{
  "abstract": "We reformulate binary classification on a manifold M as a Yang-Mills-Higgs variational problem. Labelled data is encoded as a functor from the fundamental groupoid of M to the one-object groupoid B(Z_2), whose monodromy class in H^1(M, Z_2) is a topological obstruction to realising the classifier by a sign function. The classifier-section and the connection jointly minimise a Yang-Mills-Higgs energy subject to hard data conditions: the matter sector carries the classification content, while the Yang-Mills sector is bounded below in each topological class by the Bogomolny inequality and selects the gauge background. This recovers the companion paper's harmonic interpolation as the contractible-base, flat-connection reduction. Two structural payoffs follow. First, the curvature 2-form of the selected connection supplies a dictionary with transformer attention: by the plaquette formula it is a pairwise, antisymmetric, algebra-valued form on tangent directions, which we match - as a stipulated dictionary, not a derived identity - to the antisymmetric component of the attention bilinear; the abelian/non-abelian split of curvature corresponds to the single-head/multi-head split of attention. Second, XOR on the torus is realised by the covariantly harmonic section of the double-Mobius bundle, which the variational selector picks out: solving the regularised capacitance system on all four flat Z_2-bundles gives a strict energy ordering favouring the double-Mobius class, whereas an MLP trained on the same data finds a structurally different boundary that ignores the toroidal identifications. Worked examples run an example ladder (circle, torus, S^2 monopole, S^4 instanton). The connection is throughout either flat or pinned to the Bogomolny moduli, so curvature is data-decoupled by construction; whether freeing it makes curvature respond to label proximity is posed as an open question.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.00999",
  "author_total": 1,
  "authors": [
    {
      "name": "Catalin Vasii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f324n5pr00ntw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.00999",
  "fulltext_url": null,
  "id": "f324n5pr00ntw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.00999",
  "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",
      "math.DG"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7167100264",
  "pdf_url": "https://arxiv.org/pdf/2607.00999",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f324n5pr00ntw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2607.00999",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.00999"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces",
  "title_text": "Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/f324n5pr00ntw",
  "venue": "arXiv",
  "versions": []
}