= Solution
With <spin degeneracy>, each <energy band> contains two <electrons> per <primitive unit cell>. Divalency therefore permits a filled first band, but only if its maximum lies below the second band's minimum. With negligible ionic <periodic potential>, the circular <Fermi surface> and the rectangular zone have equal enclosed area, not equal shape. Electrons occupy some states outside the first zone while some higher-<energy> first-zone states remain empty. Folded bands overlap, and the material is a <metal>.
A <periodic potential> mixes <plane waves> differing by a <reciprocal lattice vector>. At a two-wave <Bragg point>, the <nearly-free electron model> gives a direct gap $2|V_{\mathbf G}|$. Increasing appropriate <Fourier coefficients> can remove the band overlap. With the first band filled, the system is then a <band insulator>, provided the <temperature> is small compared with the insulating <band gap> in energy units.
The <rectangular free-electron band-overlap scale> follows from two different points in reciprocal space. Put $C=\hbar^2/(2m_e)$. Before gap opening, the highest first-zone energy is at the corner, while the lowest second-zone energy is at the midpoint of the nearer face:
$$
E_{1,\max}^{(0)}=C(K^2+L^2),\qquad E_{2,\min}^{(0)}=CK^2.
$$
Thus the downward shift $\delta_v$ of the first-band maximum plus the upward shift $\delta_c$ of the second-band minimum must overcome
$$
\boxed{\Delta_{\rm overlap}=CL^2=\frac{\hbar^2\pi^2}{2m_eb^2},\qquad
E_g^{\rm indirect}=\delta_v+\delta_c-\Delta_{\rm overlap}>0.}
$$
In a rigid-splitting estimate, an opening $\Delta$ lowers the first band by $\Delta/2$ and raises the second by $\Delta/2$, giving the familiar threshold $\Delta>\hbar^2\pi^2/(2m_eb^2)$. Equality is only band touching, not a positive insulating gap. This expression is the \b[overlap that must be removed], under that explicit band-shift approximation.
<Positive direct gaps can coexist with band overlap>: a local direct splitting and the global <indirect band gap> are not interchangeable. The shape and <Fourier coefficients> of the ionic potential are unspecified, so there is no universal exact critical direct-gap width determined by $a,b$ alone. For a concrete demonstration, a separable potential $2V_x\cos(2Kx)+2V_y\cos(2Ly)$ has axial direct gaps $\Delta_x=2|V_x|$ and $\Delta_y=2|V_y|$. To first order, the corner's lowest level is $C(K^2+L^2)-|V_x|-|V_y|$, and the two candidate second-band minima are $CK^2+|V_x|$ and $CL^2+|V_y|$. Therefore, within this model and neglecting higher-order band shifts, nonoverlap requires
$$
\Delta_x+\frac{\Delta_y}{2}>CL^2,
\qquad \frac{\Delta_x}{2}+\Delta_y>CK^2.
$$
Equal axial gaps would give the first-order estimate $\Delta>2CL^2/3$, rather than the rigid-splitting estimate. Such a large opening can exceed the quantitative range of weak-potential perturbation theory, but the example identifies the missing model assumption: corner levels feel both couplings. The exact general answer is the positive indirect-gap criterion above; a geometry-only direct-gap threshold would claim more than the supplied potential information supports.
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