The matrix-multiplication exponent $\omega=\omega_F$ over a fixed field $F$ is the infimum of $\tau\in\mathbb R$ for which multiplying two $n\times n$ matrices over $F$ uses $O(n^\tau)$ arithmetic operations as $n\to\infty$.
Comments
(1)
The bounds as of 29 October 2026 are $2\leq\omega_F\leq 9/4$ for every field $F$. OpenAI proved the upper bound over $\mathbb C$ in Theorem 1.1 of [1]. The manuscript is dated 2 October 2026 and was released on 6 October 2026 [14]. The all-fields statement is proved in [2], Theorem 1.1, and in [3], Section 7. These extensions, rather than the complex-field theorem alone, justify the scope of the entry.
(2)
Algorithms compute the matrix-product polynomials identically in the input entries. Additions, subtractions and multiplications, including multiplication by fixed field constants, each have unit cost [3], Section 2.
(3)
The school algorithm uses $O(n^3)$ operations. The $n^2$ linearly independent quadratic output polynomials give the lower bound $\omega_F\geq 2$ [3], Section 2. Whether $\omega_F=2$ remains open.
(4)
Selected earlier upper bounds are $\log_2 7<2.8074$ (Strassen, 1969, [4]); $\omega<2.376$ (Coppersmith and Winograd, 1990, [5]); $\omega<2.3728596$ (Alman and Vassilevska Williams, 2020 preprint, SODA 2021, [6]); $\omega<2.371339$ (Alman et al., 2024 preprint, SODA 2025, [7]); and $\omega<2.371177$ (Dupont et al., August 2026, [8]). The last result combines improved optimization with AlphaEvolve.
(5)
The exponent depends only on the characteristic of $F$, by [9], Theorem 2.8; see also [3], Section 7. It remains unknown whether different characteristics can give different exponents. The bound $9/4$ applies also to finite fields, not just characteristic zero.
(6)
The proof in [1] uses degenerations of polynomial-multiplication tensors and Strassen's asymptotic spectrum. Lü gives a polynomial-interpolation replacement for the shared-leg separation step that works in arbitrary characteristic [10]. Pass gives an explicit recursive construction using integer arithmetic [11]. Shpilka gives an elementary exposition and an explicit construction approaching the same exponent [3], Theorems 1.1 and 1.2.
(7)
The upper bound means that, for each $\varepsilon>0$ and each fixed field $F$, there are algorithms using $O_{F,\varepsilon}(n^{9/4+\varepsilon})$ operations. It does not assert an $O(n^{9/4})$ algorithm, the equality $\omega_F=9/4$, a bit-complexity bound, or a practical speedup at a specified matrix size. The operational barrier in [2], Theorem 5.4, limits the particular degeneration ingredients and combination rules used there; it is not a lower bound of $9/4$ for unrestricted matrix multiplication.
(8)
OpenAI supplies a Lean formalization over $\mathbb C$ [15]. All-fields Lean extensions are available from Jeroen Zuiddam [16] and Sela Navot [12]. The October 2026 research papers are preprints; the formalizations are additional proof artifacts, not claims of journal peer review.
Alimzhan Amanov, Maxim van den Berg and Jeroen Zuiddam, "Multiplying matrices via polynomials: exposition and barriers", 2026, Theorems 1.1 and 5.4. (arXiv)
[3]
Amir Shpilka, "Exposition of OpenAI’s Matrix Multiplication in Exponent 9/4 (and an explicit algorithm)", Electronic Colloquium on Computational Complexity, TR26-281, 27 October 2026; Theorems 1.1 and 1.2, Section 7. https://eccc.weizmann.ac.il/report/2026/281/
[4]
Volker Strassen, "Gaussian elimination is not optimal", Numerische Mathematik 13, 354–356, 1969. (doi)
[5]
Don Coppersmith and Shmuel Winograd, "Matrix multiplication via arithmetic progressions", Journal of Symbolic Computation 9(3), 251–280, 1990. (doi)
[6]
Josh Alman and Virginia Vassilevska Williams, "A Refined Laser Method and Faster Matrix Multiplication", SODA 2021, 522–539; journal version in TheoretiCS 3, 2024. First preprint: 2020. (arXiv) (doi)
[7]
Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Renfei Zhou, "More Asymmetry Yields Faster Matrix Multiplication", SODA 2025, 2005–2039. The $2.371339$ bound is in the 2024 preprint v2. (arXiv) (doi)
[8]
Emilien Dupont, Marvin Eisenberger, Borislav Kozlovskii, Abbas Mehrabian, Francisco J. R. Ruiz, Abigail See, Renfei Zhou, Josh Alman, Virginia Vassilevska Williams, and Matej Balog, "Improving the matrix multiplication exponent with modern optimization and AlphaEvolve", 2026. (arXiv)
[9]
Arnold Schönhage, "Partial and total matrix multiplication", SIAM Journal on Computing 10(3), 434–455, 1981, Theorem 2.8. (doi)
[10]
Houxi Lü, "Polynomial Separation of a Shared Tensor Leg and Matrix Multiplication in Arbitrary Characteristic", 2026. (arXiv)
[11]
Rafael Pass, "Matrix Multiplication from Polynomial Packings: A Constructive and Elementary Proof of $\omega\leq 9/4$", 2026, Theorem 1.1. (arXiv)
This is a theorem-backed enclosure, not a computed approximation to $\omega_F$. The endpoints $2$ and $9/4=2.25$ are exact. The lower bound is explained in [3], Section 2.
more
The upper bound is OpenAI's complex-field result [1], extended to every field in [2], Theorem 1.1, and [3], Section 7. The interval width measures an unresolved mathematical question, not rounding error. The literature was checked on 29 October 2026; no new local Lean compilation is claimed.
