= Solution
An isolated spin-degenerate lowest <energy band> accommodates two <electrons> per atom when there is one atom per primitive cell. Starting from one per atom, the estimate for filling the band is therefore
$$
\boxed{\text{one additional electron per Au atom}.}
$$
This corresponds to an additional areal carrier count of about $4.77\times10^{19}\ {\rm m^{-2}}$. If a genuine <band gap> separates the filled first band from the next band, there are no partially occupied band states at low <temperature>, so ordinary metallic conduction is lost.
Simply touching an edge is insufficient. In the circular approximation first contact would occur at $k_F=\pi/a$, corresponding to $n_ea^2=\pi/2\simeq1.57$ <electrons> per cell; that is a partly filled band and need not be insulating. Near zone boundaries, the periodic potential reshapes the free-electron contours. The two-electron filling estimate assumes a global gap without overlapping bands, rather than inferring an insulator merely from one local avoided crossing.
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