Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction
Abstract
We ask whether transposition changes the stationary heavy tail of the co-moving recursion xt+1=QtBQt⊤xt+Qtηt in d=3. The model has independent Haar rotations, Gaussian body-frame noise with covariance Σ, and an explicit positive metric g. The tail index α⋆ depends only on singular values, so α⋆(B)=α⋆(B⊤). The structure theorem and polar-twist mechanism are unconditional. In the g=I chart, write B=S+ω̂: Δ6(B)=−16det[ω,Sω,S2ω]. Then Bnot∼O(3)B⊤, Δ6(B)≠0, rank[ω,Sω,S2ω]=3, and controllability of (S,ω) 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, Δlog C=⟨ A,G⟩+o(|A|) as A→0, with G=2H∘[Σ,∂Σlog 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.
Related
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BibTeX
@misc{calıskan2026transposeodd,
title = {{Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction}},
author = {Nihat Çağrı Çalışkan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35923},
eprint = {2609.35923},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35923},
}
RIS
TY - UNPB
TI - Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction
AU - Nihat Çağrı Çalışkan
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35923
UR - https://arxiv.org/abs/2609.35923
AB - We ask whether transposition changes the stationary heavy tail of the co-moving recursion x_t+1=Q_tBQ_t^⊤x_t+Q_tη_t in d=3. The model has independent Haar rotations, Gaussian body-frame noise with covariance Σ, and an explicit positive metric g. The tail index α_⋆ depends only on singular values, so α_⋆(B)=α_⋆(B^⊤). The structure theorem and polar-twist mechanism are unconditional. In the g=I chart, write B=S+ω̂: Δ_6(B)=−16det[ω,Sω,S^2ω]. Then Bnot∼_O(3)B^⊤, Δ_6(B)≠0, rank[ω,Sω,S^2ω]=3, and controllability of (S,ω) 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, Δlog C=⟨ A,G⟩+o(|A|) as A→0, with G=2H∘[Σ,∂_Σlog 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.
ER -