What a complete solution must establish
“Progress toward the mass gap” can mean several different things. This map separates the obligations of the full problem and connects each one to lessons and relevant research. Select an obligation to inspect its meaning, its difficulty, and the remaining steps.
Each ingredient must belong to the same physical theory, with compatible limits and assumptions.
Full four-dimensional target remains open
A physical quantum theory
What exactly must exist?
A nontrivial quantum Yang–Mills theory on four-dimensional spacetime, with a positive physical Hilbert space, vacuum, observables, and time evolution. A Euclidean route must justify reconstruction from its correlation functions, including the relevant positivity, regularity, and covariance properties.
The difficulty
A formal path integral does not supply these objects. Gauge-fixed propagators and an indefinite auxiliary state space require a careful route to physical observables.
How the obligations connect
The continuum and large-volume limits must preserve the properties needed for reconstruction.
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This opening selection does not contain a result establishing this obligation for the full target. The absence of a linked record is not a claim that the field has made no progress.
Learn the background: Reflection positivity & reconstructionFull four-dimensional target remains open
The correct short-distance theory
Why is the limit Yang–Mills?
The continuum construction must have the prescribed Yang–Mills short-distance behavior, not merely produce some massive quantum field theory. Control must be compatible with gauge symmetry and with the renormalization trajectory.
The difficulty
Perturbative asymptotic freedom describes a weak-coupling expansion. Controlling the remainder and the full measure across all relevant scales is a separate task.
How the obligations connect
Ultraviolet estimates support removal of the regulator and identification of the resulting theory.
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Learn the background: Dimensional transmutationFull four-dimensional target remains open
Remove the ultraviolet regulator
What survives as the lattice spacing vanishes?
Construct suitable renormalized gauge-invariant observables and show that they have a controlled limit as the ultraviolet cutoff is removed. Specify the topology of convergence and how physical scales are fixed.
The difficulty
A theorem valid at one lattice spacing, or only at strong bare coupling, does not establish the required continuum scaling regime. Subsequence limits alone may leave identification or uniqueness questions.
How the obligations connect
Regulator removal must be coordinated with volume control, nontriviality, and reconstruction.
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Learn the background: Taking the continuum limitFull four-dimensional target remains open
Control infinite volume
Does the physics survive arbitrarily large boxes?
Construct the theory on the full space and control estimates as the finite-volume regulator is removed. Specify boundary conditions, the selected vacuum, and which bounds remain uniform as the box grows.
The difficulty
Finite boxes can generate a low-energy scale proportional to the inverse box size. Such a scale may vanish in infinite volume even if every finite-box theory has a positive lowest excitation.
How the obligations connect
A physical gap needs lower bounds that survive the limiting procedure, not just finite-box discreteness.
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Learn the background: The finite-volume trapFull four-dimensional target remains open
Keep a nontrivial theory
Is there still physical content in the limit?
Show that the limiting construction has genuine nontrivial gauge-invariant observables and the intended quantum dynamics. A vacuum-only theory cannot satisfy the intended existence problem.
The difficulty
Convergence is compatible with losing all observable fluctuations or drifting to an unintended limit unless this is ruled out. A normalization choice must retain the intended physical scale.
How the obligations connect
Nontriviality links identification of the limiting theory to its observable correlations and spectrum.
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This opening selection does not contain a result establishing this obligation for the full target. The absence of a linked record is not a claim that the field has made no progress.
Learn the background: The exact targetFull four-dimensional target remains open
A positive physical spectral gap
Is every physical excitation separated from the vacuum?
For the vacuum Hamiltonian H with vacuum energy set to zero, require a positive Δ such that the physical spectrum contains no energies in (0, Δ). The assertion concerns the full relevant physical spectrum, not only the states seen by one chosen operator.
The difficulty
One exponentially decaying correlator can miss a lighter state. A gap in an auxiliary stochastic generator has a different meaning. Estimates must be strong enough for the full physical theory and survive regulator removal.
How the obligations connect
Reconstruction translates suitable Euclidean spectral information into a statement about physical time evolution.
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Learn the background: What a correlator cannot seeFull four-dimensional target remains open
Every compact simple gauge group
How broad is the result?
The Clay problem asks for the construction and gap for every compact simple gauge group. A result for one group, a collection of simulated ranks, or only an infinite-rank limit does not settle that quantifier.
The difficulty
Methods based on a nontrivial center or a large-N approximation have hypotheses that must be stated. They cannot automatically be transferred to all compact simple groups.
How the obligations connect
Group assumptions apply to every other part of the construction and gap argument.
Related work in this atlas
This opening selection does not contain a result establishing this obligation for the full target. The absence of a linked record is not a claim that the field has made no progress.
Learn the background: Read a claim at its true scopeThese are connected proof obligations, not percentages of a solution or a mandatory sequence of steps. A linked result can illuminate an obligation in a restricted setting without completing it. Read the official problem statement ↗