Tensor rank of 3×3 matrix multiplication: 19 ≤ R ≤ 23, find rank 22 or raise the lower bound

Over the rationals, reals and complex numbers the bounds remain 19 ≤ R(3) ≤ 23: Laderman's 23-product algorithm (1976) and Bläser's lower bound of 19 (2003), which holds over every field. Large computer searches (flip graphs, reinforcement learning, SAT) have found many inequivalent rank-23 schemes and none of rank 22. Over the two-element field F2 the lower bound moved quickly in 2026, all in preprints: 20 (Chengu Wang, arXiv:2603.07280, March), 21 (Wang, arXiv:2609.06725, September; a structural proof by Yang and co-authors, arXiv:2609.18722), 22 (attributed to Rudich and Rousseau in the next preprint; no public source was found), and a claim that the F2 rank is exactly 23 (Tejasvi Singh Tomar, arXiv:2610.11823, 8 October 2026, revised 12 October, with Lean 4 certificates including a kernel-only build). If the F2 claim holds, no rank-22 scheme exists with integer coefficients (reduce it mod 2); a rank-22 scheme over the rationals would then need coefficients with even denominators. None of these preprints is refereed.

Little practical value: a rank-22 scheme gives an exponent of log₃22 ≈ 2.81, worse than Strassen's log₂7 ≈ 2.807, and small-matrix kernels are limited by additions and memory traffic rather than by multiplications. The value is structural: the 3×3 tensor is the smallest open case, and methods that settle it (lower-bound techniques, exhaustive certified search) are tests for the larger tensors behind ω.

Upper bound: 22 rank-one tensors (explicit coefficient vectors over the rationals) that sum to the 3×3 matrix multiplication tensor, verified by exact arithmetic. Lower bound: a proof, ideally formal, that R(3) ≥ 20 over the rationals or the complex numbers.