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

Status. The covariance studied here is a conditional 3×3 model on Sym²(V_ρ). It is not the 9×9 covariance on End(Heff) 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 πc(vⱼ) = wⱼ ⊗ wⱼ for some wⱼ ∈ V_ρ C²; assume a canonical inter-block identification together with the exchanged anti-linear involution BI on the doublet; and assume that the projected point is fixed at the level of the symmetric square. Then BI(wⱼ) = εⱼ wⱼ with εⱼ ∈ \± 1\, and the doublet coordinates satisfy the equatorial constraint |αⱼ| = |βⱼ|, with wⱼ = (rⱼ/\!√2)\,(eiφⱼ v₊ + εⱼ e-iφⱼ v₋). The sign εⱼ is not removable by a consistent choice of square root; it is carried through the argument. For the resulting 3×3 covariance on Sym²(V_ρ), the diagonal form diag(14,12,14) rⱼ⁴, of normalised spectrum [1: 1/2: 1/2], holds precisely under the two weighted conditions \[ ε_j\, r_j^4\, e2iφ_j = 0, r_j^4\, e4iφ_j = 0. \] Neither follows from the unweighted second-harmonic condition e2iφⱼ = 0, and no amplitude–phase independence is assumed. The factor of $2$ between the central and the weight-±1 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 9×9 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.