{
  "abstract": "We ask whether transposition changes the stationary heavy tail of the co-moving recursion $\\mathbf x_{t+1}=Q_tBQ_t^\\top\\mathbf x_t+Q_t\\boldsymbolη_t$ in $d=3$. The model has independent Haar rotations, Gaussian body-frame noise with covariance $Σ$, and an explicit positive metric $\\mathsf g$. The tail index $α_\\star$ depends only on singular values, so $α_\\star(B)=α_\\star(B^\\top)$. The structure theorem and polar-twist mechanism are unconditional. In the $\\mathsf g=I$ chart, write $B=S+\\widehat{\\boldsymbolω}$: $Δ_6(B)=-16\\det[\\boldsymbolω,S\\boldsymbolω,S^2\\boldsymbolω]$. Then $B\\not\\sim_{O(3)}B^\\top$, $Δ_6(B)\\ne0$, $\\operatorname{rank}[\\boldsymbolω,S\\boldsymbolω,S^2\\boldsymbolω]=3$, and controllability of $(S,\\boldsymbolω)$ are equivalent; intrinsic operator chirality is a Kalman rank condition. Unlike the exponent, the tail level need not be transpose-invariant. The exact polar-twist identity rotates the anisotropic noise frame under mirroring, showing how level asymmetry can enter. A nonzero asymmetry is established only conditionally and numerically, not by a closed-form example: under Assumption R on the SPD chart, $Δ\\log C=\\langle A,G\\rangle+o(\\|A\\|)$ as $A\\to0$, with $G=2H\\circ[Σ,\\partial_Σ\\log C]$; finite-threshold simulations are consistent with this first-order law. Strict-Haar averaging makes every lagged second-order cross-moment transpose-blind; a fourth radial moment retains an odd channel. No finite scalar-weighted radial-moment combination uniformly cancels the even quadratic form while retaining the odd covector.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35923",
  "author_total": 1,
  "authors": [
    {
      "name": "Nihat Çağrı Çalışkan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vszafffrxmgc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35923",
  "fulltext_url": null,
  "id": "1vszafffrxmgc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35923",
  "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.PR"
    ],
    [
      "arxiv",
      "q-fin.MF"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214971953",
  "pdf_url": "https://arxiv.org/pdf/2609.35923",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vszafffrxmgc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.35923",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35923"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction",
  "title_text": "Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/1vszafffrxmgc",
  "venue": "arXiv",
  "versions": []
}