{
  "abstract": "Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions $ϕ_{a,b}$ defined by $0 \\mapsto aa$ and $1 \\mapsto bb$ with $b = a + 1$. Based on a three-fold arithmetic classification of pairs of rational numbers $0 \\le x < y \\le 1$, we investigate the order relations between the corresponding Lagrange constants $\\mathcal{L}([ϕ_{a,a+1}(G(x))])$ associated with the mechanical words $G(x)$. While an inequality is established for pairs of the second type (type (2)) whenever $a \\ge 1$, the comparisons for type (1) and type (3) pairs hold for $a \\ge 2$. Consequently, for any integer $a \\ge 2$, where the order relations across all three types are completely determined, we establish an analogue of the Markoff Uniqueness Conjecture: the map $x \\mapsto \\mathcal{L}([ϕ_{a,a+1}(G(x))])$ is injective on $\\mathbb{Q} \\cap [0, 1]$. Furthermore, supported by extensive numerical computations, we propose a general conjecture on the order relations for arbitrary $b > a$, highlighting a sharp phase transition in ordering behavior at the boundary $b = a^2 + 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32450",
  "author_total": 1,
  "authors": [
    {
      "name": "Shin-ichi Yasutomi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4n2fz7ap7gbg2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32450",
  "fulltext_url": null,
  "id": "4n2fz7ap7gbg2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32450",
  "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",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214713189",
  "pdf_url": "https://arxiv.org/pdf/2609.32450",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4n2fz7ap7gbg2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.32450",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32450"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions",
  "title_text": "On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/4n2fz7ap7gbg2",
  "venue": "arXiv",
  "versions": []
}