 Indirect gap in a uniformly split zone-boundary model

Suppose a model replaces a set of free-electron boundary energies $E^{(0)}$ by lower and upper edges $E^{(0)}-V$ and $E^{(0)}+V$, with $V>0$. If these are the relevant global band edges, the indirect band gap is $(E_{\min }^{(0)}+V)-(E_{\max }^{(0)}-V)$, giving the displayed expression. A positive direct gap $2V$ at each crossing need not be enough: a band insulator needs $2V>E_{\max }^{(0)}-E_{\min }^{(0)}$. This is a band-edge estimate, not a general three-dimensional nearly-free-electron theorem; multiple Bragg planes at a corner can require a larger coupling matrix.