{
  "abstract": "We develop a graph-theoretic approach to the expected-norm matrix arithmetic--geometric mean inequality. By encoding matrix interactions with a path-supported Gram matrix, we obtain a projection family whose product is nonzero exactly when the transition graph of its index sequence is a path. This gives an intuitive explanation for failure: repeated indices can create cycles or branches that annihilate products, an effect absent for positive scalars. We prove the universal lower bound $(n)_m/n^m$ for the ratio of the with-replacement to the without-replacement mean and show that it is asymptotically sharp on the exponential scale when $m=n$. Specifically, our construction gives a ratio $\\exp(-n+O(\\sqrt n))$, matching the lower bound's leading exponential rate over unrestricted dimensions. We also determine an explicit overlap threshold, derive exact six-factor counterexamples, and show that exponential failure can coexist with vanishing pairwise commutators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07649",
  "author_total": 1,
  "authors": [
    {
      "name": "Zehua Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f2etwdk62z0gc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07649",
  "fulltext_url": null,
  "id": "f2etwdk62z0gc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07649",
  "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.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07649",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f2etwdk62z0gc.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07649",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07649"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Graph-Theoretic Approach to Matrix AM-GM: Counterexamples and Sharp Exponential Bounds",
  "title_text": "A Graph-Theoretic Approach to Matrix AM-GM: Counterexamples and Sharp Exponential Bounds",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/f2etwdk62z0gc",
  "venue": "arXiv",
  "versions": []
}