{
  "abstract": "We axiomatize the differential-comparison continuation method of [Zf] into a general machine for proving Mahler measure identities at CM points: a modular parametrization with rigorous tail bounds, a single-valued holomorphic Mahler differential on the complement of the critical image, a propagation lemma replacing all monodromy and universal-cover arguments, interval-arithmetic path certification, and an exact three-track CM evaluation. Applied to Samart's family $n_4(s)=4\\mathrm m(x^4+y^4+z^4+1+s^{1/4}xyz)$, the machine proves twelve further entries of his 2015 table [Sa15, Table 6], including the first identities in this family proved at non-degenerate CM points of class number two (attached to $\\mathbb Q(\\sqrt{-6})$ and $\\mathbb Q(\\sqrt{-15})$) and of class number four (attached to $\\mathbb Q(\\sqrt{-21})$, with four newforms and the genus group $(\\mathbb Z/2)^2$). New ingredients include exact twist levels via grossencharacter conductors (Hecke's theorem), with the interval-locked Fricke ratio as an exact root-number sign lock; a twist-level trap ($g_{16}\\otimesχ_8$ has level $64$, not $32$); a genus-character decomposition of the lattice sums in class number two; and the discovery that Samart's applicability boundary $\\{\\mathrm{Im}\\,τ=1/\\sqrt2\\}$ is not the topological boundary of the continuation region. Every identity is certified by an exact algebraic track, rigorous final-identity interval locks (half-widths $4.6\\times10^{-53}$ to $5.1\\times10^{-52}$), and independent 50--60-digit cross-checks; all scripts are public. The evaluation track also yields six new identities at Heegner points ($D=11,19,27,43,67,163$), stated as conjectures with 60-digit evidence; we close with an umbrella conjecture organizing the proved and conjectured identities, and a precise analysis of the five remaining open entries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26120",
  "author_total": 1,
  "authors": [
    {
      "name": "Huimin Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5s35xztyfp8sy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26120",
  "fulltext_url": null,
  "id": "5s35xztyfp8sy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26120",
  "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",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214082722",
  "pdf_url": "https://arxiv.org/pdf/2609.26120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5s35xztyfp8sy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26120"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart",
  "title_text": "A certified continuation machine for Mahler measure identities at CM points, with twelve new proofs of conjectures of Samart",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/5s35xztyfp8sy",
  "venue": "arXiv",
  "versions": []
}