{
  "abstract": "Qiu, Guan, and Yu conjectured that in the even-order Kaleidoscope Yang--Baxter equation every word $ΛT^{k_1}Λ\\cdots T^{k_M}Λ$, built from a square-zero matrix $Λ$ and integer powers of a shift $T$ of order $N$, vanishes once $M\\geq N/2$. We prove this for every even $N\\geq4$ and every complex parameter $u$ with $u\\neq0$ and $u^N\\neq1$, and show that the bound is sharp. For $M=N/2-1$ we determine exactly which words are nonzero; each of them has rank two. After block reduction, the factors associated with the intervening powers of $T$ lower a common flag of length $N/2$ by at least one step. The spectral idempotents of $T$ then give a presentation of the algebra generated by $T$ and $Λ$ by paths between levels, in which two paths compose only when the level changes monotonically. This algebra is independent of $u$, has dimension $N^2-N+2$, and is a pullback of two incidence algebras. Its radical is generated by $Λ$ and has nilpotency index $N/2+1$, and the algebra has infinite global dimension and infinite representation type. Replacing powers of $T$ by arbitrary functions of $T$ gives exact rank formulas and, for $N\\geq6$, rank-one elements in the top radical power. Finally, four conditions on a filtered pair of operators over an arbitrary field recover the threshold and the radical; none of them can be omitted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39262",
  "author_total": 2,
  "authors": [
    {
      "name": "Qihang Wang"
    },
    {
      "name": "Zhiyuan Yao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/998rtgq3r4r8y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39262",
  "fulltext_url": null,
  "id": "998rtgq3r4r8y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39262",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7215100464",
  "pdf_url": "https://arxiv.org/pdf/2609.39262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/998rtgq3r4r8y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:32Z",
      "native_id": "2609.39262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra",
  "title_text": "Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/998rtgq3r4r8y",
  "venue": "arXiv",
  "versions": []
}