{
  "abstract": "Let $N(H)$ be the edge--vertex incidence matrix of a finite linear $s$-uniform hypergraph whose line graph is a cograph, and define \\[ R_s(H)=(s+1)\\operatorname{rank}_{\\mathbb R}N(H)-s|E(H)|. \\] The inequality $R_s(H)\\ge0$ holds, with equality characterized by affine planes of order $s$ on every row-dependent component. Under the corresponding finite-net completion hypothesis, connected systems with bounded $R_s$ admit an explicit classification. If $F$ is connected, $R_s(F)=d\\le s-1$, the completion hypothesis holds through deficiency $d$, and $s>(d-1)^2$, then either $N(F)$ has full row rank and $|E(F)|=d$, or $F$ lies in a unique affine plane of order $s$. In the latter case, for some $0\\le h\\le d$, the system contains every line in $s+1-h$ parallel classes and exactly $d-h$ additional lines from the remaining $h$ classes. Conversely, each such system has defect $d$. Bruck's theorem gives the classification for every fixed $d$ and all sufficiently large $s$, while Metsch's completion theorem covers $d=o(s^{1/3})$. As a global consequence, if the total defect is at most $D$ in the completion range, deleting at most $D$ edges leaves a vertex-disjoint union of finite nets and affine planes. The associated column defect and row leverage are determined explicitly as well.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27777",
  "author_total": 1,
  "authors": [
    {
      "name": "Mahesh Ramani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4x59sqfj8aprv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27777",
  "fulltext_url": null,
  "id": "4x59sqfj8aprv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27777",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214169053",
  "pdf_url": "https://arxiv.org/pdf/2609.27777",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4x59sqfj8aprv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27777",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27777"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Incidence Rank and Bounded Defect for Linear Hypergraphs with Cograph Line Graphs",
  "title_text": "Incidence Rank and Bounded Defect for Linear Hypergraphs with Cograph Line Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/4x59sqfj8aprv",
  "venue": "arXiv",
  "versions": []
}