Locality forbids the barren plateau

For the $n$-qubit family built from single-qubit gates, whose dynamical Lie algebra is the product $\mathfrak g=\bigoplus_{j=1}^{n}\mathfrak{su}(2)_j$, fix the input state $\rho=|0\rangle\!\langle0|^{\otimes n}$ and the local observable $O=X_1$ (Pauli $X$ on the first qubit), with loss $\ell_\theta=\operatorname{Tr}[U_\theta\,\rho\,U_\theta^{\dagger}O]$. The loss variance is the closed-form constant $$\operatorname{Var}_\theta[\ell]=\frac{P_{\mathfrak g_1}(\rho)\,P_{\mathfrak g_1}(O)}{\dim\mathfrak{su}(2)_1}=\frac{2^{-n}\cdot 2^{n}}{3}=\frac13,$$ independent of the qubit count $n$, where $\mathfrak g_1=\mathfrak{su}(2)_1$ is the single-qubit ideal of the observable and $P_{\mathfrak g_1}$ its purity. A constant variance does not decay as $C\,b^{-n}$ with $b>1$, so the family has no barren plateau.