  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 $C^5\otimes C^5\otimes C^5$.
