HEXAGONbeta

Yang--Mills Mass Gap in 4D: Lattice SU(NN) and Every Compact Simple Group

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.PR — Probability

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗, James Montgomery Lockwood

Version 2 / Oct 18, 2026 / CC BY 4.0

Abstract

We prove that four-dimensional lattice Yang--Mills theory with any compact simple gauge group has a unique infinite-volume state, exponential clustering and an explicit mass gap at fixed lattice spacing, past the curvature limit of the ungauged theory, with the ranges βW≤0.345\beta_W\le0.345 for SU(2) and βW≤0.76\beta_W\le0.76 for SU(3). We also prove that one estimate of Dobrushin--Shlosman type in a cube of fixed physical size gives a mass gap in physical units for every continuum limit, and that this estimate, which holds at strong coupling and remains open along the continuum tuning, needs for SU(NN) a cube of side at least of order (βW/log⁡βW)1/4(\beta_W/\log\beta_W)^{1/4}.

Provenance statement

The proofs, the text, the figures, the Lean 4 formalization, the Python and shell checks the certified SU(3) computation and the plaquette-block constants were carried out by Deep Bhattacharjee with AI assistance (Claude, Anthropic) in the Claude Code environment; the Anaxia engine of the BLLM system (release v3.0.5, https://github.com/creelie/eBLLM/releases/tag/v3.0.5) first assembled the SU(2) composite, as the paper states in a footnote; release v3.1.0 of the engine (https://github.com/creelie/eBLLM/releases/tag/v3.1.0) records the Yang-Mills runs in its failure log, whose last Yang-Mills entry is the change from single links to blocks of several links, and supplied no step of the proofs; the plaquette blocks, their SU(3) version and their constants were found and checked by the authors. The paper proves, by hand: a period-two field of straight lines in Z^4 that meets every plaquette in exactly one link, so that every free link has twelve Dobrushin partners; in the gauge fixed on those links, a unique free-boundary state, reflection positive, with exponential decay and a transfer-matrix gap m >= -log(3 beta_W) for four-dimensional SU(2) at beta_W < 1/3, the ceiling of single-link Dobrushin after any forest gauge fixing; a gauge-fixed Hessian bound 29/2 (against 16) for every SU(N) and SO(N), giving Bakry-Emery, decay and a gap for |beta| < 1/29; for four-dimensional SU(2) past one third, with blocks made of the three free links of one plaquette of the gauge (sixteen block plaquettes modulo two, found by an exact-cover search), a block Dobrushin bound whose constants (Hessian eigenvalues and third derivative of the log partition function of the von Mises-Fisher law on S^3) are certified in ball arithmetic, giving uniqueness, decay and a gap m >= -(1/2) log alpha(beta_W) for beta_W <= 0.345 (alpha(1/3) <= 0.95602, alpha(0.345) <= 0.99729); for four-dimensional SU(3) the same conclusions for beta_W <= 3/4 with m >= -log(1.33319 beta_W), from an analytic spectral-gap bound for the one-link measure and a corrector of degree two in the matrix entries evaluated at every staple sum on 2341 boxes certified in ball arithmetic; and, with the same plaquette blocks for SU(3), whose constants (eigenvalues of the Hessian of the one-link log partition function in [0.056015, 0.247316], third cumulant at most 0.182126) are certified on 841 boxes in ball arithmetic, m >= -(1/2) log alpha_3(beta_W) with alpha_3(3/4) <= 0.96521, so the gap at beta_W = 3/4 rises from 1.1e-4 to 0.0177; with a second certified cover at r = 38/25 (1063 boxes, 2 r Lambda(r) <= 1.029967) and constants on 784 boxes, alpha_3(0.76) <= 0.99950, so the blocks extend the SU(3) range to beta_W <= 0.76. The combinatorics of the line field are proved in Lean 4 by decide (seven theorems, axioms propext at most). This is a result about lattice Yang-Mills theory at fixed lattice spacing; it does not prove the continuum Yang-Mills existence and mass-gap problem of the Clay Institute, and the paper says so. Both authors approved the final version. Code, checks and Lean: release v3.4.0, https://github.com/creelie/4y-m/releases/tag/v3.4.0.
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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 18, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 22, 2026
  3. v3Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 28, 2026