{
  "abstract": "For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\\max_i a_i-\\min_i a_i, Δ(n)=\\inf_D R(D).$ We prove that this infimum is attained for every $n\\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\\frac{5\\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\\sqrt5)/4$ and two equal $(3\\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\\le 0.0001273496861283553341\\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22693",
  "author_total": 1,
  "authors": [
    {
      "name": "Muxi Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cc3qxm3g10y8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22693",
  "fulltext_url": null,
  "id": "1cc3qxm3g10y8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22693",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7214033581",
  "pdf_url": "https://arxiv.org/pdf/2609.22693",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cc3qxm3g10y8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22693",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22693"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles",
  "title_text": "Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/1cc3qxm3g10y8",
  "venue": "arXiv",
  "versions": []
}