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Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.

  1. 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
  2. 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
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Open2000

Yang–Mills — Existence and the Mass Gap

Must creating an excitation above the vacuum cost a positive minimum energy? Putting a successful physical theory on rigorous mathematical foundations requires both existence and a positive gap.

Challenge passport
First posed
2000
Time it held mathematicians
26 years open · as of 2026
Starting level
Research frontier

Small experiments are possible, but a complete proof sits at the frontier of modern mathematics.

The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.

Jump to your first five minutes

AI Frontier 2026

AI Frontier 2026 — a claim is not yet a theorem

Below are claims that AI labs have published about this problem. Read what is claimed together with what is not.

Open the full ledger
  • OpenAIAnnounced 2026-10-26Sub-case

    Pure Yang–Mills existence (part of a Millennium problem) — artifacts withheld

    Lean done · artifacts withheldLab claim
    Claim
    Claims rigorous existence proofs for four-dimensional interacting quantum field theories with gauge fields and fermions (QCD and a grand-unified completion of the Standard Model), including pure Yang–Mills existence as a sub-problem. Natural-language proof 2026-09-09, Lean 10-16.
    Not claimed
    The Clay problem requires existence and a mass gap; the mass gap is not claimed. The group led by David Gross chose to withhold artifacts and write it up, so outside verification is not yet possible; AGMAI called for immediate release on October 26.
    MathVoyage's call
    The new Yang–Mills page starts at open, separating the existence announcement from the full Millennium requirements without promoting it to a mass-gap solution.
    Keep my understanding and question · private draft

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

See the puzzle visually

Construct the theory → prove a positive gap

An existence announcement does not establish the second requirement. The 2026 artifacts remain unreleased.

Problem statement

For every compact simple gauge group G, construct a nontrivial quantum Yang–Mills theory on ℝ⁴ with the axiomatic properties required by the official problem, and prove a positive mass gap Δ>0. With vacuum energy set to zero, the Hamiltonian must have no spectrum in (0, Δ).

The story of this puzzle

Writing classical field equations or approximating observables on a computer is different from constructing a quantum theory in four dimensions. The official problem requires a positive mass gap as well as existence. If excitations can have arbitrarily small positive energy there is no gap; if an interval above the vacuum is empty, there is a gap.

On October 26, 2026, OpenAI announced results including existence of pure Yang–Mills theory and a Lean formalization. The announced subproblem is existence; it does not claim to settle the mass gap. The artifacts were withheld while a group led by David Gross prepares an account. Outsiders therefore cannot yet compare the manuscript’s assumptions, axioms, gauge-group scope, and Lean statement.

This page starts at open. That editorial status reflects the absence of an established resolution of the full Clay problem, not a verdict that an unreleased proof is wrong. Use the claim panel and sources below to distinguish announcement, formalization, manuscript alignment, human explanation, and community acceptance.

Try it yourself

Mini challenge

Compare the toy energy sets A={0,1,2,3,…} and B={0,1,1/2,1/3,…}. A leaves (0,1) empty, while B contains a positive entry below every Δ>0. This illustrates a spectral gap; it does not construct Yang–Mills theory.

Beyond MathVoyage

Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.