Where the problem stands

Official status checked: 24 October 2026. Source: the Clay page’s “Unsolved” label; no status is inferred from the atlas.

The Clay Mathematics Institute lists Yang–Mills existence and the mass gap as unsolved. This site follows progress toward that target while keeping rigorous partial results, numerical evidence, controlled approximations, and new methods distinct.

This is a selected research map, not a claim to catalogue every relevant paper or rank every research program. A paper’s date, journal status, and significance answer different questions.

The target we are tracking

The official description by Arthur Jaffe and Edward Witten asks for a nontrivial quantum Yang–Mills theory on $\mathbb R^4$ for every compact simple gauge group, with the required axiomatic properties and a positive mass gap.

In the Hamiltonian language used in our course, after normalizing the vacuum energy to zero, a gap means

$$\operatorname{spec}(H)\subseteq\{0\}\cup[\Delta,\infty), \qquad \Delta>0.$$

The statement concerns physical, gauge-invariant states. An effective mass parameter in a gauge-fixed propagator does not automatically satisfy it. Neither does an auxiliary sampling algorithm’s relaxation rate.

Our main comparison setting is pure Yang–Mills at zero temperature and $\theta=0$. Other dimensions, matter content, deformations, and compactifications appear as related settings with their differences named.

What the current collection contributes

Direction	Representative record	What still has to be connected
Constructive ultraviolet control	Bałaban’s conditional stability result	The stated all-scale small-coupling hypothesis, a full limiting theory and its infrared spectrum
Rigorous lattice confinement	Chatterjee’s criterion and Cao–Nissim–Sheffield	The required regime and estimates through regulator removal
Stochastic construction	Chevyrev–Shen	Its three-dimensional local setting and the four-dimensional target
Positivity and exact identities	SU(3) bootstrap and loop equations	Certified long-distance and continuum conclusions
Numerical spectroscopy	Continuum-extrapolated spectra	Construction and a bound on the complete physical spectrum
Analytic mechanisms	Compactified semiclassics	Control beyond the small-compactification regime

These links lead to the primary references and the precise hypotheses behind each description.

Why there is no percentage-complete meter

Progress is not a single sequence of interchangeable milestones. An estimate can be strong in one dimension and unavailable in another. A calculation can determine a physical observable precisely while leaving the existence question untouched. A construction can control short distances without proving long-distance decay.

The useful comparison is a matrix of obligations: which theory, which regulator, which limits, which observables, and which notion of evidence? The proof map makes those obligations explicit. Its categories describe relevance; they should never be read as a tally of completed pieces that automatically assemble into a proof.

What would change this assessment?

A new claim should identify a specific theorem or controlled result and explain the change in scope. For a claimed solution, the crucial issue would be a complete construction of the required theory and spectral property, with every limiting step and assumption accounted for.

For an intermediate advance, the relevant change might instead be removal of a boundary restriction, estimates uniform in an additional parameter, a rigorous convergence result, or a reproducible improvement in accessible observables. Such developments deserve attention even when they do not close the entire problem.

Journal publication is recorded separately. It supplies bibliographic context; it does not convert a numerical result into a theorem or a theorem about a modified theory into a theorem about the target.

Choose your next step

If you have studied QED and gauge theory, begin with the problem in one page and then physical states. If you already work in the field, use the research atlas, benchmark protocol, and primary bibliography.

The updates page preserves substantive changes and previous statements. Each dossier gives its own actual read depth, examined locators, and source-check date. See the editorial principles.

Four distinct states

State	What this release records
Author claim	Scoped partial results and methods, stated in the exact source version; no full-solution claim is asserted by the selected records.
Technical assessment	Editorial source-scope review at the depths shown in each dossier; no independent expert proof certification.
Community acceptance	No consensus judgment is inferred from a journal label or the site’s reading. Attributable assessments would need separate sources.
Clay recognition	The official problem page labels the problem Unsolved, checked 24 October 2026.

The official prize-rules summary, checked on the same date, separates qualifying publication, elapsed time, and general acceptance from prize consideration. It does not make any of them an automatic award. The source dataset cannot change the status panel automatically.