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Let   $M_{\langle {\mathbf{n}} \rangle}\in \mathbb  C^{{\mathbf{n}}^2}{\mathord{ \otimes } }\mathbb  C^{{\mathbf{n}}^2}{\mathord{ \otimes } }\mathbb  C^{{\mathbf{n}}^2}$   denote the matrix multiplication tensor for   ${\mathbf{n}}\times{\mathbf{n}}$   matrices. We use the border substitution method [ 2 ,  3 ,  6 ] combined with Koszul flattenings [ 8 ] to prove the border rank lower bound   $\underline {\mathbf{R}}(M_{\langle {\mathbf{n}},{\mathbf{n}},{\mathbf{n}}\rangle})\geq 2{\mathbf{n}}^2 - \lceil {\text{log}}_2({\mathbf{n}})\rceil -1$  .

A $2{\mathbf{n}}^2-{\text{log}}_2({\mathbf{n}})-1$ lower bound for the border rank of matrix multiplication