{
  "abstract": "Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. For every $r\\ge1$, we prove that a product-one-free sequence of length $2r+11$ over $E_2\\times C_3^r$ cannot contain exactly $2r-1$ central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over $\\mathbb F_3$. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of $\\mathsf{d}(E_2\\times C_3^r)$ for arbitrary $r$ is not determined by this result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22972",
  "author_total": 1,
  "authors": [
    {
      "name": "Andreas Volkmann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mxq66ghwwm9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22972",
  "fulltext_url": null,
  "id": "0mxq66ghwwm9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22972",
  "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"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214055415",
  "pdf_url": "https://arxiv.org/pdf/2609.22972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mxq66ghwwm9k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A twelve-term exclusion for the small Davenport constant of $E_2\\times C_3^r$",
  "title_text": "A twelve-term exclusion for the small Davenport constant of E_2× C_3^r",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0mxq66ghwwm9k",
  "venue": "arXiv",
  "versions": []
}