{
  "abstract": "Boyd's 1998 tables of conjectural identities $m(P)=r\\,|L'(E,0)|$ between Mahler measures of two-variable polynomials and $L$-values of elliptic curves begin with the smallest possible conductor, $N=11$. The third conductor-$11$ identity, (C3), concerns the polynomial $S_0=y^2+(x^2+1)y+x^3$, which vanishes on the unit torus, and asserts that a signed split integral $I_{\\mathrm{split}}$ of $\\log|y|$ around the branch cut equals $b_{11}=L'(E_{11},0)$. We prove (C3). The proof identifies the split integral with a regulator integral along Samart's signed open chain $\\tildeγ$, closed by a small-branch arc $β_0$ into a closed anti-invariant cycle $C'$, and proves its homology class is $2γ^-$ with $γ^-$ generating $H_1(E,\\mathbb{Z})^-$: the period ratio is a-priori integral, and ball arithmetic (Arb) pins it to $2$. A direct regulator computation via Brunault's proved Siegel-unit formula---the symbol $\\{x,y\\}$ being a pair of modular units on $X_1(11)$, so Bloch's diamond theorem is not needed---then yields $I_{\\mathrm{split}}=b_{11}$; the sign is certified in interval arithmetic, and the identity agrees with the numerical value to $366$ digits. For the family $S_k=y^2+(x^2+kx+1)y+x^3$ we determine exactly the torus intersections and the modular-unit/tempered cases; further Boyd-type evaluations are recorded as conjectures. At $k=-1$ (conductor $53$) the mechanism provably fails; an appendix treats Samart's conductor-$17$ analogue conditionally. All computations are reproducible from the accompanying code; every certification step is carried out within interval arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26119",
  "author_total": 1,
  "authors": [
    {
      "name": "Huimin Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/awgtg0rkngjaj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26119",
  "fulltext_url": null,
  "id": "awgtg0rkngjaj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26119",
  "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": "W7214092083",
  "pdf_url": "https://arxiv.org/pdf/2609.26119",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/awgtg0rkngjaj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26119",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26119"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k",
  "title_text": "Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/awgtg0rkngjaj",
  "venue": "arXiv",
  "versions": []
}