Absence of Hyperuniformity in Amorphous Hard-Sphere Packings of Nonvanishing Complexity

We relate the structure factor $S(\mathbf{k} \to \mathbf{0})$ in a system of jammed hard spheres of number density $\rho$ to its complexity per particle $\Sigma(\rho)$ by the formula $S(\mathbf{k} \to \mathbf{0})=-1/ [\rho^2\Sigma"(\rho)+2\rho\Sigma'(\rho)]$. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find $\Sigma(\rho)$ and $S(\mathbf{k})$ analytically. Hyperuniformity, which is the vanishing of $S(\mathbf{k} \to \mathbf{0})$, will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperuniform when generated by dynamical rules that produce a non-extensive complexity.