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Yang--Mills Mass Gap in 4D: Lattice SU(NN) and 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 3 / 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

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 with Ultracode, 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 SU(2) composite (release v3.0.5, doi:10.5281/zenodo.23469389). Release v3.1.0 records its Yang-Mills runs (doi:10.5281/zenodo.23484160). The engine supplied no step of the proofs. The paper proves the following by hand, at fixed lattice spacing: uniqueness, clustering and a transfer-matrix gap for SU(2) up to beta_W = 0.345, for SU(3) up to beta_W = 0.76, and for every compact simple group when N|beta| < 2 kappa_G/29. It also proves that a Dobrushin-Shlosman estimate in one cube of fixed physical size, uniform along the continuum tuning, gives a mass gap for every continuum limit. It proves what that estimate requires: it holds at strong coupling; it forces the gauge-fixed links of a cube with trivial exterior to stay at mean distance at least (1-alpha)D/3 from the identity, so it fails at every fixed cube size once the coupling is large (for every compact simple group, at least the fraction 1-alpha of the Haar disorder and mean distance (1-alpha)D/2, by four colours); bounds on single lines of links that are uniform in the exterior cannot give it; and for every SU(N) it needs n^4 >= 2(1-alpha) beta_W/(5 L_N(beta_W)); and the quadratic approximation of the forest gauge itself fails it at every n. It also constructs renormalised continuum limits of these smoothed observables along subsequences and reduces their nontriviality to one ratio of lattice two-point functions. In a periodic spatial box it proves that the reflected norm fixing that renormalisation is strictly positive. It shows that a lower bound on that ratio gives tight spectral measures after renormalising at the same separation, and that a one-slab bound on the transfer matrix would give the lower bound; neither is proved. For product test functions it bounds the ratio, in both directions, by ratios of correlations of one spatially smeared slice field at two fixed physical separations, and shows that the free lattice field and two-dimensional SU(N) lattice gauge theory along its tuning meet that criterion, while the spatial flow alone does not give it. It proves that tree bounds on the truncated correlations of the flowed fields give all n-point Schwinger functions as reflection-positive distributions along a subsequence, Gaussian when the tree weight has vanishing physical volume; with amplitude scales the same bounds give a time semigroup, the ratio bound itself and, with the cube estimate, the gap for every polynomial in the fields; their order-two cases force the lattice gap below a fixed multiple of the lattice spacing, so no strong-coupling expansion gives them; for a packet smooth in time they give the ratio with the explicit rate exp(-2 sqrt(B_1) sigma) and a limit of finite energy; the tree bounds are not proved. It also proves estimates uniform in the coupling: moments of the SU(N) force, a Ward identity, a lower bound on the SU(2) plaquette variance, and that bare blocking of SU(2) links approaches Haar measure in total variation; these are local and do not give the ratio or the cube estimate. It also proves that the continuum step holds for observables smoothed by the spatial gradient flow, and computes the cube condition exactly for a Gaussian field (it fails at every size when the field is massless and holds at a fixed physical scale when it has a mass); for the free massive field it proves that conditioning on block averages of side two keeps at least the fraction 2^{-d} d^{-2} of the variance of every smeared observable, uniformly in the scaling limit. Two self-contained appendices prove the facts on spectral measures (Herglotz, Helly, Laplace transforms) and on character expansions of the Wilson weight that the continuum section uses. It does not prove the estimate along the continuum tuning or the existence of the continuum limit. It therefore does not solve the Clay Millennium problem, and the paper says so. Tools used: Anthropic Claude Opus 5.5 (Claude Code) and Claude Fable 5.1 (Ultracode); 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 v3.19.0, https://doi.org/10.5281/zenodo.23729004; all versions https://doi.org/10.5281/zenodo.23444338)
AI provenance filesAI_PROVENANCE_YangMillsEveryGroup.md · 13 KB

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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