O30 Conditional 3×3 Model on Sym²(Vρ)

Scope

O30 studies a model whose data are assumed to lie on a Veronese cone in $\operatorname{Sym}^2(V_\rho)$. This is hypothesis (H0), not a consequence of the measured $6\to3$ collapse. Hypotheses (H1)–(H3) supply the inter-block identification, weight exchange and fixed projected symmetric square.

Under those assumptions, a Veronese root satisfies $\beta_j=\varepsilon_j\overline{\alpha_j}$ with $\varepsilon_j\in\{\pm1\}$. Both branches belong to the conditional model.

Two distinct covariance objects. O28 measures a $9\times9$ covariance acting on $\operatorname{End}(H_{\mathrm{eff}})$; that measurement is an empirical result. The covariance $\mathcal{C}_c$ studied here is a conditional $3\times3$ model on $\operatorname{Sym}^2(V_\rho)$. These are different objects: O30 supplies no analytical account of the O28 spectrum, and nothing here derives it.

Exact spectral conditions

With the average taken over the full declared index set, the model covariance is diagonal with entries proportional to $(\tfrac14,\tfrac12,\tfrac14)$ if and only if

$$\langle\varepsilon r^4e^{2i\varphi}\rangle=0,\qquad \langle r^4e^{4i\varphi}\rangle=0.$$

The factor two in the diagonal entries is the square of the $\sqrt2$ coefficient in the symmetric-tensor basis. Diagonal entries alone are not eigenvalues when either off-diagonal cancellation fails.

Countermodels and zeros

Unweighted second-harmonic decorrelation is insufficient; the paper records explicit countermodels. Null points remain in the common $N_{\mathrm{total}}$ normalisation, with $\varepsilon_j:=+1$ harmlessly assigned because it is multiplied by $r_j^4$. Renormalising only over non-zero points rescales the whole covariance.

Reference

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