{
  "abstract": "We consider, in a real Clifford algebra, a family of unit bivectors $B_i = C S_i$ formed from $k+1$ pairwise anticommuting generators $C, S_1, \\ldots, S_k$ with square $-1$; we call this a \\emph{star family} with \\emph{central generator} $C$ and \\emph{leaf generators} $S_i$. For the associated rotors $U_i(θ_i) = \\cos\\frac{θ_i}{2} + B_i \\sin\\frac{θ_i}{2}$ we study two objects: the ordered chain $P_k = U_1 \\cdots U_k$ and the \\emph{star-ordered rotor loop} $Θ_k = P_k \\, Q_k$ with the \\emph{same-order reversion product} $Q_k = \\widetilde U_1 \\cdots \\widetilde U_k$, as opposed to the trivially reversed order, for which $U_1 \\cdots U_k \\widetilde U_k \\cdots \\widetilde U_1 = 1$. The scalar part of $Θ_k$ measures the \\emph{order defect} of the family. We prove two theorems. First, the scalar part of the chain factorises, $\\langle P_k \\rangle_0 = \\prod_{i=1}^k \\cos\\frac{θ_i}{2}$; this requires the star structure, and we give a counterexample. Second, we derive a closed expression for $\\langle Θ_k \\rangle_0$, valid for every $k$, whose coefficients obey a \\emph{period-four sign law} $η(m) = (-1)^{\\lfloor m/2 \\rfloor}$ and which admits the compact form $\\langle Θ_k \\rangle_0 = \\operatorname{Re}\\!\\big[(1-\\mathrm{i}) \\prod_i (\\cos^2\\frac{θ_i}{2} + \\mathrm{i}\\sin^2\\frac{θ_i}{2})\\big]$, holding for every pairwise anticommuting unit family without further hypothesis. From the compact form we obtain a polar form, a complete zero criterion, a lacunary coefficient rule, and a proof of non-separability for $k \\geq 2$. Two supplementary theorems extend the result to arbitrary permutations of the index order and to families with mixed squares $B_i^2 \\in \\{-1, +1\\}$. All proofs use only anticommutation, reversion and the symmetry of the scalar form; no physical interpretation is assumed or claimed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21593",
  "author_total": 1,
  "authors": [
    {
      "name": "Nik Jenzer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6c4yff37yprfe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21593",
  "fulltext_url": null,
  "id": "6c4yff37yprfe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21593",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.21593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6c4yff37yprfe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Star-Ordered Rotor Loops in Clifford Algebras",
  "title_text": "Star-Ordered Rotor Loops in Clifford Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/6c4yff37yprfe",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.22159164",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}