{
  "abstract": "To a coprime triple $a+b=c$ we attach two coordinates, the quality $X$ and the logarithmic mass $Y$, whose ratio $λ$ measures the balance of the triple and lies asymptotically between $2$ and $3$. In these coordinates the $abc$ conjecture and Szpiro's conjecture for Frey curves become asymptotic bounds with thresholds $X=1$ and $Y=3$, and the direct implications between them follow from the range of $λ$. We then place the approximation and power gains of Müller, Taktikos and de Weger in the same plane. Relative to a chosen presentation of the triple, one exponent factor $f$ relates the gains to the coordinates, and Szpiro's conjecture for this family is equivalent to the asymptotic bound $f\\le 3$ uniformly over all presentations. Under the two separate asymptotic gain bounds for a fixed choice of presentations, we obtain $\\limsup X\\le 9/2$ and $\\limsup Y\\le 27/2$ as the radical tends to infinity. Stronger conclusions depend on the presentation or on a joint bound for the gains. We close with numerical data and two research objectives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03212",
  "author_total": 2,
  "authors": [
    {
      "name": "R. Laniewski"
    },
    {
      "name": "K. Müller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5dez8esfzexd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03212",
  "fulltext_url": null,
  "id": "c5dez8esfzexd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03212",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220437913",
  "pdf_url": "https://arxiv.org/pdf/2610.03212",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5dez8esfzexd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.03212",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03212"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The quality plane of $abc$-triples and the decomposition into gains",
  "title_text": "The quality plane of abc-triples and the decomposition into gains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/c5dez8esfzexd",
  "venue": "arXiv",
  "versions": []
}