{
  "abstract": "Random geometric graph detection asks whether a graph generated from $n$ independent uniform points on $S^{d-1}$, with edge probabilities depending on inner products, can be distinguished from an Erdos-Renyi graph of the same edge density using only its adjacency matrix. Mao, Wu, and Xu conjectured that the cubic trace of the standardized kernel determines the detection boundary: $n^3(\\mathrm{tr}\\,K^3)^2$ tending to zero or infinity should imply impossibility or strong detection, respectively. We construct counterexamples and prove a uniform result for spherical polynomial kernels of uniformly bounded degree with mean edge density bounded away from zero and one. For the integral operator $K$ of the standardized centered kernel, define \\[ n_* = \\min\\left\\{ \\frac{1}{|\\mathrm{tr}\\,K^3|^{2/3}},\\quad \\frac{1}{\\sqrt{\\mathrm{tr}\\,K^4}},\\quad \\frac{d}{\\mathrm{tr}\\,K^2} \\right\\}, \\] with zero denominators interpreted as infinity. The total variation distance between the geometric and independent-edge models tends to zero if $n/n_* \\to 0$ and to one if $n/n_* \\to \\infty$. The three scales correspond to triangles, four-cycles, and global geometry. The pure degree-four spherical harmonic kernel has only positive nonzero eigenvalues, yet its detection scale is $n \\asymp d^5$. Throughout $d^5 \\ll n \\ll d^{16/3}$, the triangle and four-cycle signals and every fixed-degree standardized polynomial mean gap vanish, while the full graph remains strongly detectable. Thus the cubic-trace conjecture fails even without spectral sign cancellation. The impossibility proof combines higher-order spherical integration by parts, a graph expansion of cumulants, and a finite-order relative entropy comparison.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11662",
  "author_total": 1,
  "authors": [
    {
      "name": "Congyi Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9hnfwfkns3e21.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11662",
  "fulltext_url": null,
  "id": "9hnfwfkns3e21",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11662",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212251168",
  "pdf_url": "https://arxiv.org/pdf/2609.11662",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9hnfwfkns3e21.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.11662",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11662"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Counterexamples to the Mao-Wu-Xu spectral conjecture and a three-scale boundary for spherical random graphs",
  "title_text": "Counterexamples to the Mao-Wu-Xu spectral conjecture and a three-scale boundary for spherical random graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:29Z",
  "url": "https://research.draconitic.com/paper/9hnfwfkns3e21",
  "venue": "arXiv",
  "versions": []
}