{
  "abstract": "Bloch's identity $\\int_γη(f,g)=D_E((f)\\diamond(g))$, relating the regulator of a $K_2$-symbol on an elliptic curve $E/\\mathbb{Q}$ to the elliptic dilogarithm of the diamond product of the divisors, is transmitted in the Mahler-measure literature with $γ$ generating the anti-invariant subgroup $H_1(E,\\mathbb{Z})^-$ and factor $1$ (e.g. Lalin-Ramamonjisoa, Thm. 6). We observe that, when $E(\\mathbb{R})$ has one connected component ($Δ<0$), this reading is incompatible with the published intersection-product normalization (Brunault, Proposition 19): the anti-invariant subgroup then has index $2$ in the regulator quotient $H_1(E,\\mathbb{Z})/H_1(E,\\mathbb{Z})^+$. The correct statement -- factor $1$ for the quotient generator, hence factor $ι(E)\\in\\{1,2\\}$ for the subgroup generator, with $ι(E)=1\\iffΔ>0$ -- follows from Brunault's formula by an elementary index computation. We document the phenomenon twice: on 11.a3 by proved values (a Siegel-unit anchor from [Zheng, arXiv:2609.26119]; the diamond side by Brunault and Bertin), and on 20.a4 by high-precision computation of both integrals against the $q$-series for $D_E$ (subgroup/quotient ratio $2$ to $13$ digits). We also show that Touafek's Theorem 2 (involving a $Δ<0$ curve) mixes two independent normalizations (subgroup versus quotient, and a $(2π)^{-1}$ prefactor versus the bare diamond value), and we identify the two-part reinterpretation making its printed formulas mutually compatible. In the cases we have checked, no Mahler-measure or $D_E$-identity is affected: the main theorem of Lalin-Ramamonjisoa (conductor $17$, $Δ>0$) and Bertin's proved exotic relations stand untouched, and Touafek's final $D_E$-relation is not contradicted and is numerically supported, though its printed derivation retains the gap described in the text. A systematic audit is beyond the scope of this note.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05420",
  "author_total": 1,
  "authors": [
    {
      "name": "Huimin Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqg3xc7gt5wva.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05420",
  "fulltext_url": null,
  "id": "eqg3xc7gt5wva",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05420",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05420",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqg3xc7gt5wva.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05420",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05420"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient",
  "title_text": "A normalization issue in a transmitted form of Bloch's diamond formula: anti-invariant subgroup versus regulator quotient",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:46Z",
  "url": "https://research.draconitic.com/paper/eqg3xc7gt5wva",
  "venue": "arXiv",
  "versions": []
}