Barren-plateau gradient variance

For a parametrized quantum circuit whose dynamical Lie algebra is the orthogonal direct sum $\mathfrak{g} = \bigoplus_j \mathfrak{g}_j$ of its ideals, the loss variance is the reductive Ragone sum $$\operatorname{Var}_\theta[\ell] = \sum_j \frac{P_{\mathfrak{g}_j}(\rho) P_{\mathfrak{g}_j}(O)}{\dim\mathfrak{g}_j},$$ where $P_{\mathfrak{g}_j}$ is the $\mathfrak{g}_j$-purity (derived from cross-ideal Casimir orthogonality); the simple-Lie-algebra case is its single-ideal corollary. When $\dim\mathfrak{g}$ grows exponentially in the qubit count, an exponential barren plateau follows (given the Haar second-moment / Schur projection the formalization carries as a named hypothesis).