Follow the results that clarify the problem—and the assumptions that limit their reach. Each dossier connects the original paper to the concepts needed to read it.
Provenance migration · 24 October 2026. Twelve atomic claims in eight inherited dossiers separate exact source versions, assumptions, conclusions, and actual reading history. Each card reports its own source-check date and depth. This is not an exhaustive literature index or a ranking of approaches. The full four-dimensional problem remains open.
Conditional on that running-coupling hypothesis, effective densities have the form and satisfy the conditions and bounds specified in [III] section 2. Part II then states ultraviolet-stability bounds (0.1), with constants independent of renormalization step, torus and background field as specified immediately after the displayed bound.
Setting
Compact semisimple matrix Lie group G contained in U(N), as defined in [I] pp251-252; do not replace with only SU(2); None; Finite periodic torus with regularly refined lattices, from [I] introduction; no infinite-volume QFT is reconstructed here; Lattice spacing epsilon in the discrete refining sequence of [I], successive block/coarse-lattice transformations; [III] procedure stops at the unit lattice
Assumptions
The entire sequence of effective coupling constants remains in (0,gamma] with gamma sufficiently small (Part II Theorem 1). The effective-density representation and conditions are the full package of [III] section 2, not merely a generic small-coupling slogan. The associated geometric, regularity, analyticity and constant restrictions inherited from [I] and [III] remain part of the statement.
Remaining obstacle
Part II page 355 explicitly retains the coupling assumption: the proposed removal relies on the second-order perturbative result called Theorem 2 in [I], whose proof the article says had not yet been published. This review does not assert that the proof remains unpublished today; that later-history question was not resolved.
A double-trace deformation stabilizes center symmetry and permits controlled small-circle semiclassical dynamics; large-N volume independence has separate symmetry hypotheses.
Setting
SU(N); Pure gauge with double-trace deformation; R3 × S1; Continuum compactified effective description
Assumptions
Double-trace deformation; compactification and weak-coupling hierarchy; the small-circle regime cannot be extrapolated without error control.
Remaining obstacle
The deformation, twist, geometry, and regime matter. Continuing to undeformed pure Yang–Mills on R⁴ requires additional control.
Monte Carlo calculations estimate low-lying glueball masses at sampled values of N, with continuum extrapolations at fixed N and a subsequent large-N analysis.
Setting
SU(N), N = 2, 3, 4, 5, 6, 8, 10, 12; Pure gauge; Finite periodic lattice volumes with volume studies; Lattice with numerical continuum extrapolation
Assumptions
Wilson plaquette action; finite operator bases, statistical sampling and extrapolation assumptions; separate finite-volume and cutoff analyses.
Remaining obstacle
Numerical estimates depend on statistical and systematic control. They do not prove a lower bound on the complete physical spectrum or construct the continuum theory.
The framework generates direct loop equations and relations obtained by eliminating longer intermediate loops, with SU(2) applications in two, three and four lattice dimensions.
Setting
SU(2) applications; Pure gauge; Euclidean lattice; Lattice
Assumptions
Finite retained loop basis, exact input identities and elimination rules; no convergence of the full hierarchy is claimed.
Remaining obstacle
An equation-generation method is not a proof that finite truncations converge, nor a spectral or continuum construction result.
The paper reports numerical plaquette bounds from finite loop/positivity systems in two, three and four dimensions.
Setting
SU(3), with SU(2) comparisons; Pure gauge; Euclidean lattice; Lattice
Assumptions
Finite truncation and solver implementation; a numerical optimum is distinct from an independently checked rigorous certificate.
Remaining obstacle
Bounds on short-loop observables do not establish a continuum mass gap. Exact inequalities and the numerical certification of optimization outputs must be distinguished.
The source establishes uniqueness of the gauge-covariant Yang–Mills mass renormalization for local stochastic three-dimensional Yang–Mills–Higgs dynamics.
Setting
Compact Lie group in the source setup; Yang–Mills–Higgs; Three-dimensional torus; auxiliary stochastic time; Regularized stochastic dynamics
Assumptions
Compact Lie group and the specified representation on T3; non-anticipative mollifier class; finite counterterm shifts in the stated equivariant classes; local possibly explosive dynamics of Theorem 1.6.
Remaining obstacle
A stochastic counterterm is not a physical spectral gap. The result concerns three-dimensional local dynamics, not a four-dimensional continuum vacuum theory.
The method combines multiple-trace loop equations with positive-semidefinite matrices of Wilson-loop observables to restrict admissible expectation values.
Setting
SU(3), with SU(2) comparisons; Pure gauge; Euclidean lattice; Lattice
Assumptions
Specified observable basis, trace normalization and exact identities; each reflection condition retains its dimensional domain.
Remaining obstacle
Bounds on short-loop observables do not establish a continuum mass gap. Exact inequalities and the numerical certification of optimization outputs must be distinguished.
Section 3.2.2 justifies the additional twist-reflection positivity condition in two dimensions.
Setting
SU(3) in the paper’s setting; Pure gauge; Two-dimensional Euclidean lattice; Lattice
Assumptions
Two-dimensional geometry and reflection operation of section 3.2.2; no transfer to 3D or 4D is inferred.
Remaining obstacle
Bounds on short-loop observables do not establish a continuum mass gap. Exact inequalities and the numerical certification of optimization outputs must be distinguished.
Theorem 2.2 obtains a rectangular-loop bound with a static potential diverging at large separation under the defined slab-center condition.
Setting
Closed connected subgroup G of U(n), with a center-charged irreducible finite-dimensional unitary representation; Pure gauge; Lattice slabs and boxes; Lattice
Assumptions
Definition 2.1 unbroken slab center symmetry; irreducible unitary representation nontrivial on the center.
Remaining obstacle
The conclusion is a divergent static potential, not the stronger rectangular area law. Definition 2.3 uniform exponential decay is an additional premise of the separate Theorem 2.4 record, not a premise asserted here.
Theorem 2.4 obtains an area law using the exponential-correlation hypothesis of Definition 2.3, uniform under arbitrary boundary conditions.
Setting
Closed connected subgroup G of U(n), with a center-charged irreducible finite-dimensional unitary representation; Pure gauge; Lattice slabs and boxes; Lattice
Assumptions
Definition 2.3, including its quantifiers over boxes, boundary conditions and local observables; center-charged representation.
Remaining obstacle
The boundary-condition-uniform hypothesis is stronger than decay in one vacuum correlator. Its required weak-coupling continuum control is not established here.