OpenAI Claims AI Cracked Kakeya, Yang–Mills Existence and the Exponential Time Hypothesis: What's Proven, What Isn't (2026)
OpenAI says its AI proved local smoothing (Kakeya) in all dimensions, refuted ETH and proved 4D Yang–Mills existence. Claims, Lean checks, reviewers, caveats.
TL;DR: On 26 October 2026 (US time; early 27 October KST), OpenAI said its internal models had produced three results: a proof of Sogge's local smoothing conjecture in every dimension (which OpenAI says also settles the Kakeya and Stein restriction conjectures), a refutation of the Exponential Time Hypothesis (ETH), and a rigorous existence proof for 4D Yang–Mills theory, QCD and a unified completion of the Standard Model. OpenAI says all three were formalized in Lean. The math and CS write-ups and Lean files are public on GitHub. The physics materials are being held back by a review group led by Nobel laureate David Gross. The Yang–Mills mass gap is not part of the claim, so the $1M Millennium Prize isn't solved. Hours later, Anthropic posted its own, separate ETH refutation. None of the three results has had independent expert verification yet.
Everything below marked as a claim is OpenAI's (or Anthropic's) claim. None of it has been peer reviewed yet.
What OpenAI announced, and when
The announcement is an OpenAI blog post titled "Working with academic communities to build scientific understanding", dated 26 October 2026. OpenAI calls the AI manuscripts "a form of raw research data" and says it does not seek academic credit for them. Instead, it asked outside experts to act as "shepherds", and each field chose a different release model:
- Math: release the AI output and let researchers organize their own checks.
- Computer science: an external group explains the result, and the AI proof plus its Lean file are public.
- Physics: an external group writes the result up itself and decides when the AI materials get released.
The timeline, from OpenAI's post and press coverage (KST = UTC+9; US Eastern was on daylight time, UTC−4):
| Date (US / KST) | Event |
|---|---|
| 9 Sep 2026 | OpenAI says its model produced a natural-language proof of pure Yang–Mills existence |
| 18 Sep 2026 | Model extends it to QCD (quarks + gluons) |
| 16 Oct 2026 | ETH disproof found; Lean formalization of Yang–Mills existence completed |
| 17 Oct 2026 | ETH disproof formalized in Lean |
| 18–19 Oct 2026 | Local smoothing proof obtained (18th), formalized in Lean (19th) |
| 25 Oct 2026 | Date on the local smoothing manuscript in OpenAI's repo |
| 26 Oct, 4:52 p.m. ET / 27 Oct, 5:52 a.m. KST | ETH files land in OpenAI's GitHub repo; blog post goes live the same day (WebProNews) |
| 26 Oct, 11:07 p.m. ET / 27 Oct, 12:07 p.m. KST | Anthropic posts its own ETH refutation, credited to Claude |
| 27–28 Oct KST | Korean coverage; the FMKorea post goes viral |
OpenAI president Greg Brockman cooled the hype the same day: "p still probably doesn't equal np" (Emergent Mind, LLMs Blog).
The three claims at a glance
| Problem | Field | OpenAI's claimed result | Materials public? | Status | Who's checking |
|---|---|---|---|---|---|
| Local smoothing conjecture (Sogge, 1991) → Kakeya, Stein restriction | Math (harmonic analysis) | Proved in every spatial dimension n ≥ 2. Kakeya follows in all dimensions; restriction estimates "throughout the interior of the conjectured range" (paper wording) | Yes: ~59-page write-up + Lean proof on GitHub | Lean-checked by OpenAI; no expert verdict yet | No named group. Open to the whole community |
| Exponential Time Hypothesis (Impagliazzo–Paturi–Zane, 1999) | Computer science (complexity) | ETH is false: 3-SAT and #3-SAT solvable in subexponential time | Yes: write-up + Lean proof on GitHub | Lean-checked by OpenAI; open review underway | Russell Impagliazzo & Ramamohan Paturi (UC San Diego), Ryan Williams (MIT) |
| Yang–Mills existence (part of a Clay Millennium Problem), QCD, Standard Model completion | Physics (quantum field theory) | Rigorous existence of interacting 4D gauge theories with fermions, built on Balaban's approach. No mass gap claim | No: held back by the review group | OpenAI says Lean formalization of Yang–Mills existence is done; nobody outside can inspect it yet | David Gross (lead), Michael Douglas, Arthur Jaffe, Igor Klebanov, Nikita Nekrasov |
Origin years are approximate. Kakeya's original question dates to 1917; the n-dimensional Kakeya conjecture in its modern form is younger. "Yang–Mills" counts from the 1954 theory; the Clay problem itself dates to 2000.
1. Math: local smoothing, Kakeya and restriction
What it's about. Waves can focus. Think sunlight through a magnifying glass. In 1991, Christopher Sogge conjectured that the most extreme focusing has to be short-lived: if you average over a short time window, a wave is smoother than its worst instant suggests. The conjecture says exactly how much smoother, in every dimension.
Why it matters. Local smoothing sits at the top of a chain of famous problems in Fourier analysis. It implies the Bochner–Riesz conjecture, which implies Stein's restriction conjecture, which implies the Kakeya conjecture. Kakeya asks how small a set can be if it contains a unit-length needle pointing in every direction. Such sets can have zero volume, but the conjecture says they still have full (Hausdorff) dimension.
What humans had before. Local smoothing was proved in 2 spatial dimensions by Guth, Wang and Zhang (2019, published 2020). Kakeya was proved in 2D in 1971 and in 3D by Hong Wang and Joshua Zahl in 2025. Dimensions 4 and up were open.
What OpenAI claims. A proof of local smoothing for every dimension n ≥ 2, built on what it calls "a novel auxiliary partial differential equation". The manuscript's abstract claims Bochner–Riesz, Kakeya and Nikodym maximal conjectures, and that "every Euclidean Kakeya set has full Hausdorff dimension" (paper mirror).
The fine print. OpenAI's blog says the result resolves Stein's restriction conjecture "in all dimensions". The paper and repo changelog are narrower: restriction estimates "throughout the interior of the conjectured range". Whether that covers the full conjecture, endpoints included, is a question for specialists (Papex, Times of AI).
2. Computer science: the Exponential Time Hypothesis
What it's about. 3-SAT asks whether you can flip a row of on/off switches so that a list of small logic rules all hold. Brute force doubles the work with every switch added. ETH, proposed by Impagliazzo, Paturi and Zane in 1999, says you can shave the exponent but never escape it: every algorithm needs roughly 2^(c·n) time for some fixed c > 0.
Why it matters. ETH is a workhorse assumption. Researchers have used it in hundreds of papers to argue that problems in graph theory, databases, computational biology and geometry can't be solved much faster. Anthropic's paper counts more than 800.
What OpenAI claims. A deterministic algorithm that counts the satisfying assignments of fixed-width CNF formulas in subexponential time, roughly 2^(n·log log n/√log n). The exponent's share of n keeps shrinking as n grows, so ETH is false. OpenAI says it found the disproof on 16 October and checked it in Lean on 17 October.
What it doesn't mean.
- Not P = NP. The new bound is still far slower than any polynomial. OpenAI's own paper says the result "does not bear on the P vs NP question directly, nor does it yield any practical algorithms for NP-complete problems" (WebProNews).
- Not a faster SAT solver. Anthropic's paper on the same result says its bound doesn't beat exhaustive search on typical formulas "unless n > 10^10 000", and that it "says nothing about solving formulas in practice". One SAT researcher called the algorithm "galactic".
- Older hardness results don't fall automatically. CMU's Ryan O'Donnell pointed out that ETH-based results are reductions, and those reductions still work. What changes is the assumption they lean on.
Anthropic got there the same day. About six hours after OpenAI's files went up, Anthropic posted "A 2^o(n) Time Algorithm for 3-SAT", credited to Claude. It uses a different method (tensor-network contraction rather than OpenAI's polynomial-system approach) and reaches a nearly identical bound. Anthropic says its model found the core algorithm on 16 October "with 14M output tokens and no human input", and that it worked "independently and concurrently" with OpenAI. Anthropic mathematician Levent Alpöge had suggested OpenAI's solution leaned on pre-existing work (TechCrunch).
3. Physics: Yang–Mills existence, not the mass gap
What it's about. Yang–Mills theory (1954) is the math behind the strong nuclear force, and through quantum chromodynamics (QCD) it describes how quarks and gluons interact. Physicists have used it for decades with very precise experimental success. Nobody has ever proved it exists as a rigorous mathematical object in four-dimensional spacetime.
The Millennium Prize problem has two parts. The Clay Mathematics Institute's "Yang–Mills existence and mass gap" problem requires (1) proving the quantum theory exists rigorously in 4D and (2) proving it has a positive mass gap, a minimum nonzero energy for its lightest excitations. OpenAI claims part (1) only. The mass gap is still open, the $1M prize isn't claimed, and the Clay Institute has accepted nothing (TechTimes).
What OpenAI claims. It goes beyond pure Yang–Mills. OpenAI says its models produced existence proofs for QCD and for "a unified completion of the Standard Model" (the simplest grand unified theory containing the Standard Model's gauge fields and fermions), building on Tadeusz Balaban's constructive-QFT program. The key mechanism is asymptotic freedom: the strong force weakens at short distances, which keeps the lattice-to-continuum limit under control. David Gross won his share of the 2004 Nobel Prize for discovering it.
Who's checking. A group led by David Gross with Michael Douglas, Arthur Jaffe (who wrote the Millennium problem statement with Edward Witten and co-founded the Clay Institute), Igor Klebanov and Nikita Nekrasov. The group chose to hold back the AI-produced artifacts and write the result up itself. So right now nobody outside can read the physics proof or its Lean file.
Verification: what Lean does and doesn't prove
OpenAI says all three results were formalized in Lean, a proof assistant that machine-checks each logical step. That's a real step up from OpenAI's October 6 dump of 722 unverified manuscripts. It still isn't the end of checking:
- Lean confirms the formal proof is valid. It does not confirm that the formal statement says the same thing as the famous conjecture. Anthropic's README puts it bluntly: "Lean checks the proofs, but no machine checks that this file says what the paper says."
- This failure mode has already happened. A Cambridge/King's College London preprint argued that the Lean formalization of OpenAI's September Navier–Stokes claim doesn't match its written proof (Times of AI).
- The ETH review group has said its role is to examine the result openly, not to certify it.
Expert and skeptic reactions
- Ryan Williams (MIT), who is organizing the ETH review: some parts are "routine", but "there are two or three alien-looking parts, and it's exactly these parts that I saw that made me think, wow, this is something that the whole community needs to work on together." He also said "every computer scientist should probably put a little more weight on the plausibility of P = NP", and that OpenAI's algorithm probably isn't the best possible (TechCrunch).
- Alex Iosevich (University of Rochester, harmonic analysis) called the local smoothing result a "historic development" (via Al Jazeera, Quartz).
- James Leng (harmonic analyst), on X: "The proof is absolutely shocking. It is only ~25 pages and primarily uses elliptic PDEs. It notably does not use much of the mathematics developed by the harmonic analysis lineage" (Post-Cutoff).
- The Advisory Group on Mathematics and AI (members include Timothy Gowers, Martin Hairer and Edward Witten) said that since OpenAI says the physics Lean formalization is complete, "there is no scientific reason for delaying the release any further". A day earlier it had urged AI companies not to hand-pick mathematicians to get results first (Times of AI).
- Terence Tao hasn't given a verdict on these proofs. His earlier criticism was about process: after the October 6 mass release he said he saw no way to engage constructively with OpenAI (Tao's blog).
- OpenAI admits the trade-off itself: picking initial shepherds "creates asymmetry of information".
Korean community reaction
In Korea, an FMKorea "best" post titled "오픈ai 수학·컴퓨터공학·물리학 3대 난제 ai로 풀었다고 발표" ("OpenAI announces AI solved three great problems in math, CS and physics") hit about 84,000 views and 275 comments. The comments fell into three broad groups:
- Excitement about what comes next: if AI can do this, cures for diseases and other breakthroughs might be close.
- Skepticism: "it's not verified yet", with reminders that earlier AI-proof claims drew criticism.
- "Riemann next": predictions about which famous problem falls next, plus P vs NP chatter. One commenter said a Polymarket P=NP market had jumped. We couldn't confirm that.
The post got most of the basics right: OpenAI's blog banner, Lean verification, Gross and Jaffe in physics, and Impagliazzo and Williams on ETH. But a few details need fixing:
- "3D last year" refers to Kakeya, not local smoothing. Hong Wang and Joshua Zahl proved the Kakeya conjecture in 3D in 2025. Local smoothing had only been fully proved in 2D (Guth–Wang–Zhang, 2019).
- Restriction is hedged in the paper. The paper claims restriction estimates "throughout the interior of the conjectured range", which is less than the blog's "in all dimensions".
- The physics claim is bigger than Yang–Mills alone. It covers QCD and a unified Standard Model completion. The group has five members (add Douglas, Klebanov and Nekrasov), and the physics materials are not public.
- The ETH group also includes Ramamohan Paturi, one of ETH's original proposers.
- Brockman didn't hint at P = NP. He said P "still probably doesn't equal np".
- Anthropic posted a competing ETH refutation the same night, which the post didn't mention.
What it means
- If the ETH refutation holds, it's the biggest shake-up in fine-grained complexity in decades. Hundreds of "this can't be done faster, assuming ETH" results lose their footing, and Williams wants an OpenAI-funded conference to work out what replaces it.
- Cryptography is not broken. Williams says one of his first questions was whether any cryptosystem is affected. OpenAI checked and was "reasonably confident" none is. One analysis posted on Scott Aaronson's blog found the bound doesn't beat the best known factoring algorithms either. Cryptographers should still watch the follow-ups, since a better algorithm built on the new ideas could matter for key-size assumptions.
- If the local smoothing proof holds, a whole tower of harmonic analysis falls at once, with a short proof that apparently sidesteps decades of standard technique.
- The physics claim can't be judged yet because nobody outside the Gross group can read it.
- Big picture: two frontier labs say their AIs independently found the same kind of breakthrough on the same day. Whatever the review finds, the bottleneck in research is moving from finding proofs to checking and understanding them.
All three results are OpenAI's claims, pending independent expert review and peer review.
Sources
- OpenAI: Working with academic communities to build scientific understanding (26 Oct 2026)
- TechCrunch: Frontier AI labs say their models refuted the Exponential Time Hypothesis
- Anthropic / Hexagon: A 2^o(n) Time Algorithm for 3-SAT
- Times of AI: no independent verification yet
- TechTimes: OpenAI claims Yang–Mills proof; Gross reviews it
- Quartz: OpenAI said AI cracked major math, computing and physics problems
- Post-Cutoff: local smoothing / Kakeya claim details
- Papex: what's claimed, verified and still open
- WebProNews: two AI labs claim ETH is false
- Emergent Mind: what the ETH refutation means
- LLMs Blog: what experts are saying
- Terence Tao's blog: AHM statement on the October 6 release
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