{"id":"openai-math/Matrix-Multiplication-Nine-Fourths","pdf":"https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026","tags":["matrix-multiplication","algebraic-complexity","computational-complexity"],"score":0,"title":"An Upper Bound of 9/4 for the Matrix Multiplication Exponent","domain":"Theoretical computer science","family":"107","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"cs.CC","summary":"We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":7,"published":"2026-10-02","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.10494","date":"2026-10-07","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"Rectangular matrix multiplication from polynomial products","author":"Przemyslaw Uznanski"},{"url":"https://scottaaronson.blog/?p=10169#:~:text=a%20completely%20different%20approach%20than%20was%20used%20for","date":"2026-10-07","kind":"web","site":"Shtetl-Optimized","text":"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…","added":"2026-10-24","title":"The Mathocalypse"},{"url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/#:~:text=The%20new%20bound%20is%20over%20the%20complex%20numbers","date":"2026-10-07","kind":"web","site":"Combinatorics and more","text":"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…","added":"2026-10-24","title":"Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans"},{"url":"https://arxiv.org/abs/2610.18914","date":"2026-10-12","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"Polynomial Separation of a Shared Tensor Leg and Matrix Multiplication in Arbitrary Characteristic","author":"Houxi Lü"},{"url":"https://arxiv.org/abs/2610.28627","date":"2026-10-20","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"Matrix Multiplication from Polynomial Packings: A Constructive and Elementary Proof of $\\omega\\le9/4$","author":"Rafael Pass"},{"url":"https://x.com/13SergeyGusev/status/2112373928802799741","date":"2026-10-20","kind":"x","added":"2026-10-24","author":"Sergey Gusev","handle":"13SergeyGusev"},{"url":"https://arxiv.org/abs/2610.30054","date":"2026-10-21","kind":"web","site":"arXiv.org","text":"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…","added":"2026-10-24","title":"Multiplying matrices via polynomials: exposition and barriers","author":"Alimzhan Amanov, Maxim van den Berg, Jeroen Zuiddam","record":"arxiv/2610.30054"}],"formalized":true,"family_title":"Matrix multiplication with exponent at most 9/4"}