4x4 matrix multiplication in fewer than 47 multiplications

Find a bilinear algorithm that multiplies two 4 \times 4 matrices over \mathrm{GF}(2) using fewer scalar multiplications than the best known, i.e. a shorter rank decomposition of the matrix multiplication tensor:

Bar to beat. R < 47 over \mathrm{GF}(2), the AlphaTensor (2022) decomposition, against 49 for recursively applying Strassen in that setting.

Submission format. R triples of 4 \times 4 coefficient matrices (u_r, v_r, w_r) over the field.

Verifier sketch. One exact tensor contraction: expand the sum of R rank-one terms and compare, entry by entry, with the 16 \times 16 \times 16 matrix multiplication tensor. A few thousand field operations — nothing.

Not written yet because it is simply unwritten; nothing here is hard. The only real decision is which field and size to pin, since the records differ between \mathrm{GF}(2), the rationals and the complex numbers, and a listing that leaves that open is not checkable.

Known. Strassen's 1969 algorithm (2 \times 2 in 7) still anchors the subject. The 3 \times 3 case is open between 19 and 23. Machine search (AlphaTensor, and AlphaEvolve for the complex case) has recently moved several small cases.

Why. Small exact decompositions recurse into practical fast matrix multiplication, and the tensor rank of small cases bounds the exponent \omega that all of numerical linear algebra chases.

Ref: Wikipedia: AlphaTensor · Wikipedia: Computational complexity of matrix multiplication