{
  "abstract": "The conformal geometric algebra Cl(4,2) ~ R(8) carries a natural Z_2-grading defined by the dilatation bivector D=\\e_4e_0, which satisfies $D^2=1$ and splits the eight-dimensional spinor module into two four-dimensional eigenspaces $S=S_+\\oplus S_-$. We prove that the ad_D-eigenvectors of Cl(4,2), those elements with eigenvalue~$\\pm1$ under the Lie bracket $[D,\\cdot]$, are exhausted by 16 doublets of the form $T^\\pm_J=n_\\bullet e_J$, where $n_\\bullet \\in {n_\\infty,n_o}$ is a null vector and $e_J$ is a product of Lorentz-sector basis vectors indexed by $J\\subseteq\\{1,2,3,5\\}$. These doublets satisfy universal algebraic identities: $(T^\\pm_J)^2=0$ (nilpotency) and $T^+_J\\,T^-_J=(-1)^{k(k+1)/2+1}\\,η_J 2Π_+$ (single-projector proportionality), where $k=|J|$, $η_J=\\prod_{j\\in J}η_{jj}$ is the spectator metric, and $Π_+=\\tfrac12(1+D)$. The contracted sums $\\sum_{|J|=k}T^+_J T^-_J$ are shown to be controlled by the elementary symmetric polynomials $S_k$ of the Lorentz signature values $(+1,+1,+1,-1)$, with the vanishing of $S_2=0$ reflecting the $(3,1)$ signature of physical spacetime. The even-grade doublets (belonging to Cl^+(4,2) ~ Cl(4,1)) contribute $8Π_+$ to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra sl(8,R) in which the conformal algebra so(4,2) embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension Cl(p+1,q+1); there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars ... [shortened]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22275",
  "author_total": 1,
  "authors": [
    {
      "name": "Steen H. Hansen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8m8bm8vqjfqnk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22275",
  "fulltext_url": null,
  "id": "8m8bm8vqjfqnk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22275",
  "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.GM"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7214023084",
  "pdf_url": "https://arxiv.org/pdf/2609.22275",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8m8bm8vqjfqnk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22275",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22275"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors",
  "title_text": "The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8m8bm8vqjfqnk",
  "venue": "arXiv",
  "versions": []
}