Mathematical physics, constructive quantum field theory, and lattice gauge theory

Yang–Mills Existence and Mass Gap Problem

Collaboration beta

Can four-dimensional quantum Yang–Mills theory be constructed rigorously so that its vacuum is isolated from every excited state by a positive amount of energy? The source reports several exact lattice-scale footholds, but it does not claim the required continuum theory or mass gap.

spec H \subseteq {0} \cup [Δ, \infty), Δ > 0.

Clay Millennium Prize Problem: Yang--Mills and the Mass Gap

Known results and sources

The problem asks whether a nontrivial four-dimensional Yang–Mills theory exists whose vacuum is separated from all excitations by a positive gap that survives removal of the lattice cutoff.

Research problem

Exact mathematical statement

For every compact simple Lie group $G$ , construct a nontrivial four-dimensional quantum Yang–Mills theory on $\mathbb R^4$ satisfying the required quantum-field-theory axioms, with physical Hamiltonian $H$ having a unique vacuum and a strictly positive mass gap:

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

Here $0$ is an isolated vacuum energy; the containment asserts that there is no positive spectrum below $\Delta$ , without asserting that every energy above $\Delta$ belongs to the spectrum. The proof must establish existence and nontriviality of the continuum theory and a gap that remains positive in physical units as the ultraviolet cutoff is removed. A fixed-cutoff lattice gap is not enough. The current v8 source explicitly states that no complete RG iteration, continuum construction, or proof of the full problem is claimed.

Problem infographic

Problem at a glance

A problem-first scientific diagram follows a regulated four-dimensional Yang–Mills lattice toward a continuum theory and an open spectral-gap question, separating the two required achievements from finite-volume footholds and unresolved bridges. — A solution must construct a nontrivial continuum Yang–Mills theory and prove a vacuum gap that remains positive in physical units. The source reports exact finite-volume shell geometry and bounds, but the fixed-base defect expansion, continuum landing, landing-action certificate, and physical transfer remain open.

Current mathematical picture

Where work on Yang–Mills Existence and Mass Gap Problem stands

Open problem

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The current route abandons the nonlinear quaternionic polar-average coarse map because of its singularities, moving reconstruction maps, implicit corrections, and Jacobian terms. The v8 audit also invalidates the stronger interpretation of the bad-pivot numerator bound: it cannot be called a completed polymer logarithm until a family-independent fixed-base replacement ratio is proved. The immediate junction is to replace bad-component numerators by genuine activity ratios over one fixed all-good restricted measure, assemble a common shell/coarse/transition contour hull, and prove stability of an interacting pivot shell with the correct relevant-coupling recurrence. Even that would still have to be iterated to a nontrivial continuum landing action, connected to a positive finite mass certificate, and transferred through Osterwalder–Schrader reconstruction for every compact simple group.

Route status · Narrowed route

Main reductionCurrent reduction

The program separates the problem into an ultraviolet half and an infrared half. First, a constructive gauge-covariant renormalization-group iteration must produce a nontrivial continuum theory and land at a fixed physical scale without hiding massless shell modes. Second, the landing action must satisfy a finite boundary-completed auxiliary-Hamiltonian certificate whose gap transfers through the shell martingale and Osterwalder–Schrader reconstruction to a physical mass gap. The active ultraviolet subroute uses a block-tree quotient, straight retained links, 45 pivot-plaquette shell coordinates per coarse site, an exact reference normalization, and an exact conditional bad-set numerator decomposition whose fixed-base polymer logarithm remains open.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.Task status · Ready to work on

Work mapped so far

Yang–Mills Existence and Mass Gap Problem in numbers

4.6kretained lines of mathematical investigation4,561 in the current working snapshot

Argument development: 3,545 · 78%
Explored or eliminated routes: 52 · 1%
Computational analysis: 255 · 6%
Open obligations: 272 · 6%
Definitions and setup: 437 · 10%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byArthur Jaffe, Edward WittenProposed2000Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.

Suggested move: Start with shell-only marks: define core, guard, transition, and hard radii; prove the conditional minimizer stays in the guard cell; and establish a denominator lower bound with no unmatched power of the coupling.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.Suggested move: Start with shell-only marks: define core, guard, transition, and hard radii; prove the conditional minimizer stays in the guard cell; and establish a denominator lower bound with no unmatched power of the coupling.

Ready to work on

Assemble one common shell/coarse/transition hull and prove the interacting pivot-shell stable-manifold theorem with the corrected factor-two relevant recurrence.Suggested move: After the shell-only ratio is controlled, add each coarse mark through its unique three witness pivots, close every residual halo exactly once, and bound the signed generated-interaction norm through the required derivative orders.

Ready to work on

Iterate the ultraviolet construction to a nontrivial continuum landing action, prove its positive finite mass certificate, and transfer that gap to the Osterwalder–Schrader Hamiltonian for every compact simple group.Suggested move: First extract the correctly normalized one- and two-loop coefficients and contract irrelevant interactions; then establish universal landing, certify the landing action, and prove physical-unit clustering and reconstruction.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen problem

The Clay Millennium problem remains open. No construction is known of a nontrivial quantum Yang--Mills theory on R^4 for every compact simple gauge group with the required axiomatic strength, and no positive Hamiltonian spectral gap has been proved for such a constructed theory. Rigorous fixed-lattice strong-coupling results, two- and three-dimensional constructions, supersymmetric or matter-coupled theories, and numerical spectra do not settle either complete four-dimensional target.

[3][2][9]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Authoritative summaryDouglas's peer-reviewed review and Clay's maintained problem page continue to report the axiomatic four-dimensional construction and mass gap as unsolved, while identifying stochastic quantization and rigorous strong-coupling lattice analysis as active advances.[9][3]
2024
Peer reviewedChandra, Chevyrev, Hairer, and Shen constructed a renormalized local-in-time stochastic Yang--Mills--Higgs flow in three dimensions with gauge covariance and a Markov process on gauge orbits up to possible finite-time blow-up; it is not a global four-dimensional Yang--Mills measure or mass-gap result.[8]
2023
Peer reviewedShen, Zhu, and Zhu proved, in explicit strong-coupling ranges for lattice SO(N) and SU(N), uniqueness of the infinite-volume measure, functional inequalities, and exponential correlation decay. The lattice spacing is fixed, so no four-dimensional continuum limit follows.[7]
2003
Peer reviewedLévy constructed the Yang--Mills measure on compact orientable surfaces, a rigorous two-dimensional continuum special case that does not supply the four-dimensional Clay construction or its mass-gap theorem.[6]

Mathematical neighborhood

Related results and reusable starting points

Current focusYang--Mills existence and mass gap

Dependency or reductionAxiomatic continuum existence on R^4

The first Clay obligation is a nontrivial four-dimensional continuum quantum field theory satisfying axioms at least as strong as Wightman or Osterwalder--Schrader and the required short-distance behavior. A gap statement for an unconstructed target theory is insufficient.

[2][9]

Dependency or reductionPositive Hamiltonian mass gap

After constructing the theory, one must prove that its Hamiltonian has no spectrum in an interval (0, Delta) for some Delta greater than zero. Existence without this spectral theorem is not a solution.

[2]

Dependency or reductionLattice-to-continuum and infinite-volume limits

A lattice route must control both infinite volume and lattice spacing tending to zero, preserve nontriviality and the axioms, and keep gap estimates uniform enough to survive the continuum limit. Fixed-spacing theorems do not complete this bridge.

[2][4]

Solved special caseStrong-coupling lattice Yang--Mills

For lattice SO(N) and SU(N) gauge theories in stated strong-coupling ranges, there is a unique infinite-volume measure with functional inequalities and exponential decay. The theorem is at fixed lattice spacing and does not prove the continuum target.

[7]

Solved special caseTwo-dimensional Yang--Mills measure

The Yang--Mills measure is rigorously constructed on compact orientable two-dimensional surfaces. Dimension two is structurally special and this construction is not the four-dimensional Clay theory or its paired mass-gap result.

[6]

Weaker or relaxed formThree-dimensional stochastic Yang--Mills--Higgs

Renormalized stochastic quantization is constructed locally in time for three-dimensional Yang--Mills--Higgs, with a gauge-orbit Markov process up to possible blow-up. It neither supplies a global invariant measure nor reaches four-dimensional pure Yang--Mills.

[8]

Related problemNumerical glueball spectrum

Monte Carlo glueball spectra in pure SU(3) lattice gauge theory give quantitative physical evidence for massive excitations but are not a constructive or spectral proof for the continuum axiomatic theory.

[10]

Related problemConditional Seiberg--Witten formalization

The Lean-encoded Seiberg--Witten solution concerns a supersymmetric N=2 SU(2) theory and derives conclusions from explicit physical postulates. It is a neighboring formal-methods experiment, not a construction of four-dimensional pure Yang--Mills.

[12][13]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal statement · partial resource linkedFormal Conjectures issue 2365: Yang-Mills Existence and Mass Gap
The open issue proposes a Lean formalization and is labeled as needing prerequisites; it does not link a merged Lean declaration or proof of the Millennium statement.
[11]
formal library support · partial resource linkedLean encoding of the Seiberg--Witten solution
A neighboring supersymmetric genus-one construction is encoded from explicit physical postulates. Its paper states that it does not construct the interacting theory and does not claim the pure Yang--Mills Millennium result.
[12][13]
computation · not independently reproducedAnisotropic-lattice SU(3) glueball spectrum
Morningstar and Peardon used Monte Carlo calculations at several lattice spacings and volumes to estimate the low-lying pure-gauge SU(3) glueball spectrum. This collection did not rerun it, and numerical spectrum evidence is not proof of the Clay statement.
[10]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA proof-assistant library for compact simple Lie groups, principal bundles, connections, curvature, gauge transformations, and gauge-invariant observables at the needed analytic level.
Formalization targetFormal Wightman and Osterwalder--Schrader quantum-field-theory axioms, operator-valued distributions, reflection positivity, locality, Hilbert-space reconstruction, and Hamiltonian spectral theory.
Formalization targetA rigorous formal treatment of renormalization and the ultraviolet continuum limit in four-dimensional nonabelian gauge theory, including asymptotic freedom and nontriviality.
Formalization targetFormal infinite-volume and lattice-spacing limits with uniform estimates connecting lattice correlation decay to the reconstructed continuum Hamiltonian gap.
Formalization targetAn exact merged formal statement that quantifies over every compact simple gauge group and keeps continuum existence and the positive mass gap as separate required conclusions.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction1 of 91
lemma7 of 97

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementYang–Mills Existence and Mass Gap Problem
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementThe Gaussian shell has a uniform spectral bandintermediate
retained route statementThe compact shell admits global product-Haar pivot coordinatesintermediate
retained route statementFine Wilson energy controls retained coarse curvatureintermediate
retained route statementThe small-curvature pivot cell is uniformly strongly convexintermediate
retained route statementBad-component numerators are bounded, but the polymer logarithm is not closedintermediate
retained route statementEach coarse plaquette has three unique witness pivotsintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetProve the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.open
Research targetAssemble one common shell/coarse/transition hull and prove the interacting pivot-shell stable-manifold theorem with the corrected factor-two relevant recurrence.open
Research targetIterate the ultraviolet construction to a nontrivial continuum landing action, prove its positive finite mass certificate, and transfer that gap to the Osterwalder–Schrader Hamiltonian for every compact simple group.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
Narrowed routeSource-reported limitationThe current route abandons the nonlinear quaternionic polar-average coarse map because of its singularities, moving reconstruction maps, implicit corrections, and Jacobian terms. The v8 audit also invalidates the stronger interpretation of the bad-pivot numerator bound: it cannot be called a completed polymer logarithm until a family-independent fixed-base replacement ratio is proved. The immediate junction is to replace bad-component numerators by genuine activity ratios over one fixed all-good restricted measure, assemble a common shell/coarse/transition contour hull, and prove stability of an interacting pivot shell with the correct relevant-coupling recurrence. Even that would still have to be iterated to a nontrivial continuum landing action, connected to a positive finite mass certificate, and transferred through Osterwalder–Schrader reconstruction for every compact simple group.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve the buffered fixed-base restricted-ensemble replacement theorem with a family-independent good measure and uniform activity-ratio bounds.

Yang–Mills Existence and Mass Gap Problem · ready to start

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Sources and references13 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Conservation of Isotopic Spin and Isotopic Gauge Invarianceoriginal source · C. N. Yang, R. L. Mills · Physical Review · 1954 · DOI 10.1103/PhysRev.96.191 · accessed Aug 7, 2026
2
Quantum Yang--Mills Theoryoriginal source · Arthur Jaffe, Edward Witten · Clay Mathematics Institute · 2000 · accessed Aug 7, 2026
3
Yang--Mills & the Mass Gapmaintained problem list · Clay Mathematics Institute · accessed Aug 7, 2026
4
Gauge field theories on a latticepeer reviewed result · Konrad Osterwalder, Erhard Seiler · Annals of Physics · 1978 · DOI 10.1016/0003-4916(78)90039-8 · accessed Aug 7, 2026
5
Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensionspeer reviewed result · Tadeusz Balaban · Communications in Mathematical Physics · 1987 · DOI 10.1007/BF01215223 · accessed Aug 7, 2026
6
Yang--Mills Measure on Compact Surfacespeer reviewed result · Thierry Lévy · Memoirs of the American Mathematical Society · 2003 · ARXIV math/0101239 · DOI 10.1090/memo/0790 · accessed Aug 7, 2026
7
A stochastic analysis approach to lattice Yang--Mills at strong couplingpeer reviewed result · Hao Shen, Rongchan Zhu, Xiangchan Zhu · Communications in Mathematical Physics · 2023 · DOI 10.1007/s00220-022-04609-1 · accessed Aug 7, 2026
8
Stochastic quantisation of Yang--Mills--Higgs in 3Dpeer reviewed result · Ajay Chandra, Ilya Chevyrev, Martin Hairer, Hao Shen · Inventiones Mathematicae · 2024 · DOI 10.1007/s00222-024-01264-2 · accessed Aug 7, 2026
9
The Yang--Mills Millennium problemsurvey or monograph · Michael R. Douglas · Nature Reviews Physics · 2026-01-12 · DOI 10.1038/s42254-025-00909-2 · accessed Aug 7, 2026
10
The glueball spectrum from an anisotropic lattice studypeer reviewed result · Colin J. Morningstar, Mike Peardon · Physical Review D · 1999 · ARXIV hep-lat/9901004 · DOI 10.1103/PhysRevD.60.034509 · accessed Aug 7, 2026
11
Formal Conjectures issue 2365: Yang-Mills Existence and Mass Gapformalization · Google DeepMind · GitHub · 2026-02-19 · accessed Aug 7, 2026
12
Axioms for physical reasoning: codifying the Seiberg--Witten solution in Leanpreprint · Michael R. Douglas · arXiv · 2026-07 · ARXIV 2607.06379 · accessed Aug 7, 2026
13
Seiberg--Witten Lean repositoryformalization · Michael R. Douglas · GitHub · 2026 · accessed Aug 7, 2026

Important qualifications

This record keeps the two Clay obligations separate: axiomatic construction of a nontrivial four-dimensional continuum quantum Yang--Mills theory and proof of a positive Hamiltonian spectral gap for that theory.
The selected literature is not an exhaustive history of constructive quantum field theory, renormalization, lattice gauge theory, or physics evidence.
Fixed-lattice strong-coupling theorems, two- or three-dimensional constructions, supersymmetric or matter-coupled models, and numerical glueball spectra are retained only with their stated scope and do not resolve the four-dimensional pure-theory target.
No recent self-posted full-solution manuscript was promoted to a milestone because the maintained Clay page and the 2026 peer-reviewed review continue to report the problem as unsolved. This is not a claim that every manuscript was exhaustively reviewed.
The cited Monte Carlo computation was not rerun or independently reproduced in this collection and is not treated as proof of either Clay component.
The scoped formalization search found an open proposal issue and a conditional neighboring supersymmetric formalization, but no merged exact formal statement or proof of the Clay target. This does not establish nonexistence in every proof assistant or private project.
No unreviewed source material, submitted mathematical claim, contributor estimate, attachment, code, or packet computation was inspected or used as external authority. This record has no proof, novelty, review, acceptance, visibility, publication, or deployment authority.

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