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abstract: 'Recent discoveries of supposedly pure $\alpha$-tetragonal boron require to revisit its structure. The system is also interesting with respect to a new type of geometrical frustration in elemental crystals, which was found in $\beta$-rhombohedral boron. Based on density functional theory calculations, the present study has resolved the structural and thermodynamic characteristics of pure $\alpha$-tetragonal boron. Different from $\beta$-rhombohedral boron, the conditions for stable covalent bonding (a band gap and completely filled valence bands) are almost fulfilled at a composition B$_{52}$ with two $4c$ interstitial sites occupied. This indicates that the ground state of pure $\alpha$-tetragonal boron is stoichiometric. However, the covalent condition is not perfectly fulfilled because non-bonding in-gap states exist that cannot be eliminated. The half occupation of the $4c$ sites yields a macroscopic amount of residual entropy, which is as large as that of $\beta$-rhombohedral boron. Therefore, $\alpha$-tetragonal boron can be classified as an elemental crystal with geometrical frustration. Deviations from stoichiometry can occur only at finite temperatures. Thermodynamic considerations show that deviations $\delta$ from the stoichiometric composition (B$_{52+\delta}$) are small and positive. For reported high-pressure syntheses conditions $\delta$ is predicted to be about 0.1 to 0.2. An important difference between pure and C- or N-containing 