{
  "abstract": "We introduce the \\emph{second order Zarankiewicz number} $z_2(m,n)$ for irreducible doubly simple biquadratic forms with $|E_1|=z(m,n)$, together with the intermediate recursive-line and signed parameters $z_{RL}(m,n)$ and $z_{SL}(m,n)$. They satisfy the unconditional hierarchy \\[ \\operatorname{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), \\] where $z_{RL}$ is defined by the strengthened recursive rectangle criterion $(RW3^+)$ together with the conditions $(S)$ and $C_4$-freeness of $G_1$, and $z_{SL}$ is defined by the signed criterion $(RW3^\\pm)$. We show that $(RW3^+)$ is sound and strictly weaker than the literal weak cross-cell test $(W3)$ on the weak-admissible class, and that $(RW3^\\pm)$ is likewise sound. At the smallest four-column cases we obtain \\[ z_2(5,4)=z_{SL}(5,4)=z_{RL}(5,4)=13>12=z_{wL}(5,4), \\] and \\[ z_2(6,4)=z_{SL}(6,4)=z_{RL}(6,4)=16>14=z_{wL}(6,4). \\] In three columns this yields \\[ z_2(m,3)=z_{RL}(m,3)=2m \\qquad \\text{for all } m\\ge 3, \\] with strict separation from $z_{wL}(m,3)$ for every $m\\ge10$, and exact gap $\\lfloor(m-3)/3\\rfloor$ for $m\\ge16$. Further finite computations give $z_{RL}(5,5)=17$, $z_{RL}(7,4)=19$, and $z_{RL}(7,7)\\ge32>28=z_{wL}(7,7)$. Along $N=2p$ with $p$ an odd prime, we obtain the cubic asymptotic separation \\[ z_2\\!\\left(\\binom{N}{2},N\\right)-z_{wL}\\!\\left(\\binom{N}{2},N\\right)\\ge \\left(\\frac{1}{16}-o(1)\\right)N^3. \\] For the exceptional incidence case $p=3$ we prove $z_{SL}(15,6)=z_2(15,6)=60$. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all $m\\ge15$. The conjectural equality $z_2=z_{RL}$ is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30555",
  "author_total": 2,
  "authors": [
    {
      "name": "Johan Löfberg"
    },
    {
      "name": "Liqun Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18he7wvb0e3z9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30555",
  "fulltext_url": null,
  "id": "18he7wvb0e3z9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30555",
  "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": "W7204894795",
  "pdf_url": "https://arxiv.org/pdf/2608.30555",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18he7wvb0e3z9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.30555",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30555"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Second Order Zarankiewicz Number",
  "title_text": "Second Order Zarankiewicz Number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/18he7wvb0e3z9",
  "venue": "arXiv",
  "versions": []
}