{
  "abstract": "We prove the Baker--Rumely conjecture that the tau invariant of a metrized graph admits a positive lower bound proportional to its total length, with an absolute constant. We also construct simple cubic metrized graphs whose normalized tau invariants tend to $59/7260<1/108$, disproving the proposed universal constant $1/108$. The lower bound is independent of the genus, the number of edges, and the distribution of edge lengths. Its proof combines a second-moment inequality for Euclidean lattices with a partition of the edge coordinates of a cycle lattice into three independent sets. The counterexamples have only two edge lengths and admit an elementary resistance calculation. Through the tropical moment identity, the lower bound also gives a uniform estimate for the non-archimedean terms in height formulas for Jacobians.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04994",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruihua Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7724xrwp563v8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04994",
  "fulltext_url": null,
  "id": "7724xrwp563v8",
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  "landing_url": "https://arxiv.org/abs/2610.04994",
  "language": null,
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  "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.AG"
    ],
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      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.NT"
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      "openalex",
      "2608|Algebraic Geometry and Number Theory"
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  "openalex": "W7220668706",
  "pdf_url": "https://arxiv.org/pdf/2610.04994",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7724xrwp563v8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04994"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Uniform positivity of the tau invariant",
  "title_text": "Uniform positivity of the tau invariant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7724xrwp563v8",
  "venue": "arXiv",
  "versions": []
}