{
  "abstract": "We prove the irrationality of \\[ \\Li_2(-1/4),\\qquad \\Li_2(1/5),\\qquad \\Li_2(-1/3). \\] The three arguments form a natural progression. The first is an endpoint completion of the five-parameter Rhin--Viola method. Rhin and Viola's 2019 treatment already supplies the negative-argument continuation, permutation invariance and factorial divisor; at $z=-4$ we use three bounded shifts, an elementary fixed-contour estimate and a four-term recurrence to remove the remaining nonvanishing problem in the complex-saddle regime. For $1/5$ we pass to the six-parameter Viola--Zudilin family and apply the Rhin--Viola factorial transformations term by term inside two binomial expansions of the same integral. This inherited divisor crosses the arithmetic threshold that the unrefined five-parameter construction does not reach in our computations. The same crossbreed, combined with the negative-argument continuation, proves the result at $-1/3$. All proof-critical finite inequalities are accompanied by exact executable certificates. Supplementary computations propagate $10{,}000$ exact primitive coefficient pairs for each of $1/5$ and $-1/3$, with complete gcd removal and no failure of the certified smallness or adjacent nonproportionality checks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06895",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Prellberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5y5x4t4c48a5y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06895",
  "fulltext_url": null,
  "id": "5y5x4t4c48a5y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06895",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212055273",
  "pdf_url": "https://arxiv.org/pdf/2609.06895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5y5x4t4c48a5y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at $-1/4$, $1/5$, and $-1/3$",
  "title_text": "Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at −1/4, 1/5, and −1/3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5y5x4t4c48a5y",
  "venue": "arXiv",
  "versions": []
}