The Clay Mathematics Institute Millennium statement of the quantum Yang-Mills problem (Jaffe and Witten 2000) requires a quantum field theory on R4, for any compact simple gauge group G, with local operators in correspondence with the gauge-invariant local polynomials in the curvature F, short-distance correlators agreeing with asymptotic freedom and the prescribed operator-product-expansion singularities, a stress tensor, Wightman / Osterwalder-Schrader axiomatic properties, and a strictly positive mass gap. The statement is a content specification on the quantum field theory and does not commit to a primitive observable. It constrains what the constructed theory must deliver while leaving the construction's starting point open. The construction uses a Dirac matter field Ψ on R4 valued in C4Dirac ⊗ V, with V a finite-dimensional Hermitian bundle carrying a faithful unitary representation of the compact simple gauge group G. The connection one-form Aaμ is determined as the retarded-kernel image of the matter current on V, with the non-abelian Yang-Mills equation selected as the unique consistent nonlinear completion under the Barnich-Brandt-Henneaux local-BRST-cohomology classification. The substrate Fock space is the standard Dirac Wightman QFT on R4. The gauge-sector observables (the gauge-invariant F-polynomials, the stress tensor, the gauge-sector contribution to the Hamiltonian) are composite operators built from the connection composite on the same Fock space. The Dirac matter is construction scaffolding rather than physical matter content at the Clay-target level. The Clay-target physical Hilbert space is the gauge-invariant matter-vacuum sector of the substrate Fock space, equivalently the heavy-matter decoupling limit, on which the Clay clauses are evaluated. Each Clay clause derives from the substrate construction. The local-operator algebra for gauge-invariant F-polynomials is the composite algebra on the substrate Fock space, with the retarded-kernel functional supplying the nonlinear nonlocal composite structure. The short-distance correlators on the target sector reproduce the one-loop β-function coefficient bGpure = −(11/3) C2(G) through the matter-vacuum classical-equation-of-motion identification with standard pure Yang-Mills, and the perturbative OPE singularities through the substrate's heat-kernel-regulated matter-mode trace. The stress tensor combines the matter Dirac stress-energy with the gauge-sector composite extension, with the matter-gauge exchange cancelling on the coupled equations of motion. Matter-sector Wightman locality is established at the field level, and gauge-sector composite locality reduces to a Ward-identity argument on the gauge-invariant composite algebra. The mass gap follows from asymptotic freedom plus the colour-singlet character of the target sector at the level of the standard QCD-physics argument. The substrate-ontological starting point eliminates three principal obstructions the standard primitive-connection construction faces. Because Aaμ is a composite operator on the matter Fock space rather than a primitive observable, gauge fixing on a primitive connection is not required, the Gribov ambiguity is absent, and the Strocchi-Wightman locality-positivity obstruction on a Lorentz-covariant primitive gauge field has no target here (Wightman positivity at the matter-field level is the standard Dirac-matter positivity). Gauge symmetry is realised on Ψ as local-basis-freedom on V, and the construction is uniform across compact simple G. Clauses (C1)-(C9) follow on the matter-vacuum target sector, with the operator-theorem fill-in items WP1-WP9 named explicitly in §10 of the manuscript.
Daniel Fook Hao Tan (Fri,) studied this question.