{
  "abstract": "We exhibit an explicit pair $(A,B)$ of integral symmetric $9\\times9$ matrices, defining a nonsingular pair of quadratic forms over $\\mathbb{Q}$, such that no member of the rational pencil $λq_A+μq_B$ has Witt index $4$ over $\\mathbb{Q}$. This refutes a conjecture from \\cite{Que16a}, which predicted that every nonsingular pair in $n$ variables generates a pencil containing a form of Witt index $\\lceil (n-1)/2\\rceil$. The obstruction is purely $2$-adic and affects the whole pencil at once: every member has Witt index exactly $3$ over $\\mathbb{Q}_2$. The proof is finite and elementary: a parity argument on $\\det(λA+μB)$, a congruence lemma reducing $\\mathbb{P}^1(\\mathbb{Q}_2)$ to the twelve classes of $\\mathbb{P}^1(\\mathbb{Z}/8)$, and a verification at each class, in which the anisotropy verdict is certified in two independent ways. All scripts are provided in the GitHub repository.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07692",
  "author_total": 1,
  "authors": [
    {
      "name": "Tony Quertier"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6pr34b0g8cdqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07692",
  "fulltext_url": null,
  "id": "6pr34b0g8cdqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07692",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212064253",
  "pdf_url": "https://arxiv.org/pdf/2609.07692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6pr34b0g8cdqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A pencil of quadratic forms in nine variables with no member of Witt index four",
  "title_text": "A pencil of quadratic forms in nine variables with no member of Witt index four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/6pr34b0g8cdqe",
  "venue": "arXiv",
  "versions": []
}