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Yang--Mills Mass Gap on the Lattice and One Cube from the Continuum

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, Tristan Romanov

Version 6 / 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 and an explicit transfer-matrix gap in a forest gauge, up to βW=0.351\beta_W=0.351 for $\SU(2)$ and βW=0.77\beta_W=0.77 for $\SU(3)$. We prove that a Dobrushin--Shlosman estimate in one cube of fixed physical size gives a mass gap in every continuum limit, and that the cube needs a side of order (βW/log⁡βW)1/4(\beta_W/\log\beta_W)^{1/4} lattice spacings. We prove that the curvature of Ba{\l}aban's block averages costs action independent of the scale, that a translation of the Haar measures makes it rare at every scale for the abelian theory, and that for $\SU(2)$ every such translation transported along a forest reaches its constant on the gauge orbit of the vacuum, which fixes the one estimate that remains open.

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). The programs were written by Deep Bhattacharjee, James Montgomery Lockwood and Tristan Romanov; T. Romanov constructed the certificates of Proposition 88 and Theorem 191. All four 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. New in this version: the plaquette blocks to beta_W = 0.351 (SU(2)) and 0.77 (SU(3)), the fixed-set subsection and the coarse-defect subsection with the Haar tilt and its covariant form on a forest. 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 v4.31.0, https://doi.org/10.5281/zenodo.23895262; all versions https://doi.org/10.5281/zenodo.23444338)
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Claude CodeVersion Fable 5.1 Medium

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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 versionSupersededOct 30, 2026
  5. v5Submitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 31, 2026
  6. v6Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentNov 02, 2026