Explicit Lattice Yang–Mills Mass Gap for Every Compact Simple Group Reducing the Continuum Gap to One Cube

Three papers and a Lean 4/Mathlib certificate on four-dimensional lattice Yang–Mills theory, by Deep Bhattacharjee, Ushashi Bhattacharya, James Montgomery Lockwood and Tristan Romanov. (1) Spacing-Uniform Lattice Inequalities Cannot Produce the Yang–Mills Mass Gap: a lattice gap uniform in lattice units forces trivial continuum two-point functions. (2) Mass Gap of Lattice Yang–Mills Theory up to the Dobrushin Ceiling: the sharp Hessian bound 4d, the sharp quarter for von Mises–Fisher laws on the three-sphere, and a lattice mass gap for SU(2) up to beta_W = 2/9. (3) Explicit Lattice Yang–Mills Mass Gap for Every Compact Simple Group Reducing the Continuum Gap to One Cube (Hexagon preprint 2610.00341). In a forest gauge, fixed on straight lines that meet each plaquette once: a unique state, reflection positivity, exponential clustering and a transfer-matrix gap for SU(2) up to beta_W = 0.345 and SU(3) up to beta_W = 0.76; for every compact simple group a gap for N|beta| < 2 kappa_G/29, with kappa_G the least Ricci curvature, extended to N|beta| < 2 kappa_G/28.840002 by T. Romanov's computer-assisted certificate (checked independently here and by J. M. Lockwood; its finite data are checked by the Lean 4 kernel). A conditional continuum step: a Dobrushin–Shlosman condition on one cube of fixed physical size, uniform along the tuning, gives a mass gap in physical units for every continuum limit. The paper shows what this condition needs and where it fails (large beta at a fixed cube size, the Gaussian field, U(1)), and builds a block-averaging route towards it: a constrained quadratic gap 537/5000 (Romanov's exact certificate), a covariant cut-off of Balaban's averaging that composes on fields of small curvature, chessboard bounds on large plaquettes under the SU(2) Wilson measure, and effective forms that are local uniformly in the scale. On the fibres of the cut-off average the constrained law is log-concave on a chart and its covariances decay exponentially with constants depending only on the group; every small-field point of a fibre lies in that chart, and the mass outside it is bounded in mean over the fibres (Theorem 126). Along its private links each cut-off average is a submersion with a constant independent of the field (Lemma 127), and for SU(2) its law under Haar measure has bounded density (Lemma 129), so a Peierls argument inside the fibre bounds the mass off the chart on every fibre through a field with small plaquettes (Proposition 130, Corollary 131); on every fibre, whatever the boundary links, the law concentrates near the least action of the fields whose averaging sites are near the identity (Lemma 132, Proposition 133, Corollary 134); comparing the fibres with the same exterior through the law of the averages, on every exterior with a completion whose plaquettes near W are small the fibres where that least action is of order one have exponentially small mass, and every other fibre carries the chart decay (Proposition 135, Corollary 136). Under the SU(2) Wilson measure, along logarithmic tuning, the fraction of bad unit cubes in a cube of fixed physical size is small with probability tending to one, uniformly over every exterior (Proposition 150). On every window of bounded size and under every exterior, the plaquettes near W are small with probability tending to one along the tuning, so the exterior has a calm completion (Corollary 152); on cubes of fixed physical size both the fibres of small mass and the exteriors with no calm completion are the typical case (Proposition 153), so they need estimates uniform in the size of the window. The constants of the fibre chart are explicit for every compact G in U(r): Cauchy estimates on a complex neighbourhood give c_3 = max(1.5·10^4 √r, dim G) and c_6 = 6.81·10^5 dim G c_3^2, so for SU(2) the Brascamp–Lieb bound on the chart holds for β ≥ 2.3·10^17 (Proposition 120); weighted operator norms make the rate and amplitude of the decay explicit and the same for every compact group, alpha_G = 2.5·10^-4 and c_9 = 1.2·10^4 (Proposition 125). An appendix records two steps of arXiv:2606.19362 that fail as stated: a gap inferred from the two-point function of one observable, and a compactly supported spectral cutoff given a positive Laplace representation. Lean: 86 theorems and lemmas, no sorry, standard axioms only. Estimates uniform in the size of the window for the fibres of small mass and the exteriors with no calm completion, the composition across scales and the existence of the continuum limit are not proved, so the continuum Yang–Mills existence and mass-gap problem remains open.