arXiv:2611.09790v1 Announce Type: cross Abstract: We consider Box-Cox transformations of correlation matrices, $\Phi_\alpha(C)=\mathrm{vecl}\{(C^{\alpha}-I)/\alpha\}$, with Jacobian $J_\alpha$. The family nests the raw correlations at $\alpha=1$, the generalised Fisher transformation (GFT), $C\mapsto\mathrm{vecl}(\log C)$, at $\alpha=0$, and the negated off-diagonal concentration coordinates at $\alpha=-1$. The inverse problem is a strictly convex program for every $\alpha$, so every member is injective. The map is onto $\mathbb{R}^{d}$, $d=n(n-1)/2$, if and only if $\alpha\le0$, such that $\alpha=0$ defines the boundary for unrestricted parametrizations. For $\alpha>0$ the image is bounded. The conditioning obeys the bound $\kappa(J_\alpha)\le\kappa(C)^{|1-\alpha|}$, where $\kappa(C)=\lambda_{\max}(C)/\lambda_{\min}(C)$, and on the surjective range the exponent is smallest at $\alpha=0$. Under Gaussian sampling with $n\geq3$, the asymptotic covariance of the transformed sample correlations is the identity at independence for every member, but only $\alpha=0$ retains the identity to first order in the correlations, whereas the leading joint third cumulant at independence vanishes only at $\alpha=1/3$. Finally, applying the transformation to the diagonal rescaling of $C$ that makes the result hollow adds a new family to the existing correlation parametrizations. For every $\alpha$ the rescaling solves a strictly convex problem and the inverse is explicit, and the scaled coordinates are unrestricted exactly when $\alpha=0$, where they recover the known global normal coordinates, centred at independence, for the efficient information metric of the correlations.