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Matrix multiplication with exponent at most 9/4

Proves

\omega\le9/4

over ℂ, giving

O_\varepsilon(n^{9/4+\varepsilon})

arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n

^{a}

with a > 0.465 permits

n^{2+o(1)}

rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.

Manuscripts in this family · 3

An Upper Bound of 9/4 for the Matrix Multiplication Exponent

Oct 2, 2026·paper.pdfLean

We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.

Full abstract

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Complex Matrix Multiplication Below 2.258 and Rectangular Bounds

Sep 24, 2026·Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdfLean

— secondary writeup 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)\lt 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.

Full abstract

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Staggered extraction for exact matrix multiplication over every field

Sep 24, 2026·Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026.pdfLean

We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.

Full abstract

PDF on GitHub LaTeX source