{
  "abstract": "Positive cyclic quantum mutations at a third root of unity induce an explicit weight on polygon flip graphs. We study the resulting discrepancy between minimum flip count and minimum accumulated normalization loss. Every pentagon three-to-two replacement strictly decreases weight, and in connected types $A_1$, $A_2$, and $A_3$ every minimum-weight path is a flip geodesic. In type $A_4$, however, a six-flip path has smaller weight than the unique five-flip geodesic. A compatible positive projective gluing construction amplifies this example to an exponential success-probability separation in every connected type $A_n$, $n\\ge4$. We also give a quantitative perturbation bound preserving the separation. The low-rank minimality statement has an exhaustive finite certificate; the pentagon inequality and the all-rank construction are analytic. In the associated sequential postselection model, differences of path weights determine success-probability ratios independently of the input state. The comparison holds for separately normalized exact mutation steps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40661",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaomin Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3v5qwqttkaj0y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40661",
  "fulltext_url": null,
  "id": "3v5qwqttkaj0y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40661",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40661",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3v5qwqttkaj0y.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40661",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40661"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Weighted flip graphs from cyclic quantum mutations: pentagon inequalities and exponential separations",
  "title_text": "Weighted flip graphs from cyclic quantum mutations: pentagon inequalities and exponential separations",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/3v5qwqttkaj0y",
  "venue": "arXiv",
  "versions": []
}