A Conditional 3×3 Model on Sym²(Vρ): Equatorial Fibres from a Fixed Symmetric Square, and the Exact Decorrelation Conditions for its Spectrum

Abstract

Status. The covariance studied here is a conditional $3\times3$ model on ${\operatorname{Sym}}^2(V_\rho)$. It is not the $9\times9$ covariance on ${\operatorname{End}}({H_{\mathrm{eff}}})$ measured in O28, and it does not provide an analytical account of its spectrum. Assume that the admissible data lie on the Veronese cone of rank-one symmetric squares, so that each observed projection is $\pi_c(v_j) = w_j \otimes w_j$ for some $w_j \in V_\rho \cong \mathbb{C}^2$; assume a canonical inter-block identification together with the exchanged anti-linear involution ${\mathrm{BI}}$ on the doublet; and assume that the projected point is fixed at the level of the symmetric square. Then ${\mathrm{BI}}(w_j) = \varepsilon_j w_j$ with $\varepsilon_j \in \{\pm 1\}$, and the doublet coordinates satisfy the equatorial constraint $|\alpha_j| = |\beta_j|$, with $w_j = (r_j/\!\sqrt{2})\,(e^{i\phi_j} v_+ + \varepsilon_j e^{-i\phi_j} v_-)$. The sign $\varepsilon_j$ is not removable by a consistent choice of square root; it is carried through the argument. For the resulting $3\times3$ covariance on ${\operatorname{Sym}}^2(V_\rho)$, the diagonal form $\operatorname{diag}(\tfrac14,\tfrac12,\tfrac14)\langle r_j^4 \rangle$, of normalised spectrum $[1: \tfrac{1}{2}: \tfrac{1}{2}]$, holds precisely under the two weighted conditions \[ \bigl\langle \varepsilon_j\, r_j^4\, e^{2i\phi_j} \bigr\rangle = 0, \qquad \bigl\langle r_j^4\, e^{4i\phi_j} \bigr\rangle = 0. \] Neither follows from the unweighted second-harmonic condition $\langle e^{2i\phi_j}\rangle = 0$, and no amplitude–phase independence is assumed. The factor of $2$ between the central and the weight-$\pm1$ eigenvalues is the squared symmetric-tensor normalisation. The Carnot grading labels the central direction $Z=[X,Y]$ that carries it; it is not the algebraic cause of the factor. The $9\times9$ covariance ratio measured in O28 is an empirical result, and this paper supplies no analytical account of it. O28 is cited here as external motivation only; no derivation runs from it to this model.