Yang-Mills Existence and Mass Gap

Yang-Mills existence and mass gap is one of the seven Millennium Prize Problems announced by the Clay Mathematics Institute (CMI) on 24 May 2000, each carrying a $1 million prize.^[3] It asks for a mathematically rigorous construction of quantum Yang-Mills theory, the key ingredient of the Standard Model of particle physics, in four-dimensional space-time,^[13] and a proof that the theory has a "mass gap": a strictly positive lower bound on the energy of every excitation above the vacuum.^[1] The official problem description was written by Arthur Jaffe and Edward Witten.^[1] As of 30 October 2026, CMI's website lists the problem as unsolved.^[2][3]

The problem has two parts. The existence part asks for a four-dimensional interacting quantum field theory satisfying standard axioms such as those of Wightman or Osterwalder and Schrader. No such theory had been constructed when the problem was posed.^[1] The mass gap part asks for a proof of a property that physicists infer from experiment and from lattice computer simulations, but for which no proof is known.^[2] On 26 October 2026, OpenAI announced that its internal models had produced a proof of the existence of pure Yang-Mills theory and of quantum chromodynamics (QCD) in four dimensions. OpenAI described the Yang-Mills result as part of the Millennium problem and did not claim a mass gap. The proof had not been released as of 30 October 2026, and the physicists reviewing it wrote that "confinement and the mass gap remain very important open challenges."^[19][20]

Statement of the problem

Jaffe and Witten's official description states the problem as follows:^[1]

Yang-Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on R⁴ and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in [45, 35].

The two works cited in that sentence are Streater and Wightman's PCT, Spin and Statistics, and All That (1964), which sets out the Wightman axioms, and Osterwalder and Schrader's "Axioms for Euclidean Green's functions" (1973 and 1975).^[1][9][10]

The description spells out several further requirements:^[1]

In a footnote, Jaffe and Witten exclude "weak-existence (compactness)" as a solution of the existence part unless other techniques also establish properties of the limit, such as the mass gap and the axioms.^[1]

They explain the importance of the gap in terms of clustering: with a mass gap, correlations between local observables at points x and y decay exponentially in the distance |x - y|. They also list natural extensions that are not part of the prize problem, among them proving quark confinement, proving the existence of other four-dimensional gauge theories that include extra fields while preserving asymptotic freedom, and extending the results from R⁴ to arbitrary four-manifolds.^[1]

Prize rules

Under the CMI rules revised on 26 September 2018, CMI does not accept direct submissions. Before it will consider a proposed solution, the solution must be published in a qualifying outlet, at least two years must have passed since publication, and the solution must have "received general acceptance in the global mathematics community."^[4] Of the seven problems, CMI lists only the Poincaré conjecture as solved.^[3]

Physical background

Yang-Mills theory

C. N. Yang and R. L. Mills introduced the theory in 1954 in "Conservation of Isotopic Spin and Isotopic Gauge Invariance."^[5] It generalizes Maxwell's electromagnetism, whose gauge group is the abelian group U(1), to a compact non-abelian gauge group G. The curvature becomes F = dA + A ∧ A, and the resulting field equations are nonlinear, unlike Maxwell's.^[1] Classically, Yang-Mills fields describe massless waves traveling at the speed of light. Jaffe and Witten describe this as a "serious obstacle" to using the theory for the weak and strong nuclear forces, which are short-ranged.^[1]

For the weak force, the obstacle was removed in the Glashow-Salam-Weinberg electroweak theory, with gauge group SU(2) × U(1) and an added Higgs field. For the strong force the answer came from a property of the quantum theory itself: asymptotic freedom.^[1]

Asymptotic freedom and QCD

In 1973 David Gross and Frank Wilczek, and independently H. David Politzer, showed that non-abelian gauge theories become weakly coupled at short distances.^[6][7] The three shared the 2004 Nobel Prize in Physics "for the discovery of asymptotic freedom in the theory of the strong interaction."^[8] Asymptotic freedom made it possible to describe the strong force by a gauge theory with G = SU(3) coupled to quark fields, called quantum chromodynamics.^[1]

Jaffe and Witten list three quantum properties QCD must have to match the observed strong interaction, each very different from the classical theory:^[1]

They write that experiment and computer simulations give "strong encouragement" that QCD has these properties, but that there is no convincing theoretical computation, whether or not mathematically complete, demonstrating any of the three in QCD as opposed to a simplified truncation of it.^[1]

Evidence from lattice gauge theory

Kenneth G. Wilson introduced a lattice formulation of gauge theory in 1974.^[11][20] With a compact gauge group and a finite lattice, the functional integral becomes a finite-dimensional integral. Constructing the continuum theory then requires showing that expectations of gauge-invariant observables converge as the lattice spacing goes to zero and the volume goes to infinity. The Wilson action also preserves reflection positivity, which is needed to recover a quantum-mechanical Hilbert space.^[1]

Monte Carlo simulations of lattice gauge theory began in the late 1970s. Jaffe and Witten cite Creutz's 1980 Monte Carlo study of SU(2) gauge theory as an example.^[1] Lattice studies of pure SU(3) gauge theory, which contains only gluons, compute the spectrum of glueballs. Colin Morningstar and Mike Peardon's 1999 anisotropic-lattice study, for example, put the lightest (scalar, 0⁺⁺) glueball at 1730 (50)(80) MeV, using a scale of r₀⁻¹ = 410(20) MeV to convert from lattice units.^[12] The committee reviewing OpenAI's 2026 claim wrote that lattice work has provided "strong evidence for the existence of the continuum limit of the quantum Yang-Mills theory, for confinement and the mass gap."^[20] Such results are numerical evidence, not proof, and CMI's summary says no proof of the gap is known.^[2]

Axiomatic framework

A solution has to produce an object that satisfies precise axioms, not just a recipe for calculations. Jaffe and Witten describe two equivalent frameworks:^[1]

• Wightman axioms (Minkowski signature). Gårding and Wightman gave mathematically precise axioms for quantum field theories on four-dimensional Minkowski space. The Hilbert space carries a representation of the Poincaré group with positive energy (H ≥ 0). There is a vacuum vector, unique up to a phase. Field operators at points that no light signal can connect commute (or anticommute for two fermionic fields). Haag and Kastler introduced a related scheme for local functions of the field.
• Osterwalder-Schrader axioms (Euclidean signature). Osterwalder and Schrader found the condition of "reflection positivity," which lets a Euclidean-invariant field theory be continued back to a Lorentz-invariant theory on Minkowski space.^[9][10] This built on work by Kurt Symanzik and by Nelson. Much constructive work builds Euclidean theories as non-Gaussian probability measures on spaces of distributions.^[1]

In Michael Douglas's 2004 status report for CMI, establishing these axioms is "the 'existence' part of the problem," while the mass gap concerns the falloff of correlation functions with distance.^[13]

Constructive quantum field theory

"Constructive quantum field theory" is the program of building interacting field theories that satisfy the axioms. It produced the first rigorous examples in the late 1960s and 1970s, but almost all of them live in two or three space-time dimensions.^[1]

Balaban's program used renormalization transformations that integrate out high-momentum degrees of freedom on the lattice and rescale. Gauge choices vary locally and change from scale to scale.^[1] In 2013 J. Dimock published "The renormalization group according to Balaban, I. Small fields" in Reviews in Mathematical Physics, an exposition of the method.^[15] OpenAI's October 2026 announcement says its work "builds on Balaban's approach" and links to Dimock's first paper in that series.^[19]

In 2000, Jaffe and Witten summarized the state of the field: "at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions." They added that there was "ample indication" that known methods could be extended to construct Yang-Mills theory on a four-torus T⁴, but that even so, "no present ideas point the direction to establish the existence of a mass gap that is uniform in the volume," nor to take the infinite-volume limit from T⁴ to R⁴.^[1]

Status reports and recent approaches

In his April 2004 report for CMI, Douglas wrote that he knew of "no significant progress towards a mathematically rigorous construction of the quantum Yang-Mills theory" in the preceding years. In his view the state of the art remained the work of Balaban and of Magnen, Rivasseau and Sénéor. He pointed to two related lines of work: two-dimensional nonlinear sigma models, and four-dimensional supersymmetric Yang-Mills theories, including Nikita Nekrasov's instanton-counting derivation of the Seiberg-Witten solution.^[13]

Douglas published an updated review, "The Yang-Mills Millennium problem," in Nature Reviews Physics in January 2026. Its summary says the definition problem requires proving that the theory has a continuum limit, which "requires renormalization, which has never been made rigorous in the needed generality." On the mass gap, it says "it is widely felt that no clear path is known towards proving it." It names rigorous stochastic quantization and the rigorous strong coupling expansion as recent mathematical approaches, both part of probability theory.^[14] Examples include Sourav Chatterjee's 2018 survey "Yang-Mills for probabilists," which recasts parts of the problem as questions in probability,^[16] and work by Ajay Chandra, Ilya Chevyrev, Martin Hairer and Hao Shen that defines state spaces and Markov processes for the stochastic Yang-Mills heat flow in two dimensions and for the stochastic quantisation equation of Yang-Mills-Higgs theory in three.^[26][17]

AI-assisted efforts in 2026

By autumn 2026, some researchers were using large language models to try to complete Balaban's program.^[23] Scientific American quoted New York University mathematician Roland Bauerschmidt: Balaban's papers "are not complete and are extraordinarily difficult to read, but experts did not seriously doubt that they could be completed with sufficiently many years devoted to it."^[24]

Alpöge and Sussman repository

Ethan Sussman, a mathematician at Northwestern University, and Levent Alpöge of Anthropic began collaborating after Sussman emailed Alpöge on 8 September 2026, according to Scientific American.^[24] On 6 October 2026 they opened a public GitHub repository described as "an attempt with Levent Alpoge to get Claude to finish Balaban's program to construct Yang--Mills."^[18] The repository's disclaimer says it contains work in progress "written largely by LLM" that "has not been satisfactorily vetted," with novel ideas to be attributed to Claude. It contains a manuscript that, "if correct," constructs quantum electrodynamics in three dimensions in infinite volume, and a three-dimensional Yang-Mills manuscript that, if correct, verifies the Osterwalder-Schrader axioms for tr[F²]. In a note in the repository, Alpöge wrote that they appeared to have a four-dimensional existence argument on R⁴ that was "extremely long and basically not understandable as is."^[18] Sussman told Scientific American he believed the pair had been "maybe even as close as a few weeks away" when OpenAI made its announcement.^[24]

OpenAI's October 2026 claim

What was announced

On 26 October 2026 OpenAI published "Working with academic communities to build scientific understanding," announcing three results, one each in mathematics, computer science and physics. The other two concerned the local smoothing and Kakeya conjectures and a claimed refutation of the Exponential Time Hypothesis.^[19] The physics result, titled "Existence of quantum chromodynamics and completions of the Standard Model," is described as follows:^[19]

• a "mathematically rigorous existence proof of certain interacting quantum field theories containing gauge fields and fermions in four dimensions, including Quantum Chromodynamics (QCD) and a unified completion of the Standard Model of particle physics";
• the Standard Model case concerns "the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model," since the Standard Model itself "cannot be extrapolated to arbitrarily short distances";
• "as a subproblem," a proof of the existence of pure Yang-Mills theory, "which is part of the Quantum Yang-Mills Theory Clay Mathematics Institute Millennium Prize problem."

The announcement does not mention the mass gap. It credits the result to OpenAI's internal models, says the work "builds on Balaban's approach in constructive quantum field theory," and calls asymptotic freedom "the key mechanism underlying the existence of these theories as mathematically rigorous objects."^[19] OpenAI did not name the model in the post. Scientific American reported that it was the same internal model behind OpenAI's September 2026 Navier-Stokes claim.^[24]

The shepherding committee

OpenAI said that for the physics result it would form an external group and "defer to them on the release of materials," and that the group had decided "to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves."^[19] Under its stated principles, members are unpaid, sign no nondisclosure agreements and do not speak for OpenAI. OpenAI says it does not seek academic credit for the result.^[19]

The committee is chaired by David Gross, with Michael R. Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov.^[20] Jaffe co-wrote the official Millennium problem description, and Douglas wrote CMI's 2004 status report and the 2026 Nature Reviews Physics review.^[1][13][14]

In a statement posted on the Kavli Institute for Theoretical Physics website, the committee wrote:^[20]

• that it was "delighted to learn of the recent proof by OpenAI of the existence of quantum Yang-Mills theory, including coupling to fermions," which "appears to be an important step towards putting QFT on a rigorous mathematical footing";
• that Balaban's ideas were among those that "enabled OpenAI to follow the theory all the way to infinite distances and prove the existence of its proper continuum limit. But confinement and the mass gap remain very important open challenges";
• that "before we release it, we would like to be convinced that it is correct." Its first step will be to check that the definitions and problem statement in OpenAI's Lean formalization match those standard in physics and mathematics, and that the formalization aligns with the informal proof. It will then release the Lean proof;
• that it will later release its assessment "together with the full 1147-page LLM proofs";
• that it has "no responsibility to OpenAI."

What the claim does and does not cover

Under the official statement, a proof of existence without a mass gap would not solve the Millennium problem. CMI continued to list the problem as unsolved.^[2][3]

Reactions

• AGMAI. On 26 October the Advisory Group on Mathematics and Artificial Intelligence (AGMAI), an independent group whose members include Edward Witten, called on OpenAI to publish "without delay" the natural-language and Lean outputs. It argued that since OpenAI said it had delayed release only until the Lean formalization was complete, "there is no scientific reason for delaying the release any further." A day earlier, AGMAI had recommended that AI companies stop designating particular mathematicians to receive, check and present AI output, whether or not others get access at the same time.^[21] See Advisory Group on Mathematics and Artificial Intelligence.
• Committee member Michael Douglas called the result "a major advance" in comments to Scientific American, saying that mathematicians had made steady progress on rigorous quantum field theory for over 50 years but that "real-world theories like quantum chromodynamics always seemed out of reach" until now.^[24]
• Peter Woit wrote on his blog Not Even Wrong, hosted by Columbia University's mathematics department, that "it's hard to know exactly what they're claiming to have proved." In a reply to a commenter he called the omission of the mass gap "intentionally deceptive" and criticized the decision to keep the material private, noting that the committee's statement gave no information on the claimed Standard Model completion.^[22]
• Levent Alpöge wrote on X, as quoted by Scientific American, that "instead of contacting [Sussman] to work with him, as has been suggested publicly many times, the people at OpenAI chose to do this."^[24]
• Tristan Buckmaster of New York University disputed OpenAI's timeline, telling Scientific American: "There is no way it took one month to Lean-verify Yang-Mills. It takes, at most, a couple of days."^[24]

Scientific American reported that OpenAI claimed only the existence part of the problem, and The Next Web reported that the physics group chose not to release the AI-produced material for now.^[23][25]

Status as of 30 October 2026

No manuscript, theorem statement or Lean code for OpenAI's physics claim had been made public. The shepherding committee said it would release the Lean proof only after checking that its definitions matched standard ones. No independent verification had been reported, and CMI listed the Yang-Mills problem as unsolved.^[2][20] Even if the existence claim holds up, the mass gap would remain open, as would quark confinement, which Jaffe and Witten list as an extension beyond the prize problem.^[1][20]

Continue reading

• OOpenAI Navier-Stokes Proposed SolutionOpenAI Navier-Stokes Proposed Solution is a proposed resolution of the Navier-Stokes existence and smoothness Millennium Prize Problem announced by OpenAI on September 8, 2026.Listed in See also
• EExponential Time HypothesisThe exponential time hypothesis (ETH) is a conjecture in computational complexity theory about how fast the 3-SAT problem can be solved.Listed in See also
• OOpenAI Mathematics CollectionThe OpenAI Mathematics Collection is a public collection of mathematical manuscripts and proof artifacts produced by an internal OpenAI model. OpenAI announced the release on October 6, 2026.Listed in See also
• MMathematical reasoning in AIMathematical reasoning in AI is the ability of computer systems to solve mathematical problems: carrying out multi-step calculations, proving theorems, and answering competition or research questions that…Listed in See also
• AAutomated Theorem ProvingAutomated theorem proving (ATP) is the use of computer programs to find or check mathematical proofs, expressing both the statement and the proof in a formal_logic so that correctness can be decided…Listed in See also
• AAI for ScienceAI for science is the use of artificial intelligence, particularly deep learning and large language models, to accelerate scientific discovery across biology, chemistry, physics, materials science…Listed in See also

More to read

• OpenAI
• Lean (Theorem Prover)
• Advisory Group on Mathematics and Artificial Intelligence
• Kakeya Conjecture
• Local Smoothing Conjecture
• Terence Tao

Topics:AI for ScienceMathematicsOpenAI

References

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18. ^1 ^2 ^3Ethan Sussman. Yang--Mills-LLM-experiment (GitHub repository, created 6 October 2026). Retrieved 30 October 2026.
19. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9 ^10 ^11 ^12 ^13 ^14OpenAI. Working with academic communities to build scientific understanding. 26 October 2026.
20. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8 ^9 ^10 ^11 ^12Michael R. Douglas, David J. Gross (Chair), Arthur Jaffe, Igor R. Klebanov and Nikita Nekrasov. Proof by OpenAI of the Existence of Quantum Yang-Mills theory: Statement by the shepherding committee. Kavli Institute for Theoretical Physics, October 2026.
21. ^1 ^2Advisory Group on Mathematics and Artificial Intelligence. Recommendation for Immediate Release of Yang-Mills AI Output (26 October 2026) and "Disadvantages of Designating Mathematicians for AI Output Releases" (25 October 2026). Retrieved 30 October 2026.
22. ^Peter Woit. Secret Unified Physics Slop. *Not Even Wrong*, 26 October 2026 (with later updates).
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24. ^1 ^2 ^3 ^4 ^5 ^6 ^7 ^8Joseph Howlett. OpenAI just scooped mathematicians' work on another million-dollar problem. *Scientific American*, 27 October 2026.
25. ^Ana Maria Constantin. OpenAI says its AI model has disproved the Exponential Time Hypothesis. The Next Web, 26 October 2026.
26. ^Ajay Chandra, Ilya Chevyrev, Martin Hairer and Hao Shen. "Langevin dynamic for the 2D Yang-Mills measure." 2020. arXiv:2006.04987

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