The recent publication in Nature of "Discovering faster matrix multiplication algorithms with reinforcement learning" by Fawzi et al. has shown a method for discovering fewer element multiplication steps in multiplying two matrices. The method is, to over simplify considerably, statistical, yet the resulting multiplication/addition schemas are combinatorial - i.e. the method may well be magic/oracular but the result is easily checkable.
From figure 3 in the paper, it gives minimum previously known mults for 2x2, 3x3, 4x4, and 5x5 matrices, 7,23,49, and 98 respectively, and the better ones found by their method for 4 and 5, namely 47 and 96 (for binary matrices).
It would be nice to know if there has been any research into determining if there are any exact lower bounds for these constants.
Is it known that 7 is the minimum for 2x2? That seems likely given the amount of investigation into it, but I don't think I've heard of any proof that that is the minimum. Maybe with an astronomical number of additions, one can get 6 mults or on the other hand there could be a proof that beyond some number of additions there is nothing more that can be done.
How about 23 mults for 3x3? There could easily be fewer mults but it is beyond our current search capabilities and maybe new techniques like RL/AlphaZero could be developed to hand it to us for checking.
A cursory search of OEIS doesn't turn up anything likely for 1,7,23. But of course, after 1, those values are in question.
Like Ramsey numbers, it may be hard to get exact lower bounds even for small constants. I'm just wondering what is known.