{
  "abstract": "Let (A,b) be a finite abelian group with a nondegenerate symmetric pairing into Q/Z, and let r(A) be its least number of generators. For infinitely many primes ℓ, we realize (A,b) ⊥ ⟨κ/ℓ⟩ as the entire paired Jacobian of a simple planar biconnected graph, with one nonzero integer κ independent of ℓ. The primes avoid any prescribed finite set. Every vertex has degree two or three. The graphs have arbitrarily large prescribed girth, treewidth at most two, and cycle rank max{1,r(A)}, which is optimal. For each cycle rank they are subdivisions of one fixed planar multigraph. We construct a family whose complementary orders form a primitive arithmetic progression; an explicit cofactor determines the complementary pairing. Dirichlet's theorem gives the prime-order conclusion. A separate construction retains outerplanarity, allowing parallel edges and unbounded degree. Finally, for r ≥ 3, the paired Jacobian of K_(2^r) is the orthogonal sum of 2^(r−1)−1 hyperbolic planes over Z/2^r Z. The case K_8 contradicts the exceptional-pairing conjecture of Gaudet, Jensen, Ranganathan, Wawrykow and Weisman. Preprint. MSC 2020: 05C25, 05C50, 05C10, 11E08, 11N13, 20K01.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Anass Nifa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4nw8g9a6f221y.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.22865118",
  "fulltext_url": null,
  "id": "4nw8g9a6f221y",
  "issns": [],
  "landing_url": "https://hal.science/hal-05769897v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-co"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05769897/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4nw8g9a6f221y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:43Z",
      "native_id": "hal-05769897",
      "source": "hal",
      "url": "https://hal.science/hal-05769897v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-co",
      "subfield": "combinatorics"
    }
  ],
  "title": "Prime-order complements in planar graph Jacobians",
  "title_text": "Prime-order complements in planar graph Jacobians",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:43Z",
  "url": "https://research.draconitic.com/paper/4nw8g9a6f221y",
  "venue": "HAL",
  "versions": []
}