{
  "abstract": "The equivalence classes of boundary conditions of a gauge theory on a two-dimensional orbifold are the fibres of a marginal map, indexed by an affine semigroup: one generator per alphabet label, graded by weight, embedded by its local data at the fixed points. This note identifies that semigroup. Without weights the configuration has a name and a literature, whose results about our cases are attributed here: over $\\mathbb{Z}_2$ it is the cut configuration of an explicit graph in the sense of Sturmfels-Sullivant --- the four-cycle for $T^2/\\mathbb{Z}_2$, the wheel $W_4$ for $S^1/\\mathbb{Z}_2\\times S^1/\\mathbb{Z}_2$ --- verified as an equality of configurations; over $\\mathbb{Z}_m$ with equal cone orders, the group-based phylogenetic model on a claw tree; with unequal orders, a mixed-order variant we do not find in the literature; for higher products, the binary hierarchical model of a cross-polytope boundary complex. The product orbifold's ring is a row of a 2008 table --- codimension, degree, minimal generators, normality --- every invariant of which our machinery reproduced without knowing of it. What none of the three covers is the alphabet with weights, which arise from induction to higher-dimensional irreducibles of a non-abelian space group and from conjugate-pair recombination over real or quaternionic ground. That sector is adjacent to, but not identified with, the non-abelian direction Sturmfels and Sullivant raised in 2005, and is where our contributions sit: gluing trees for the weighted alphabets and the orthogonal and symplectic columns, and the group-based model on the tripod, a complete intersection exactly when the finite abelian group has order at most three. The first group beyond $\\mathbb{Z}_3$ separates local from global: the $\\mathbb{Z}_4$ tripod is a complete intersection on the Zariski-open set the phylogenetics literature works in, and not globally.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02630",
  "author_total": 1,
  "authors": [
    {
      "name": "Carles Marín"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bms020qecsvqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02630",
  "fulltext_url": null,
  "id": "bms020qecsvqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02630",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7207694707",
  "pdf_url": "https://arxiv.org/pdf/2609.02630",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bms020qecsvqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:52Z",
      "native_id": "2609.02630",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02630"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them",
  "title_text": "An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/bms020qecsvqf",
  "venue": "arXiv",
  "versions": []
}