{
  "abstract": "Relations compose through the entity they share - Alice is Bob's mother and Bob is Carol's father, so Alice is Carol's grandmother - and the order matters, since a father's mother is not a mother's father; verifying a composed relation means closing the loop. We show that if each entity's coordinate frame may be rotated independently (per-entity frame freedom), three things follow. In the pure-gauge sector, relations that share no cycle cannot interact at all. The only gauge-invariant score of a triangle is a class function of its holonomy $H = T_{ij} T_{jk} T_{ki}$ - a Wilson loop. And the only equivariant update of an edge is $U = Ψ(H)\\, T_{ij}$ with $Ψ$ conjugation-equivariant. Edge attention by Wilson loops is therefore not one design among many but the unique form the symmetry permits. Three communities hold pieces of this form: the commuting-operator embeddings of knowledge graphs are its abelian (commuting) sector; AlphaFold's triangle update and the Edge Transformer have its shape - an edge updated through the third vertex - without its constraints; and lattice QCD builds it equivariantly on lattices. We instantiate the equivariant sub-family on general relational graphs as holonomy attention. With no training and composition alone, it infers unobserved $S_4$ relations on sparse graphs at 0.88 with a fifth of the edges corrupted, at or slightly below specialist synchronization algorithms, where every commutative architecture is capped at 0.083; its holonomy score flags corrupted triangles at AUC 0.99 where commuting scores are capped at 0.76 by a second selection rule, and it separates clusters without labels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37656",
  "author_total": 3,
  "authors": [
    {
      "name": "Aakash Sarkar"
    },
    {
      "name": "Ananya Mukherjee"
    },
    {
      "name": "Motoshige Sato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtd3vvza4cya6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37656",
  "fulltext_url": null,
  "id": "dtd3vvza4cya6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37656",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "physics.comp-ph"
    ],
    [
      "openalex",
      "1702|Advanced Graph Neural Networks"
    ]
  ],
  "openalex": "W7224119346",
  "pdf_url": "https://arxiv.org/pdf/2610.37656",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtd3vvza4cya6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:50Z",
      "native_id": "2610.37656",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37656"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    }
  ],
  "title": "Attention as Lattice Gauge Theory: Relations Interact Only Through Wilson Loops",
  "title_text": "Attention as Lattice Gauge Theory: Relations Interact Only Through Wilson Loops",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:50Z",
  "url": "https://research.draconitic.com/paper/dtd3vvza4cya6",
  "venue": "arXiv",
  "versions": []
}