{
  "abstract": "Giuliano Basso asked whether every $n$-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex $v$, then the coefficient of its boundary at $v$ has the same parity as $°(v)$. Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum $G$ of finite length and a boundaryless current $T\\in\\I_1(G)$ with $\\set(T)=G$, while no $S\\in\\I_1(G)$ with $\\set(S)=G$ can have multiplicity one $\\Hh^1$-almost everywhere. We classify all full-support integral cycles on $G$. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is $3/2$. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio--Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14515",
  "author_total": 1,
  "authors": [
    {
      "name": "Deguang Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1fwgz83a97gq9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14515",
  "fulltext_url": null,
  "id": "1fwgz83a97gq9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14515",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213330598",
  "pdf_url": "https://arxiv.org/pdf/2609.14515",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1fwgz83a97gq9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14515",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14515"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Odd branching obstructs multiplicity-one integral current structures",
  "title_text": "Odd branching obstructs multiplicity-one integral current structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1fwgz83a97gq9",
  "venue": "arXiv",
  "versions": []
}