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Mass Gap of Four-Dimensional Lattice Yang--Mills Theory for Every Compact Simple Group

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

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

Version 4 / 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 transfer-matrix gap at couplings past the curvature limit of the ungauged measure, up to βW=0.345\beta_W=0.345 for $\SU(2)$ and βW=0.76\beta_W=0.76 for $\SU(3)$, in a gauge that fixes one link of every plaquette. We prove that a Dobrushin--Shlosman estimate in one cube of fixed physical size, or any exponential decay of renormalised correlations in physical units, gives a mass gap for every continuum limit. The cube estimate is open along the continuum tuning; for $\SU(N)$ it needs cubes of side at least of order (βW/log⁡βW)1/4(\beta_W/\log\beta_W)^{1/4}. For block averages, a cut-off of Ba{\l}aban's average composes across all scales on fields of small curvature for every compact group, and under the $\SU(2)$ Wilson measure the large fields form clusters of vanishing physical size.

Provenance statement

Deep Bhattacharjee produced the proofs, the text, the figures, the Lean 4 formalization and the Python and shell checks with AI assistance (Claude Opus 5.5 in Claude Code, Claude Fable 5.1, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra). All three authors read every proof and approved the final text. The Anaxia engine of the BLLM system first assembled the combination of the forest gauge with the sharp quarter behind the SU(2) theorem (doi:10.5281/zenodo.23469389; run records doi:10.5281/zenodo.23484160) and supplied no step of the proofs. The paper does not prove the cube estimate along the continuum tuning or the existence of the continuum limit, so it does not solve the Clay Millennium problem, and it says so. Tools used: Anthropic Claude Opus 5.5 (Claude Code) and Claude Fable 5.1; OpenAI GPT-6.1 Sol Ultra and GPT-6 Astra Ultra; BLLM Anaxia engine v3.0.5 and v3.1.0; Lean 4 (v4.33.0) with Mathlib; Python (NumPy, SciPy, python-flint with FLINT and Arb); C; shell. arXiv identifier: none. Formalization: https://github.com/creelie/4Y-M/blob/main/YangMills/StrongCoupling/GaugeFix.lean (release v4.8.0, https://doi.org/10.5281/zenodo.23786523; all versions https://doi.org/10.5281/zenodo.23444338)
AI provenance filesAI_PROVENANCE_YangMillsEveryGroup.md · 7 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 High

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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 versionSupersededOct 28, 2026
  4. v4Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 30, 2026