Still open — the rank of 3 × 3 matrix multiplication

The 2 × 2 question is closed: rank 7. One size up it is not. For 3 × 3 matrices the tensor is 9 × 9 × 9 with 27 ones, and nobody knows its rank:

• at most 23 — Laderman found 23 blocks in 1976;
• at least 19 — Bläser’s proof, 2003.

DeepMind’s published 3 × 3 entry has 23 blocks in both arithmetics: AlphaTensor matched Laderman and did not lower it.

What is at stake is the exponent. Recursing on 3 × 3 blocks with R multiplications costs n^(log₃ R): rank 21 would beat Strassen (n^2.771 against n^2.807); rank 22 would not (n^2.814). Nobody has found either, and nobody has proved 23 is the fewest.