{
  "abstract": "We formulate the low-energy theory of superconducting pairing in the geometric algebra $Cl_{3,0}$. Conventional BCS mean-field theory and its extensions ($s$-, $p$-, $d$-wave, etc.) have algebraic counterparts in an eight-dimensional real multivector space. The construction is a two-level projection: matrix blocks are represented as Clifford multivectors, and their grade-0 projected geometric products extract the scalar blocks from which Hamiltonians are built. The result is an algebraic reconstruction, physically equivalent to the standard complex-matrix formulation. In this reconstruction, the dichotomy of pairing channels acquires an algebraic root: it follows from the fermionic identity $B^\\dagger(V)=-B^\\dagger(V^{\\mathsf T})$, and the Fierz rearrangement is realized by the map $Φ:G\\mapsto Ge_{31}$ (with $e_{31}^2=-1$). This yields a kernel/matrix dictionary: at the effective-kernel level grades $0\\oplus3$ correspond to spin singlets and grades $1\\oplus2$ to triplets; at the pairing-matrix level the singlet plane is $\\mathrm{span}\\{e_2,e_{31}\\}$. The BCS limit has a double grade identity: kernel grade 0 (scalar glue) and vertex grade 2, a bivector ($iσ_2\\leftrightarrow e_{31}$). We state three algebraic results: (i) a decomposition theorem for BdG quantum geometry with a null-texture criterion, applied to the chiral-state controversy in CsV$_3$Sb$_5$; (ii) a Fierz sign duality in the four-dimensional Minkowski algebra, with a chirality-reversal extension to two-node Weyl-semimetal models; (iii) an error bound and condition-number structure for tomographic reconstruction under a linear-response probe model. Illustrations include a glue-generation mechanism, a one-loop algebraic renormalization flow whose feeding constant coincides in magnitude with the $\\mathfrak{su}(2)$ Casimir, closed-form $μ=0$ inter-node spectra, and a UTe$_2$ two-phase analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33806",
  "author_total": 1,
  "authors": [
    {
      "name": "Youping Dai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vh22eh664pn0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33806",
  "fulltext_url": null,
  "id": "8vh22eh664pn0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33806",
  "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",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "cond-mat.str-el"
    ],
    [
      "arxiv",
      "cond-mat.supr-con"
    ],
    [
      "openalex",
      "3104|Physics of Superconductivity and Magnetism"
    ]
  ],
  "openalex": "W7214762393",
  "pdf_url": "https://arxiv.org/pdf/2609.33806",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vh22eh664pn0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.33806",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33806"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Superconducting Pairing Symmetry on Geometric-Algebra Foundations via the Scalar-Projection Method",
  "title_text": "Superconducting Pairing Symmetry on Geometric-Algebra Foundations via the Scalar-Projection Method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/8vh22eh664pn0",
  "venue": "arXiv",
  "versions": []
}