An Upper Bound of 9/4 for the Matrix Multiplication Exponent
We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.
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a completely different approach than was used for O(n 2.373 ) and so forth Ω ( n 3 ) \Omega(n^3) lower bound on the determinantal complexity of the permanent , improving the previous best bound which was quadratic. A randomized polytime algorithm to approximately count the number of perfect…
The new bound is over the complex numbers , a substantial advance toward the conjecture . Percolation: criticality and nonuniqueness. The release includes absence of infinite critical clusters for bond percolation on infinite connected locally finite quasi-transitive graphs with , extending the…
We improve the shared-leg separation construction in Proposition 3.1 of OpenAI, *An Upper Bound of $9/4$ for the Matrix Multiplication Exponent* (October 2, 2026). If $M$ tensor blocks have a common first space and independent matched sectors on their other two spaces, then $M+n-1$ copies of their…
We give a self-contained constructive proof that the matrix multiplication exponent satisfies $ω\le9/4$. This bound was recently claimed by OpenAI~\cite{original} through non-constructive arguments in Strassen's asymptotic spectral framework, counting exact scalar arithmetic operations over the…
— Sergey Gusev (@13SergeyGusev) Oct 20, 2026
We give an exposition of OpenAI's recent proof that the exponent $ω_\mathbb{C}$ of matrix multiplication over the complex numbers satisfies $ω_\mathbb{C}\leq9/4$. Our version of the proof gives $ω_\mathbb{F}\leq9/4$ over every field $\mathbb{F}$. We also treat the rectangular matrix multiplication…
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Rectangular matrix multiplication from polynomial productsarXiv.orgPrzemyslaw UznanskiOct 7, 2026
In this note, we extend the matrix-multiplication analysis by OpenAI to rectangular products and prove $ω(1,k,1)\le2$ for $0\le k\le\frac{1}{2}$, and $ω(1,k,1)\le1+k+\frac{1}{4k}$ for $k\ge\frac{1}{2}$. In particular, $ω(1, \frac{1}{2},1)=2$ and the dual exponent satisfies $α\ge\frac{1}{2}$. The…