Altermagnetism and weak magnetism in the insulating distorted perovskite antiferromagnet <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>NaOsO</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math>

The ${\mathrm{GdFeO}}_{3}$-type perovskite antiferromagnet ${\mathrm{NaOsO}}_{3}$, calculated here to be altermagnetic for all three typical collinear antiferromagnetic orders, was suggested early on to be a Slater-type insulator, due in large part to its continuous metal-insulator transition and its small energy gap. Below the N\'eel temperature, the gap opens along with ``weak magnetism'', accompanied by a sharp change in the magnetic susceptibility and resistivity. Without explicit correlation in the band structure calculation, and neglecting spin-orbit coupling (SOC), already a gap opens. Inclusion of a modest on-site Coulomb repulsion ($U\ensuremath{\sim}1$ eV) is sufficient to eliminate a SOC-induced small band overlap, opening a gap similar to the experimentally observed gap of around 100 meV. Combined evidence supports the viewpoint that ${\mathrm{NaOsO}}_{3}$ lies in an unusual crossover region between Slater and Mott insulator. The unreported altermagnetism in ${\mathrm{NaOsO}}_{3}$ is demonstrated and its consequences are considered. The origin of the very weak magnetism has been investigated using a combination of ab initio calculations and symmetry analysis of the magnetic space group, confirming the origin lying in the Dzyaloshinskii-Moriya SOC buttressed by altermagnetic order. After determining the easy axis, our calculation leads to an Os spin canting angle of about ${3}^{\ensuremath{\circ}}$, accounting for the observed weak magnetism and sharp change in the susceptibility. The altermagnetism spin-split bands (up to $\ensuremath{\sim}100$ meV) are accompanied by a chiral-split magnon spectrum in both acoustic and optical modes in the THz range, and lead to significant anomalous Hall conductivity upon hole doping.