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

We study the stationary heavy tail of the co-moving recursion xₜ₊₁ = Q_t B Q_t^T x_t + Q_t η_t in d = 3, with independent Haar rotations and Gaussian body-frame noise of covariance Σ, relative to an explicit positive metric g. The tail index α★ is a singular-value functional, so α★(B) = α★(Bᵀ). The central unconditional result is a structure theorem. In the g = I chart, writing B = S + b_ω, the conditions Δ₆(B) = −16 det[ω, Sω, S²ω] ≠ 0, B ≁O(3) Bᵀ, rank[ω, Sω, S²ω] = 3, and controllability of (S, ω) are equivalent. Thus intrinsic operator chirality is exactly a Kalman rank condition. The tail level, unlike the exponent, is not forced to be transpose-invariant. An exact polar twist identity rotates the anisotropic noise frame under mirroring, which is the exact and unconditional mechanism by which a mirror asymmetry can enter the level. That a nonzero asymmetry is actually realised is established here conditionally and numerically rather than by an unconditional closed-form example: under Assumption R, on the SPD chart, Δ log C = ⟨A, G⟩ + o(‖A‖) as A → 0, with G = 2H∘[Σ, ∂_Σ log C], and the finite-threshold simulations are consistent with this first-order law. Strict-Haar averaging nevertheless makes every lagged second-order cross-moment transpose-blind, while a fourth radial moment retains an odd channel. No finite scalar-weighted radial-moment combination cancels the even quadratic form uniformly while retaining the odd covector. The exact metric-ray inverse and common-Gram repair remain population/oracle identification statements.