HEXAGONbeta

Mass Gap of Four-Dimensional Lattice Yang--Mills Theory in a Forest Gauge

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

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗

Version 2 / Oct 18, 2026 / CC BY 4.0

Abstract

We fix the gauge of four-dimensional lattice Yang--Mills theory on a field of straight lines that meets every plaquette in exactly one link, so that each free link interacts with twelve others instead of eighteen. In this gauge the theory has a unique infinite-volume state with exponential decay of correlations and a positive mass gap at fixed lattice spacing, for SU(2) beyond the limit of every single-link argument in a gauge of this kind, by blocks made of the three free links of a plaquette, for every SU(N) beyond the curvature limit of the ungauged theory, and for SU(3) on a range more than two and a half times as large as was known before, where the same blocks raise the gap at the end of that range by two orders of magnitude and carry the range past it to βW=0.76\beta_W=0.76.

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.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

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