{
  "abstract": "The second-order Zarankiewicz number $z_2(m,n)$ and the biquadratic sum-of-squares rank $\\mathrm{BSR}(m,n)$ are related by the unconditional hierarchy \\[ \\mathrm{BSR}(m,n)\\ \\ge\\ z_2(m,n)\\ \\ge\\ z_{SL}(m,n)\\ \\ge\\ z_{RL}(m,n) \\ \\ge\\ z_{wL}(m,n)\\ \\ge\\ z(m,n). \\] We introduce the \\emph{second-order augmented Zarankiewicz number} $z_{2A}(m,n)$, obtained from $z_2(m,n)$ by deleting the requirement that the configuration be \\emph{limited}, so that \\[ \\mathrm{BSR}(m,n)\\ \\ge\\ z_{2A}(m,n)\\ \\ge\\ z_2(m,n). \\] Although the defining class is enlarged, $z_{2A}$ still obeys the universal cell bound of Löfberg and Qi, because that bound uses only the \\(C_4\\)-freeness of the one-edge graph. We prove \\[ \\mathrm{BSR}(4,4)\\ \\ge\\ z_{2A}(4,4)\\ =\\ 11\\ >\\ 10\\ =\\ z_2(4,4) \\ =\\ z_{RL}(4,4), \\] the first recorded separation between the second-order number and its augmented variant. This result also gives a better lower bound for $\\mathrm{BSR}(4,4)$. The lower bound is witnessed by an explicit non-limited $4\\times4$ configuration of displayed length $11$ whose recursive-line closure satisfies $(\\mathrm{RW}3^+)$; the matching upper bound excludes $12$ by the universal cell bound together with an exact finite classification of the $161$ twelve-square configurations, and \\(z_2(4,4)=z_{RL}(4,4)=10\\) is the exact value of Xu and Yan.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04448",
  "author_total": 2,
  "authors": [
    {
      "name": "Liqun Qi"
    },
    {
      "name": "Chunfeng Cui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a2jt8ca3433ce.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04448",
  "fulltext_url": null,
  "id": "a2jt8ca3433ce",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04448",
  "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": "W7220707435",
  "pdf_url": "https://arxiv.org/pdf/2610.04448",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a2jt8ca3433ce.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04448",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04448"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Second-Order Augmented Zarankiewicz Number",
  "title_text": "The Second-Order Augmented Zarankiewicz Number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/a2jt8ca3433ce",
  "venue": "arXiv",
  "versions": []
}