different fields of science. This section also provides an overview and analysis of recent experiments, which is necessary because the current situation is very complicated. Through this analysis, we are able to delineate a clear approach for identifying pure $\alpha$-T boron. After describing the calculation method (Sec. \[sec:method\]), the article studies the electronic structures of individual interstitial sites (IS) in Sec. \[sec:bondnature\]. By considering all the involved IS, the driving force for the deviation from stoichiometry is clarified in Sec. \[sec:supercell\]. Based on this analysis, a deviation is predicted for various synthesis conditions (Sec. \[sec:high-pressure\]). An assessment on the characterization of recently discovered $\alpha$-T boron is attempted in Sec. \[sec:lattice-para\]. Behind the deviation from stoichiometry, we can say more about why $\alpha$-T boron is not satisfied with a simple structure B$_{52}$. Section \[sec:frustration\] is devoted to discuss this topics from a view point of geometrical frustration. In the last section, we summarize our results.

Theoretical background and experimental facts {#sec:background}
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For a long time solid-state theory of boron crystals had a fundamental problem. For many boron-rich crystals, band structure calculations showed metallic behavior, whereas in experiments all the crystals were found to be semiconductors. We call this problem the 