We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczy\'{n}ska-Buczy\'{n}ski and results of Jelisiejew-\v{S}ivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for $1$-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $C^5\otimes C^5\otimes C^5$.

翻译：我们确定一套简洁的最小边界级的微粒的公式,以美元为单位,以美元=5美元计算,以美元=5美元计算,以美元为单位计算,以美元=5美元计算,以美元=5 600美元计算,以美元=5 000美元计算,以C'm\otimes C'm\otimes C'm\otimes C'm\otimes C'm_m\otimes C'm_m=m美元计算。我们利用两个最近的事态发展,即Buczy\'{{n}定义的111-equarrations 和Jelisiejew-v{v}Sivic对各种通勤矩阵的结果,我们采用了一个新的简洁的高压器变数,即111-algemaric,利用它来强化弗里德兰的正常形态,使1美元贬值的微粒子能够满足斯特拉斯的方程式。我们用111-algebra 来确定野生最低边界级的微粒,并将其分类为$C5\otimec times C'otimes C'otime5美元=5美元。