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Mass Gap of Lattice Yang--Mills Theory up to the Dobrushin Ceiling

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physics

Contributed by Deep Bhattacharjee ↗, Ushashi Bhattacharya ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

We prove that the Hessian of the Wilson action on dd-dimensional lattice gauge fields is bounded by 4d4d, sharply, so lattice $\SU(N)$ Yang--Mills theory has a unique infinite-volume state with a positive mass gap for ∣β∣<1/(8d)|\beta|<1/(8d). Every von Mises--Fisher law on the three-sphere moves by at most a quarter of its parameter in Kantorovich distance; consequently four-dimensional $\SU(2)$ theory has a mass gap for βW<2/9\beta_W<2/9, the ceiling of all single-link Dobrushin arguments. A gauge-invariant finite-box criterion for decay and the area law is also given. The continuum mass gap remains open.

Provenance statement

The proofs, the text, the figures, the Lean 4 formalization and the Python and shell checks were carried out by Deep Bhattacharjee with the help of AI assistance (Claude, Anthropic) in the Claude Code environment. The paper proves, by hand and without conditions: the sharp constant 4d in the Hessian bound for the Wilson action of SU(N), SO(N) and U(N); Bakry-Emery, log-Sobolev and Poincare inequalities, uniqueness and a lattice mass gap for |beta| < 1/(8d), and that no uniform curvature argument goes further; the "sharp quarter", W_1(nu_b, nu_b') <= |b - b'|/4 for von Mises-Fisher laws on S^3 in the chordal metric (by McShane extension, Green's formula in the ball, closed Bessel sums, positivity of all Taylor coefficients of one entire function, and a Riccati comparison), and hence uniqueness and a transfer-matrix gap m >= -log(9 beta_W/2) for four-dimensional SU(2) for all beta_W < 2/9, which is the ceiling of all single-link Wasserstein-Dobrushin arguments; and a gauge-invariant Dobrushin-Shlosman finite-box criterion. The finite and algebraic steps of the sharp quarter and of the thresholds are proved in Lean 4 with Mathlib (25 theorems in YangMills/StrongCoupling/), depending only on propext, Classical.choice and Quot.sound, with no sorry or added axiom. The analytic steps (Bessel identities, harmonic extension, the Riccati comparison) are proved by hand in the paper and are not formalized; a Python script checks them numerically and in exact arithmetic (49 checks). 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. Tools used: Anthropic - Claude (Claude Code); Lean community - Lean 4 (v4.33.0) with Mathlib; Python (SymPy, mpmath, NumPy, SciPy); shell. arXiv identifier: none (not posted on arXiv). Formalization: https://github.com/creelie/4Y-M/tree/main/YangMills/StrongCoupling (archived on Zenodo with the repository; release v1.0.0 https://doi.org/10.5281/zenodo.23444338, later versions v2.0.0 and v2.0.1 listed on that record)
AI provenance filesAI_PROVENANCE_YangMillsDobrushinCeiling.md · 4 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 18, 2026