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Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction — arXiv:2609.35923

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.35923
License (as archived): CC0-1.0
Archived: 2026-10-07T18:04:20.928Z
SHA-256 of the archived text: e8dd33b5829fb744ed89feaad94095517112f7a5da5b5fd7939aec21b82a90a2
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Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction — arXiv:2609.35923 — arXiv (en), https://arxiv.org/abs/2609.35923 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ce304bef03f0 (Metraipolis archive dc_ce304bef03f0, sha256 e8dd33b5829fb744ed89feaad94095517112f7a5da5b5fd7939aec21b82a90a2)

BibTeX

@misc{dcce304bef03f0,
  title = {Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction — arXiv:2609.35923},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.35923},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ce304bef03f0; sha256 e8dd33b5829fb744ed89feaad94095517112f7a5da5b5fd7939aec21b82a90a2},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.35923. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We ask whether transposition changes the stationary heavy tail of the co-moving recursion $\mathbf x_{t+1}=Q_tBQ_t^\top\mathbf x_t+Q_t\boldsymbol\eta_t$ in $d=3$. The model has independent Haar rotations, Gaussian body-frame noise with covariance $\Sigma$, and an explicit positive metric $\mathsf g$. The tail index $\alpha_\star$ depends only on singular values, so $\alpha_\star(B)=\alpha_\star(B^\top)$. The structure theorem and polar-twist mechanism are unconditional. In the $\mathsf g=I$ chart, write $B=S+\widehat{\boldsymbol\omega}$: $\Delta_6(B)=-16\det[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]$. Then $B\not\sim_{O(3)}B^\top$, $\Delta_6(B)\ne0$, $\operatorname{rank}[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]=3$, and controllability of $(S,\boldsymbol\omega)$ 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, $\Delta\log C=\langle A,G\rangle+o(\|A\|)$ as $A\to0$, with $G=2H\circ[\Sigma,\partial_\Sigma\log C]$; 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-30 in math.PR).
Categorie arXiv: Probability (math.PR) ; Mathematical Finance (q-fin.MF).
Pagina del lavoro: https://arxiv.org/abs/2609.35923 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T17:37:10.649Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-30?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).