Gauge fields on the SQG substrate and the Yang–Mills mass gap: what is proved, what is computed, and what remains open

The Clay problem on Yang–Mills theory asks for a construction of four-dimensional quantum Yang–Mills theory satisfying the axioms of quantum field theory, together with a proof of a positive mass gap. This note records what the substrate of the SQG programme (stacked honeycomb layers times a time chain) contributes to this question, and what it does not. Proved: the Wilson lattice theory on the substrate is reflection positive, so it has a positive transfer matrix and a non-negative Hamiltonian. At strong coupling it confines and has a mass gap in lattice units. The effective gauge operator of the SQG condensate along the scale axis has a gap: equal to the continuum edge or a geometric tower below it, by a sharp criterion. Computed (pre-registered Monte Carlo): Z3 and SU(3) on the substrate and on the cubic lattice show confinement at strong coupling and a finite-temperature transition at N_t = 2. On the substrate the transition is shifted by a common factor 1.17 in the coupling. A first look at the glueball channels at N_t = 8 finds the tensor channel heavier than the scalar by a factor 1.1–1.7 on both lattices (continuum 1.4), a consistency check only. Derived and checked: the strong-coupling structure of the substrate. To leading order every plaquette type has the same expectation value w(beta); the first correction orders the types by the smallest closed surfaces through them (hexagon: prisms with 8 faces; rectangles: cubes with 6), with the ratio 3/2 of the corrections between z–t and bond–t plaquettes, found as 1.52 ± 0.10 in the Z3 runs. The leading string-tension anisotropy follows (honeycomb plane 2/(3√3) of the z–t plane). At weak coupling the one-loop plaquette variances are the leverages of the plaquettes in the space of exact 2-forms: 1/2 on the cubic lattice, 0.592 (hexagon), 0.546 (bond rectangles), 0.564 (z–t) on the substrate, mean 5/9; the tadpole part of the one-loop matching accounts for a quarter of the observed shift of beta_c, and no Lambda-ratio is claimed. Not addressed: the continuum limit, where the Clay problem lies. The note claims no solution of it.