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AI Frontier 2026 · As of 2026-10-27

2026: The Year Machines Began Solving Open Problems

September 8: Navier–Stokes. October 6: 722 manuscripts. October 26: ETH. A place to look back, claim by claim, at what was asserted, what has been checked, how mathematicians answered, and what 'solved' means in mathematics.

16 claims10 Lean public2 disputed or withdrawn

No problem status on this site has changed on the strength of a claim alone.

Human research voyage: Guth–Katz · de Grey · Córdoba–Martínez-Zoroa · Wang–Zahl →Beyond the attempt list — a workshop season across 22 pages →

Five layers behind the word 'solved'

Headlines use one word; mathematics has at least five layers. How far each claim has climbed is the unit of record on this page.

  1. 1

    A company announced it

    Blog post, manuscript, reasoning summary. The fastest and loudest layer.

  2. 2

    The Lean code compiles

    A formal statement is mechanically checked. What the statement says is still open.

  3. 3

    The Lean statement matches the paper

    People must compare code and prose. Navier–Stokes was challenged here.

  4. 4

    People read and explain it again

    Workshops, seminars, rewrites. The layer where a manuscript becomes mathematicians’ language.

  5. 5

    The community accepts it as a theorem

    Peer review, textbooks, two years of scrutiny. Clay prizes are discussed only here.

Timeline

  1. Claim

    OpenAI disproves the grid intuition for Erdős unit distances

    An internal model produced a construction with more unit-distance pairs than the grid, checked by outside mathematicians. MathVoyage recorded it in content 4.19.

    OpenAI — model disproves a discrete-geometry conjecture (2026-05-20)

  2. Claim

    OpenAI publishes ten advances — sphere packing, non-sofic groups, Ramsey numbers

    Ten results with human-drafted manuscripts and model-made Lean proofs. None coincides with a MathVoyage problem.

    OpenAI — Ten advances in mathematics and theoretical CS (2026-08-01)

  3. Claim

    Anthropic raises the simple-zero proportion bound to 67.25%

    A proportion theorem, not the hypothesis. Spec 95 fixed ten trillion, 67.25%, and 100% as distinct evidence layers.

    Anthropic — 67.25% of zeta zeros on the critical line (2026-08-10)

  4. Claim

    Alpöge and Buckmaster post Lean-verified finite-time blow-up for forced Euler

    About twelve hours before OpenAI's announcement; Buckmaster added a statement alleging that their progress had leaked to OpenAI.

    OpenAI — On the Navier–Stokes Millennium Prize Problem (2026-09-08)BBC Korean — a 90-year problem in 88 hours (2026-09-09)Wikipedia — Navier–Stokes priority controversy

  5. Claim

    OpenAI claims Navier–Stokes statements C and D

    A proof plus Lean formalization that a fluid at rest under smooth forcing develops a singularity in finite time; about 10,000 agents over 88 hours. OpenAI said it would not claim the prize.

    OpenAI — On the Navier–Stokes Millennium Prize Problem (2026-09-08)OpenAI Navier–Stokes manuscript PDFopenai/NavierStokesAndEuler Lean repository

  6. Community

    Twenty-eight Fields Medalists sign 'A Severe Misalignment of AI in Mathematics'

    Argues that using open problems as benchmarks harms the field and its community, while not rejecting AI; it objects to announcements lacking write-ups, isolated methods, and citations.

    Fields Medalists' declaration — A Severe Misalignment of AI in Mathematics (2026-09-11)

  7. Verification

    Clay Mathematics Institute: apparently settled, verification deliberately unhurried

    The problem's status became 'Active'. Prize rules require publication and two years of scrutiny; no laureate is listed as of the access date.

    Clay Mathematics Institute — Navier–Stokes problem pageWikipedia — Navier–Stokes priority controversy

  8. Governance

    OpenAI announces 100+ further resolutions and an independent advisory group

    A nine-member group hosted at the Institute for Advanced Study; it explicitly does not advise on research pacing.

    TechCrunch — OpenAI forms math advisory group (2026-09-21)AGMAI — Advisory Group on Mathematics and AI (guidelines 2026-09-29; posts 10-06, 10-25, 10-26)

  9. Governance

    AGMAI publishes guidelines for responsible release of AI-generated mathematics

    Based on 600+ survey responses, proposes pathway 2A (scholarly articles) and 2B (unfinished output) requiring external repositories, disclosed prompts and reasoning, attempt counts and costs, Lean, and workshop funding.

    AGMAI — Advisory Group on Mathematics and AI (guidelines 2026-09-29; posts 10-06, 10-25, 10-26)

  10. Claim

    OpenAI releases 722 manuscripts in 372 families on GitHub

    Unreleased internal model, about three hours of compute per result, about 4,000 problems attempted; includes quasi-Riemann, Hilbert's tenth over ℚ, non-five-colorability of the plane, and Kakeya in 3 and 4 dimensions.

    OpenAI — Sharing AI progress in mathematics (2026-10-06)openai/math repository READMEopenai/math CONTENTS.md — per-family result map

  11. Dispute

    Cambridge and KCL researchers report Lean–paper mismatches in the Navier–Stokes proof

    Points to at least two places, including one where the code proves a weaker estimate than the paper states; it does not call the result wrong and argues machine checks cannot replace peer review.

    Bastounis, Circelli & Hansen — Navier-Stokes lost in translation (arXiv:2610.08144)ZDNet Korea — withdrawals and fixes after the 722 release (2026-10-09)

  12. Verification

    Three withdrawals and fourteen fixes within a day; 300/719 formalized

    A sign error in a Weil-class manuscript invalidated two dependent papers. OpenAI keeps withdrawn papers with notices instead of deleting them.

    openai/math history.md — withdrawals, fixes, additionsZDNet Korea — withdrawals and fixes after the 722 release (2026-10-09)

  13. Community

    Association for Human Mathematics urges mathematicians to stop working with OpenAI

    Frames the 700-file release as a demonstration of power rather than scholarship; Tao reposted the statement on his blog.

    Association for Human Mathematics statement — reposted on Tao's blog (2026-10-07)Chosun Ilbo — 722 AI problem solutions; where is human understanding? (2026-10-08)

  14. Community

    'Mathocalypse'? — the question of who will do the reading

    Public commentary from mathematicians asks first about intelligibility, who bears verification labor, early-career researchers, and credit for prior work. Korean coverage described people checking the list before the proofs.

    The Conversation — Is this the 'mathocalypse'? (Melissa Lee, Monash)Dong-A Science — OpenAI releases 722 problem manuscripts at once (2026-10)

  15. Governance

    AGMAI advises against designating mathematicians to receive AI output first

    Cites delayed and unequal access, opaque selection, and borrowed credibility, recommending pathways 2A/2B instead.

    AGMAI — Advisory Group on Mathematics and AI (guidelines 2026-09-29; posts 10-06, 10-25, 10-26)

  16. Claim

    OpenAI announces local smoothing, an ETH refutation, and Yang–Mills existence

    The math output was released directly; the ETH result is being examined by a group led by Impagliazzo, Paturi, and Williams with public artifacts; the physics group led by Gross is withholding artifacts to write it up.

    OpenAI — Working with academic communities to build scientific understanding (2026-10-26)openai/math history.md — withdrawals, fixes, additions

Claim ledger

Claims that touch the 16 expedition problems and 25 workshop problems, plus a few landmark cases that show the verification layers. The status label is what this site currently assigns to the problem.

Problems not addressed

Problems on this site not found in openai/math family titles or summaries as of 2026-10-27. "Not addressed" is not evidence that machines cannot solve them.

Looking back

Five layers behind the word 'solved'

The autumn's headlines use 'solved' as one word, but in mathematics it splits into at least five layers. A company announced it. The Lean code compiles. The statement Lean proved matches the paper's statement. A person can read the proof and explain it again. The community accepts it as a theorem and writes it into textbooks. The Navier–Stokes episode stalled at the third layer: the code passed, yet an outside audit found places where it proves a weaker estimate than the paper states.

The Clay Institute's rules institutionalize the fifth layer: a prize is discussed only after publication and two years of scrutiny. 'Apparently settled' and 'settled' are therefore different sentences, and that is why no problem status on this site has changed.

The three withdrawals within a day on October 7 show these layers are not decoration. One sign error in an unformalized manuscript brought down the two manuscripts built on it. Calling the repository 'raw research data' is not modesty; it is the right classification.

How the mathematical community responded

There is no single reaction. The September 11 declaration by 28 Fields Medalists did not reject AI; it rejected using open problems as benchmarks. A famous problem's value lies not in the answer but in what follows it—scrutiny, simplification, textbooks, students—and mass announcements without write-ups or citations cut that chain.

The nine-member advisory group hosted at the Institute for Advanced Study wrote procedure instead of protest: external repositories, disclosed prompts and reasoning, attempt counts and costs, Lean where possible, workshop funding. Its October 6 statement called the release the beginning of understanding, not its completion; on October 25 it opposed handing output to designated mathematicians first because that creates unequal access.

Sharper voices exist. The Association for Human Mathematics called the 700-file release a demonstration of power and urged mathematicians to stop collaborating. Buckmaster, who had been working on Navier–Stokes, raised priority and leak concerns, which OpenAI contested with an investigation. In Korea, mathematicians at KAIST and IBS conveyed the tension between 'enormous if verified' and 'people are checking the list before the proofs'.

And there is the quiet reaction: people reading a result in class, people rewriting a manuscript from scratch. It makes no headlines, but the fifth layer is built by exactly these people.

What changes, and what does not

What changes: the scarcity of proofs may vanish. When hundreds of candidate proofs arrive in a week, the bottleneck is reading, not production. Who reads, how that labor is credited, and whether an unread proof is a theorem at all become live questions. When an early-career researcher's problem is 'finished' overnight, the rules of a career shake with it.

What changes: the grammar of credit. OpenAI wrote that it seeks no academic credit, yet manuscripts that omit prior work can erase credit in practice. People had to point out that the core Navier–Stokes idea rests on the 2023 non-smooth-forcing construction of Córdoba and Martínez-Zoroa.

What does not change: Lean proves formal statements; it does not translate problems. Forced versus unforced fluids, ETH versus P ≠ NP, zero-free regions versus the Riemann hypothesis—these distinctions are made by people, not machines. So is deciding why mathematics asks a question and which question deserves to come next.

What does not change: mathematics' clock. Someone rereading the September 2026 announcement in autumn 2028 will be the one to say what it was. Until then this page records each status as a claim.

What MathVoyage can do

A map of verification status. News peaks on announcement day and fades; mathematics runs the other way. This ledger records, per claim, Lean status, outside audits, withdrawals, and peer review with dates, and changes status only when it changes. It extends to every problem the evidence ladder that showed ten trillion, 67.25%, and 100% as different scales for Riemann.

A map of what people understand. Linking each claim to the human work and method lineages it stands on—de Grey's 1,581-vertex graph, Wang–Zahl in three dimensions, Guth–Katz polynomial partitioning, Córdoba–Martínez-Zoroa forcing—gives a machine's result a place in human history.

Traces without accounts. True to this site's principles, a place to leave 'one sentence I understood' and 'one question I still have' per claim, without login, would gather the quiet reactions that build the fifth layer. Reading what a claim does not claim is practice anyone can do.

New questions for the workshop. Workshop problems absent from OpenAI's list—Collatz, odd perfect numbers, perfect cuboids, Brocard—are not problems machines failed at; they are problems not yet on the attempt list. What 2026 changed is not the difficulty of problems but the human role of choosing which to ask, and why.

Questions to carry

  1. Q1If alternative C of the Clay statement holds, would you say 'the Navier–Stokes problem is solved'?

    Hint

    Which fluid—forced or unforced—is closer to an airplane wing or the weather?

  2. Q2If a Lean-checked proof can be explained by no one, is it a theorem?

    Hint

    Decide for yourself at which of the five layers you would start using the word 'theorem'.

  3. Q3If the plane truly cannot be five-colored, is the only remaining question 'six or seven'?

    Hint

    Consider why the answer might depend on whether color classes must be measurable.

  4. Q4If ETH is false, what happens to cryptography—nothing, or everything?

    Hint

    Compare subexponential and polynomial time using 2^{√n} and n^3.

  5. Q5If you were an early-career mathematician and your three-year problem appeared on a list one morning, what would you ask next?

    Hint

    Recall where the declaration says the resources 'students and ideas' grow.

Sources and access dates

Positions are summarized rather than quoted. Secondary sources were used only to cross-check dates.