{
  "abstract": "We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm $B(x)=x+\\sum_{n\\geq1}b_nx^{n+1}$, obtained from the universal formal group of complex cobordism by the modulus square construction: $$ b_n=\\frac{C_n}{d_n},\\qquad d_n=\\frac{n+1}{2}\\operatorname{lcm}(1,\\ldots,2n+2), $$ where $C_n\\inΩ_{\\mathrm U}^{-4n}$ is primitive and undecomposable. We prove that $C_n$ belongs to the coefficient ring $Λ$ of the universal two-valued law and to the subring $Λ_{\\mathrm{St}}$ generated by quotient Stong manifolds, and give an explicit integral Stong manifold formula for it. The rational lift of $b_n$ to quaternionic cobordism modulo torsion has exact denominator $2d_n$. Consequently, $C_n$ has no integral quaternionic lift, whereas $2C_n$ does. Every Chern number of $C_n$ is divisible by $d_n$, and $c_{2n}(C_n)=d_n$. We introduce odd genera $\\mathrm{Bc}_N$. For $N\\geq n$, $\\mathrm{Bc}_N$ detects the odd part of $d_n$. Its restriction to the Stong ring is integral exactly for $N\\leq3$, while the universal odd genus $\\mathrm{Bc}_{\\infty}$ takes values there with only odd denominators. For every $n\\geq5$, we construct undecomposable classes in $Λ^{-4n}$ with equal Ochanine genera and top Chern numbers but distinct $\\mathrm{Bc}_3$ values. Finally, the oriented extension of $\\mathrm{Bc}_3$ is integral on closed spin manifolds of real dimension below $24$ and on closed string manifolds of real dimension at most $24$. The spin bound is sharp: an Anderson-Brown-Peterson spin $24$-manifold has nonintegral $\\mathrm{Bc}_3$ genus.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15878",
  "author_total": 2,
  "authors": [
    {
      "name": "Victor Buchstaber"
    },
    {
      "name": "Mikhail Kornev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1fkyh27q5nybk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15878",
  "fulltext_url": null,
  "id": "1fkyh27q5nybk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15878",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7213240315",
  "pdf_url": "https://arxiv.org/pdf/2609.15878",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1fkyh27q5nybk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15878",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15878"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Logarithm of the Universal Two-Valued Formal Group",
  "title_text": "Logarithm of the Universal Two-Valued Formal Group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1fkyh27q5nybk",
  "venue": "arXiv",
  "versions": []
}