On the Tensor Rank of 3×3 Matrix Multiplication: Barriers and Open Problems

The tensor rank of 3×3 matrix multiplication has remained bounded between 19 and 23 since Laderman’s 1976 algorithm. This gap has persisted for nearly five decades of research. Blankline Research has assembled a dedicated team to investigate whether tensor ranks of 19, 20, 21, or 22 are achievable. This report presents our current findings. We identify four anchor products that form an irreducible orthogonal structure and introduce the w-vector routing problem, which obstructs compound term compression. Using SMT solvers and exhaustive search, we further prove that Laderman’s algorithm is locally optimal. Together, these results characterize precise structural barriers that must be overcome in order to close the remaining rank gap.