and not as good — for now — in making new theory" |
| Who verifies | Lean is the de facto standard in math; outside labs in biology. In both, human reading and interpretation lag behind |
| Cost | A few hundred dollars per Erdős problem; ~$2,000 for Astra's ten results; Navier–Stokes reportedly cost millions in compute |
| Open questions | Credit, handling of unpublished work and AI usage data, press-release-first announcements, how to value a result that exploits a loophole in the problem statement |
2026 is the year it became undeniable that AI can solve open problems. But as Navier–Stokes shows, "solved" and "understood" are far apart. The focus is shifting from how many problems AI can close to who builds the machinery that turns those answers into usable knowledge — readable papers, proper citations, protection of unpublished work, and experimental validation. For companies using AI in R&D, which AI you feed unpublished data into is already a practical question.
Sources
- OpenAI, "The unit distance problem" (May 20) https://openai.com/index/model-disproves-discrete-geometry-conjecture/
- OpenAI, "Ten advances in mathematics and theoretical computer science" (Aug 1) https://openai.com/index/ten-advances-in-mathematics/
- OpenAI, "On the Navier–Stokes Millennium Prize Problem" (Sep 8) https://openai.com/index/navier-stokes-solution/
- OpenAI, "Advisory Group on Mathematics and Artificial Intelligence" (Sep 21) https://openai.com/index/advisory-group-on-mathematics-and-ai/
- OpenAI, "GPT-5.2 derives a new result in theoretical physics" https://openai.com/index/new-result-theoretical-physics/
- OpenAI, "Introducing new capabilities to GPT-Rosalind" (Sep 11) https://openai.com/index/introducing-new-capabilities-to-gpt-rosalind/
- Anthropic, Riemann zeta research post https://www.anthropic.com/research/riemann-zeta
- Anthropic, "Claude accelerates protein design and analytical chemistry" (Aug 18) https://www.anthropic.com/research/Claude-accelerates-protein-design
- Anthropic, "Formalizing Fermat's Last Theorem" https://www.anthropic.com/research/formalizing-fermats-last-theorem
- Kevin Buzzard, "FLT: Anthropic has beaten me to it" (Sep 4) https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-has-beaten-me-to-it/
- Quanta Magazine, "Why the Legendary Erdős Problems Are Falling to AI" (Aug 3) https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/
- Scientific American, "Did OpenAI solve the wrong Navier-Stokes problem?" (Sep 21) https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/
- NPR, "AI solved one of math's hardest problems…" (Sep 22) https://www.npr.org/2026/09/22/nx-s1-5968588/openai-navier-stokes-problem-mathematicians-learn-little
- arXiv:2609.20803 (Vicol et al., limits of OpenAI's construction) https://arxiv.org/abs/2609.20803
- Wikipedia, "Navier–Stokes priority controversy" (used as an index of coverage) https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority_controversy
- Fields Medalists, "A Severe Misalignment of AI in Mathematics" https://mathandai.org/
- arXiv:2603.04735 (Brenner et al., cosmic strings) https://arxiv.org/abs/2603.04735
- Brookhaven National Laboratory (Potts model) https://www.bnl.gov/newsroom/news.php?a=122742
- Nature, "Accelerating scientific discovery with Co-Scientist" (May 19) https://www.nature.com/articles/s41586-026-10644-y
- Nature News, "Drug firms' secret data supercharge AI protein models" (Sep 14) https://www.nature.com/articles/d41586-026-02882-x
- Reuters, "Anthropic quietly sets up biology lab…" (Sep 18) https://www.reuters.com/world/anthropic-quietly-sets-up-biology-lab-it-ramps-ai-drug-program-2026-09-18/
- Arc Institute, "Virtual Cell Challenge 2026" https://arcinstitute.org/news/virtual-cell-challenge-2026
Names, institutions and figures follow the cited sources. "Solved" reflects the AI companies' wording at announcement and does not imply formal recognition by the Clay Mathematics Institute or other bodies.