{
  "abstract": "We study the algebraic structure underlying Sp-orientations in the $\\eta$-periodic motivic stable homotopy category $SH(k)[\\eta^{-1}]$. Borel classes determine a geometric formal ternary law, but the HW-Hurewicz map shows that its universal coefficients generate a proper $W(k)$-subalgebra $\\Lambda\\subsetneq (MSp[\\eta^{-1}])_*$, although $\\Lambda[1/2]=(MSp[\\eta^{-1}])_*[1/2]$. Thus the failure of generation is purely 2-primary. To capture part of the missing information, we introduce framed involutions. The spectrum $MSp[\\eta^{-1}]$ carries a canonical framed involution, yielding a Quillen-type idempotent with telescope $MSL[\\eta^{-1}]$ and a canonical splitting $(MSp[\\eta^{-1}])_* \\cong \\mathcal{R}_{fr} \\otimes_{\\mathbb{Z}} (MSL[\\eta^{-1}])_*$, where $\\mathcal{R}_{fr}$ is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring $\\mathcal{W}^{\\eta}$, and prove that $\\mathcal{W}^{\\eta}$ is isomorphic to the Lazard ring $L$ after inverting 2. If $W(k)\\cong \\mathbb{Z}$, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map $\\phi:\\mathcal{W}^{\\eta}\\to (MSp[\\eta^{-1}])_*$ that is injective and becomes an isomorphism after inverting 2. Integrally, however, additional secondary power series are needed to recover the full cobordism coefficient ring.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.06795",
  "author_total": 1,
  "authors": [
    {
      "name": "Tao Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kx5bpeh9wtag.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.06795",
  "fulltext_url": null,
  "id": "3kx5bpeh9wtag",
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  "landing_url": "https://hal.science/hal-05687153v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
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      "math.AT"
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      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
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  "openalex": "W7167808831",
  "pdf_url": "https://hal.science/hal-05687153/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kx5bpeh9wtag.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:31Z",
      "native_id": "hal-05687153",
      "source": "hal",
      "url": "https://hal.science/hal-05687153v2"
    },
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2607.06795",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.06795"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On $η$-periodic Formal Ternary Laws",
  "title_text": "On η-periodic Formal Ternary Laws",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/3kx5bpeh9wtag",
  "venue": "HAL",
  "versions": []
}