{
  "abstract": "The recursive-line Zarankiewicz number maximizes the number of squares in a structured irreducible sum-of-squares representation encoded by an augmentation of an extremal $C_4$-free bipartite graph. We determine its four-column behavior under the strengthened recursive definition in the manuscript of Löfberg and Qi dated 9 September 2026. Combining AI-assisted discovery with exact certificate verification and finite exclusion computations, we determine eighteen of the nineteen values for $2\\le m\\le20$ and isolate the only unresolved case to $37\\le\\zr(14,4)\\le38$. More significantly, we prove the first eventual exact formula in the four-column setting: \\[ \\zr(m,4)=\\floor{\\frac{5m+6}{2}}\\qquad(m\\ge15). \\] The upper bound follows from the classical identity $z(m,4)=m+6$ and a sharp cell count. For the matching lower bound, we construct a two-row extension chain from an explicit $20\\times4$ seed and derive the odd orders by a fixed deletion. Analytic propagation, together with two independently audited symbolic certificate tables, proves the construction for arbitrary chain length rather than merely for a finite computational range. Thus every extremal configuration has no holes when $m$ is even and exactly one hole when $m$ is odd, and the same exact formula holds for the second-order number $z_2(m,4)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11093",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhiwei Chen"
    },
    {
      "name": "Yannan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1j1mmbbztjg9e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11093",
  "fulltext_url": null,
  "id": "1j1mmbbztjg9e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11093",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212406395",
  "pdf_url": "https://arxiv.org/pdf/2609.11093",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1j1mmbbztjg9e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.11093",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11093"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Recursive-Line Zarankiewicz Numbers with Four Columns",
  "title_text": "Recursive-Line Zarankiewicz Numbers with Four Columns",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1j1mmbbztjg9e",
  "venue": "arXiv",
  "versions": []
}