{
  "abstract": "A $p$-ary bent partition of $\\mathbb{F}_p^n$ is a partition into $K$ nonempty cells such that every balanced assignment of its cells to $\\mathbb{F}_p$ produces a bent function. It was asked whether every possible depth $K$ is a power of $p$; for general $p$, previous affirmative results required regularity or cell-symmetry hypotheses. We prove the stronger unconditional statement that, for every nonzero $h$, exactly $p^n/K$ points remain in the same fine cell under translation by $h$. Thus the fine cells form a partitioned difference family and the fine label map is zero-difference balanced. Consequently $K\\mid p^n$, so $K=p^t$; nonempty cells further give $1\\le t<n$. In even dimension, the classical cell-size theorem yields $K\\mid p^{n/2}$. Together with the known odd-dimensional ternary three-fibre parameter restriction, this gives the global bound $t\\le\\lfloor n/2\\rfloor$. The proof is an exact finite average over balanced coarsenings. The main counting identity and selected consequences are formalized and kernel-checked in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28133",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhaorui Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cx8xk3ar8n350.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28133",
  "fulltext_url": null,
  "id": "cx8xk3ar8n350",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28133",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7204754723",
  "pdf_url": "https://arxiv.org/pdf/2608.28133",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cx8xk3ar8n350.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28133",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28133"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fine Difference Structure and Prime-Power Depth of Bent Partitions",
  "title_text": "Fine Difference Structure and Prime-Power Depth of Bent Partitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/cx8xk3ar8n350",
  "venue": "arXiv",
  "versions": []
}