Complex Matrix Multiplication Below 2.258 and Rectangular Bounds

Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and $\omega (1,0.709,1)<2.092$. The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.