Explainer
OpenAI claims a proof that Yang–Mills theory exists
OpenAI says its models have proved that four-dimensional quantum Yang–Mills theory exists, one half of a Millennium Prize Problem; the mass gap is not claimed. The 1,147-page proof has not been released and is being checked by a committee chaired by David Gross.
Key points
- OpenAI says its models proved that four-dimensional quantum Yang–Mills theory and QCD exist, half of a Millennium Prize Problem. The mass gap is not claimed.
- The mathematical task: start from Wilson’s lattice integral, which is well defined, take the limits of zero grid spacing and infinite volume, and show the result is an interacting quantum theory of fields that meets the standard axioms.
- None of the 1,147 pages has been released. A committee chaired by David Gross will publish them once it has checked them, while leading mathematicians call for immediate release.
- If correct, it would be a landmark in mathematics, but it would change no prediction of particle physics.
In this article
- The news What OpenAI announced
- What is a quantum field?
- What happens when the grid disappears?
- Where do forces come from?
- What if the phase is a matrix?
- What does the commutator do?
- How do we make it quantum?
- Is that integral defined?
- How do we make it finite?
- How should the coupling change as the grid shrinks?
- What does constructing the theory mean?
- How can the grid be removed?
- Is the limit a quantum theory?
- Could the limit lose the interaction?
- What does the mass gap add?
- Back to the claim Where OpenAI’s claim sits
- Back to the claim Release, and a dispute about it
- Back to the claim A parallel public effort
- Back to the claim What would change
- Back to the claim What to watch next
The news
What OpenAI announced
On 26 October 2026 OpenAI said its internal models had proved that quantum Yang–Mills theory exists in four space-time dimensions, and that quantum chromodynamics (QCD), the theory of the strong nuclear force, does too. A formal proof of the Yang–Mills result was checked in Lean, a computer proof assistant, on 16 October. This is one half of a Millennium Prize Problem; the other half, the mass gap, is not claimed. None of the 1,147 pages has been released: a committee chaired by the Nobel laureate David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, is checking them first.
What exactly would have been proved? The cards below answer that one question at a time, starting from quantum mechanics: what a quantum field is, why Yang–Mills theory is hard to define, how one tries to define it, and what “exists” has to mean. After them we return to the claim.1,2,11
What is a quantum field?
You know how to quantize one particle: a Hilbert space and a Hamiltonian \(H\). How would you quantize something spread through space, such as the electric field? The natural attempt is to give every point its own quantum system and couple each to its neighbours. On a grid of sites \(x\), with one oscillator per site and \([\varphi_x,\pi_y]=i\delta_{xy}\), the simplest choice is \[ H = \sum_x \Bigl[\tfrac12\pi_x^2 + \tfrac12\sum_{j}(\varphi_{x+e_j}-\varphi_x)^2 + \tfrac12 m^2\varphi_x^2\Bigr]. \] The middle term ties neighbours together, so a disturbance travels as a wave. \(H\) splits into independent normal modes, and one unit of excitation of a mode is what we call a particle. On a finite grid this is ordinary quantum mechanics and nothing is in doubt. The trouble starts when we ask for real space, which has no grid.
What happens when the grid disappears?
Shrink the spacing toward zero and watch a single site: its fluctuations grow without bound. From now on we take time imaginary, \(t=-i\tau\), which turns the oscillating \(e^{-itH}\) into the decaying \(e^{-\tau H}\) and quantum amplitudes into averages. In four-dimensional space-time the field without mass or interactions then has \[ \langle\varphi(x)\varphi(y)\rangle = \frac{1}{4\pi^2\,|x-y|^2}, \] which blows up as \(y\to x\). So \(\varphi(x)\) is not an operator at a point but a distribution, meaningful only when averaged over a small region. That is harmless until particles interact. Interactions multiply fields at the same point, as in \(\varphi(x)^4\), and a product of distributions at a point has no meaning. This is where the infinities of quantum field theory come from. Keep it in mind; first we need to know which interacting field Yang–Mills theory is, and why that one.
Where do forces come from?
Quantum mechanics already has a symmetry: multiplying a wavefunction by a phase \(e^{i\chi}\) changes nothing observable. Suppose we demand more, that the phase may be chosen independently at every point, \(\psi(x)\mapsto e^{i\chi(x)}\psi(x)\). Derivatives then break the symmetry, because \(\partial_\mu(e^{i\chi}\psi)=e^{i\chi}(\partial_\mu\psi+i\,\partial_\mu\chi\,\psi)\) has an extra term. The only repair is a new field \(A_\mu\) that shifts by \(\partial_\mu\chi\) to cancel it, so that \(\partial_\mu-iA_\mu\) behaves. Part of \(A_\mu\) can be changed at will by choosing \(\chi\); the part that cannot is its curvature, \[ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu . \] This is electromagnetism: the components of \(F\) are the electric and magnetic fields, and its particles are photons. Demanding the symmetry, called gauge symmetry, forced a force into existence. Phases commute, and as a result photons do not act on one another directly.5
What if the phase is a matrix?
Particles can carry an internal state with several components, such as the three “colours” of a quark. The analogue of a phase is then a unitary matrix that mixes the components, from a compact group such as \(\mathrm{SU}(2)\) or \(\mathrm{SU}(3)\). In 1954 Chen Ning Yang and Robert Mills asked what happens if this matrix, too, may vary from point to point. The same argument goes through and forces a potential \(A_\mu\), now matrix-valued. But matrices do not commute, and the curvature gains a term: \[ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu,A_\nu]. \] That one commutator is the whole difference from electromagnetism. With \(\mathrm{SU}(3)\) and quarks added, this is quantum chromodynamics (QCD), the theory of the strong force; its particles, the analogues of photons, are gluons.5
What does the commutator do?
Classically the field is governed by its action, the total size of its curvature, \[ S[A] = \frac{1}{2g^2}\int_{\mathbb R^4}\sum_{\mu,\nu}\|F_{\mu\nu}(x)\|^2\,d^4x, \] where \(\|F\|^2=-\tr(F^2)\) is unchanged when the matrix is rotated, and the number \(g\), the coupling, will set the strength of the interaction. To see what the commutator does, write \(A=g\,a\) and expand. Without it the action is quadratic in \(a\): linear wave equations, whose waves pass through each other undisturbed, a free theory. The commutator adds terms cubic and quartic in \(a\), of size \(g\) and \(g^2\), so waves of the field scatter off each other: gluons interact with gluons even with no matter present. Yang–Mills theory is therefore interacting, and we saw where that leads once the grid disappears: products of rough fields at a point.6
How do we make it quantum?
For one particle there is a route from an action to quantum mechanics that is a theorem. In imaginary time the Feynman–Kac formula writes \(e^{-\tau H}\) as an average over paths, built on Wiener measure, with each path weighted by \(e^{-S}\) for its classical action. The natural move is to do the same for fields, averaging over all possible fields: \[ \langle\mathcal O\rangle = \frac1Z\int \mathcal O(A)\,e^{-S[A]}\,\mathcal DA , \] where \(Z\) makes the weights sum to one. Every prediction of the theory, every correlation and every particle mass, would be an average of this kind. If this integral were defined, the theory would be. Is it?6
Is that integral defined?
No, for three reasons. The symbol \(\mathcal DA\) suggests a Lebesgue measure on the space of all fields, and none exists in infinite dimensions; for paths, Wiener measure absorbs the quadratic part of \(e^{-S}\), and for fields only the quadratic part can be absorbed that way. The rest of \(S\) multiplies fields at a point, and the typical field is a distribution, so it is meaningless. And \(S\) does not change when the matrix is changed point by point, so each physically distinct field is counted infinitely often. Physicists compute correct numbers anyway, by subtracting infinities according to fixed rules and picking one representative of each family of equivalent fields, but these recipes do not produce a measure. Hence the official description of the Millennium Prize problem, by Arthur Jaffe and Edward Witten: there is not yet even a precise definition of quantum gauge theory in four dimensions. To get one, first make everything finite.6
How do we make it finite?
Kenneth Wilson’s 1974 answer is the starting point of every construction: go back to a grid, of spacing \(a\) in a finite box. The potential’s job was to compare phases at neighbouring points, so put that comparison directly on each edge \(\ell\), as a matrix \(U_\ell\) from the group, roughly \(e^{aA_\mu}\). Multiplying the four matrices around a small square gives \(U_p\), which differs from the identity by the curvature there. The weight becomes \[ d\mu_a(U) = \frac1Z \exp\Bigl(-\beta\sum_p\bigl(1-\tfrac1N\operatorname{Re}\tr U_p\bigr)\Bigr)\prod_{\ell} dU_\ell , \qquad \beta=\frac{2N}{g^2}. \] All three obstacles are gone. There are finitely many variables, each in a compact group with its natural finite measure; nothing is multiplied at a point; and the symmetry is exact and, the group being compact, costs only a finite factor. As \(a\to0\) the weight reduces to the Yang–Mills action. The price is the grid, and the problem is now to remove it.6
How should the coupling change as the grid shrinks?
Shrink \(a\) while holding \(g\) fixed and the predictions at fixed distances drift, because fluctuations at the newly resolved short scales feed into them. To hold the physics fixed, \(g\) must be tuned with the scale. With \(\mu\) the energy scale, roughly one over the distance, the leading-order answer is \[ g^2(\mu) = \frac{1}{2b_0\ln(\mu/\Lambda)}, \qquad b_0>0 . \] The sign of \(b_0\) decides everything. Because it is positive, the coupling goes to zero at short distances, a property called asymptotic freedom, found in 1973 by Gross, Frank Wilczek and David Politzer. That is good news for the limit: near \(a=0\) the theory is almost free, and calculations that treat gluons as nearly non-interacting work and match accelerator experiments. At long distances the coupling grows, and only computer simulations work. The constant \(\Lambda\) is a mass scale the theory generates on its own.5,6
What does constructing the theory mean?
We can now state the task. Start from Wilson’s well-defined averages \(\langle\mathcal O\rangle_{a,L}\) in a box of size \(L\), tune \(g\) with \(a\) as asymptotic freedom dictates, and show that \[ \langle\mathcal O\rangle = \lim_{L\to\infty}\;\lim_{a\to0}\;\langle\mathcal O\rangle_{a,L} \] exists. The limit \(a\to0\), called the continuum limit, is a short-distance problem, where the coupling is weak. The limit \(L\to\infty\) is a long-distance problem, where the coupling is strong and no general method was known. Both are needed: a theory in a box is not a theory of space-time.6
How can the grid be removed?
Not all at once. Average the field over small blocks, giving a new theory on a grid twice as coarse, and repeat, removing fluctuations one length scale at a time; physicists call this the renormalization group. Each step can be controlled because the coupling at that scale is small. The danger is accumulation. In four dimensions the coupling at the \(k\)-th step falls only slowly, \[ g_k^2 \approx \frac{1}{2b_0\,k\ln2}, \] so errors of size \(g_k^2\) at every step would add up like the harmonic series and diverge. Controlling them is the technical heart of the problem. In the 1980s Tadeusz Balaban did this rigorously in a finite box, in papers running to many hundreds of pages, but did not finish a construction. The committee says Balaban’s ideas were among those that enabled OpenAI’s proof.2,6,7
Is the limit a quantum theory?
Suppose both limits exist. What we then hold is a family of averages in imaginary time, but we started from quantum mechanics: a Hilbert space and a Hamiltonian. Can we get back? Osterwalder and Schrader gave conditions under which we can. The averages must be symmetric under rotations and translations of \(\mathbb R^4\), which the grid broke and the limit must restore. And they must satisfy reflection positivity: for any observable \(\mathcal O\) at positive times, with \(\Theta\) reflecting time, \[ \langle\,\overline{\Theta \mathcal O}\;\mathcal O\,\rangle \ge 0 . \] This says a certain inner product is positive, which is exactly what is needed to build a Hilbert space from the averages, and with it a Hamiltonian \(H\ge0\) and a vacuum state. Wilson’s grid theory satisfies it at every spacing, so it survives the limit.5,6
Could the limit lose the interaction?
There is a cheap way to meet every condition so far. A free theory, the quadratic part of the action alone, obeys all the axioms and is easy to construct. Its correlations are all fixed by pairs: \[ \langle\varphi_1\varphi_2\varphi_3\varphi_4\rangle = \langle\varphi_1\varphi_2\rangle\langle\varphi_3\varphi_4\rangle + \langle\varphi_1\varphi_3\rangle\langle\varphi_2\varphi_4\rangle + \langle\varphi_1\varphi_4\rangle\langle\varphi_2\varphi_3\rangle . \] So the Millennium problem asks for an interacting theory, one whose correlations differ from this. The danger is real. In 2021 Michael Aizenman and Hugo Duminil-Copin proved that natural grid versions of the simplest interacting four-dimensional theory, the oscillator field we began with plus a \(\varphi^4\) term, become free as the spacing goes to zero: the interaction is lost in the limit. Asymptotic freedom is why experts expect Yang–Mills theory to escape this fate, but expecting is not proving. Interacting theories have been built in two and three dimensions; four is the hard case.6,12
What does the mass gap add?
Existence is half of the problem. The other half asks whether the Hamiltonian the axioms deliver has a gap \(\Delta>0\) above the vacuum: \[ \operatorname{spec}(H) \subset \{0\}\cup[\Delta,\infty). \] As in quantum mechanics, a gap makes correlations decay exponentially, like \(e^{-\Delta|x|}\), so the force has a finite range. That is what is observed: the strong force reaches only about the size of a proton although gluons are massless in the equations, and quarks and gluons are never seen alone (confinement). Why is it so hard to prove? Any mass the theory has must be a multiple of \(\Lambda = \mu\,e^{-1/(2b_0 g^2(\mu))}\), a function of \(g\) whose derivatives at \(g=0\) all vanish, so no expansion in powers of \(g\) can detect it. The Millennium problem asks for existence and the gap, for every compact simple matrix group.6
Back to the claim
Where OpenAI’s claim sits
In these terms, the committee says OpenAI followed the theory “all the way to infinite distances” and proved the existence of its continuum limit: both limits, \(a\to0\) and \(L\to\infty\), which would go beyond what Jaffe and Witten thought known methods could reach. The claim also covers QCD. The mass gap is explicitly not claimed.
Public documents do not yet say which axioms the limit satisfies, how it is shown not to be free, whether every compact simple matrix group is covered or only \(\mathrm{SU}(3)\) and its relatives, or exactly what theorem the Lean files prove. These are not technicalities: the physicist Peter Woit and the mathematical physicist John Baez have pointed out that a finite-box construction, or a free limit, would fall far short. The committee says its first task is to check exactly these points.1,2,6,8
Back to the claim
Release, and a dispute about it
OpenAI asked Gross on 16 October to take custody of the proof; Gross chose the other members. The committee says it works for the academic community, not OpenAI, and will release the Lean proof once validated and the full proofs later, with its assessment. The Advisory Group on Mathematics and Artificial Intelligence, nine mathematicians including Timothy Gowers, Martin Hairer and Edward Witten, called on 26 October for immediate release, arguing that the Lean version is complete and there is no scientific reason for delay. The committee says it does not want to release a result that might be wrong.2,3,9
Back to the claim
A parallel public effort
Ethan Sussman of Northwestern University and Levent Alpöge, who works at Anthropic, had been using Claude to complete Balaban’s programme in public. Their repository contains AI-written manuscripts, in their words not yet satisfactorily vetted, claiming constructions of the quantum theory of electromagnetism with electrons, and of Yang–Mills theory, in three dimensions and outlining the four-dimensional case. Scientific American reports they thought they were weeks from four dimensions when OpenAI announced. The episode raises an unsettled question: how to assign credit when AI systems finish programmes people began decades ago.3,10
Back to the claim
What would change
For physics predictions, nothing: they come from calculations in powers of \(g\), reliable at short distances where \(g\) is small, and from computer simulations on a grid, neither of which depends on an existence proof. For mathematics, a correct proof would be a landmark, the first interacting four-dimensional quantum field theory satisfying the standard axioms, and a setting in which the mass gap and confinement can be posed as theorems. For quantum computing there is an indirect link: simulating Yang–Mills-type theories on a grid is a leading proposed application of quantum computers, and such simulations must be extrapolated to zero grid spacing. Our reading is that a proof that the limits exist supports that extrapolation in principle, while saying nothing about how fast it converges or how many qubits it needs.3,4,6
Back to the claim
What to watch next
First the precise statement: definitions, axioms, gauge groups and the theorem the Lean files prove. Then the committee’s assessment, the 1,147 pages, and independent reading by mathematicians who build quantum field theories rigorously and know Balaban’s work. The three-dimensional results of the Sussman–Alpöge effort test the same methods in a simpler setting.
This article draws on OpenAI’s announcement as reported, the committee’s statement, the advisory group’s statements, the official problem description and news reports; we have not seen the proof. Update, 8 November: we have found no release of the proofs, the Lean files or a formal statement since publication on 28 October; the claim remains unverified.1,2,9
Sources & further reading
- OpenAI announcement, 26 October 2026 ↗
- Statement by the shepherding committee (Douglas, Gross, Jaffe, Klebanov, Nekrasov), October 2026 ↗
- Scientific American, OpenAI just scooped mathematicians’ work on another million-dollar problem, 27 October 2026 ↗
- The Decoder, OpenAI asks outside experts, including a Nobel laureate, to check proofs its AI produced, 27 October 2026 ↗
- Clay Mathematics Institute: Yang–Mills and the Mass Gap ↗
- Jaffe and Witten, Quantum Yang–Mills Theory (official problem description) ↗
- Scientific American, Dueling proofs from OpenAI and Anthropic astound scientists, 27 October 2026 ↗
- Peter Woit, Not Even Wrong: Secret Unified Physics Slop, 26 October 2026, and comments ↗
- Advisory Group on Mathematics and Artificial Intelligence: statements of 25 and 26 October 2026 ↗
- Sussman and Alpöge, Yang–Mills LLM experiment (GitHub repository) ↗
- Tech Times, OpenAI claims Yang–Mills proof; Nobel laureate whose discovery enables it now reviews it, 27 October 2026 ↗
- M. Aizenman and H. Duminil-Copin, Marginal triviality of the scaling limits of critical 4D Ising and φ⁴₄ models, Annals of Mathematics 194, 163 (2021); arXiv:1912.07973 ↗