Matrix multiplication with exponent at most nine quarters

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Family 107 · Theoretical computer science
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-09

Multiplying matrices is one of the first algorithms encountered in linear algebra. Its familiar cubic running time looks almost inevitable: there are many output entries, and each requires a long dot product. Family 107 claims a substantial improvement in the asymptotic arithmetic exponent, together with related bounds for rectangular matrices and for fields of positive characteristic.

1. The question

For two square matrices $A$ and $B$ of size $n$, their product $C=AB$ has entries

$$C_{ij}=\sum_{k=1}^{n}A_{ik}B_{kj}.$$

Computing each entry separately uses about $n^3$ multiplications. But these computations share structure. Can one combine the input entries, perform fewer multiplications, and reconstruct every output exactly?

The central question concerns the power of $n$ needed as matrices become arbitrarily large. It does not ask which implementation wins for a matrix on a particular laptop. A small improvement in an exponent can have consequences throughout theoretical algorithms, even when its constants make direct implementation unattractive.

The strongest manuscript in this family, M1, claims that over the complex numbers the exponent is at most $9/4=2.25$. Two companions provide rectangular bounds and guarantees over other fields. Their different domains matter: the strongest numerical claim is not stated uniformly over every field.

2. Background and essential definitions

Arithmetic operations and exponents

An arithmetic operation is an addition, subtraction, or multiplication of field elements. In this model, one such operation has unit cost. Over the complex numbers, this idealizes exact arithmetic; it does not account for the number of bits needed to represent coefficients or control rounding errors.

The matrix multiplication exponent $\omega$ is the infimum of attainable asymptotic powers. The assertion $\omega\le 9/4$ means that, for every fixed $\varepsilon>0$, there is an algorithm using

$$O_{\varepsilon}\!\left(n^{9/4+\varepsilon}\right)$$

arithmetic operations. The hidden constant and chosen construction may depend on $\varepsilon$. The claim does not require an algorithm with exactly $O(n^{9/4})$ cost.

The ordinary algorithm gives an exponent at most three. The standard lower bound is two, corresponding to the quadratic amount of output information in a generic product. Reaching exponent two remains a much stronger target than the bound claimed here.

Rectangular multiplication

Suppose instead that $A$ has $n$ rows and approximately $n^a$ columns, while $B$ has approximately $n^a$ rows and $n$ columns. Write $w(a)$ for the exponent of this multiplication. Its output is still an $n$-by-$n$ matrix.

The dual exponent is

$$\alpha=\sup\{a\in[0,1]:w(a)=2\}.$$

It measures how wide the inner dimension may become while the arithmetic exponent remains two. Larger $\alpha$ is better. Algorithms often produce rectangular products naturally, so this is useful information that a square bound alone does not supply.

Tensors as records of multiplication

A tensor here can be understood as a bookkeeping device for a bilinear calculation. Introduce formal variables $x_{ij},y_{jk},z_{ki}$ and record matrix multiplication by

$$T_n=\sum_{i,j,k=1}^{n}x_{ij}y_{jk}z_{ki}.$$

The coefficient of $z_{ki}$ records the desired entry of the product. The three groups of variables are called the tensor’s legs.

A rank-one term multiplies one linear combination from each group. Tensor rank is the smallest number of such terms needed to express the tensor. Low-rank identities encode economical bilinear algorithms. Combining identities through tensor products corresponds to applying small constructions recursively to large problems.

3. Development history and previous partial results

Strassen’s 1969 algorithm changed the problem by multiplying two-by-two matrices with seven multiplications instead of eight. Recursive blocking gives exponent $\log_2 7$, approximately 2.807. The important discovery was that computing outputs separately wastes algebraic relationships.

Later methods studied many products together. Bini’s work on approximate bilinear algorithms explained how suitable limiting identities can produce exact asymptotic algorithms through interpolation. “Approximate” in that context concerns an intermediate algebraic construction, not permission for incorrect final entries.

Schönhage’s asymptotic sum inequality made simultaneous independent products useful for bounding the exponent. Strassen’s laser method sought large collections of such products inside tensor powers. Coppersmith and Winograd combined low-rank constructions with sets avoiding arithmetic progressions, obtaining a square bound below 2.375477.

Subsequent advances increasingly depended on how candidate products were selected, separated, and recovered. Higher tensor powers offered more possibilities but also more opportunities for different products to share variables. The refined laser method and asymmetric extraction methods reduced these losses.

The manuscripts compare their results with the 2026 optimization work of Dupont and collaborators, which reports a bound below 2.371177. For rectangular multiplication, they also discuss Vassilevska Williams, Xu, Xu, and Zhou, whose bounds include $\alpha\ge0.321334$.

This history explains both aspects of the family: a large claimed improvement in characteristic zero, and a smaller but field-uniform improvement using a more familiar extraction framework.

4. What the new manuscripts claim

M1, dated 2 October 2026, states

$$\omega_{\mathbb C}\le\frac94.$$

Its theorem gives exact matrix multiplication in the arithmetic-operation model with an arbitrarily small additional exponent, as defined above.

M2, dated 24 September, claims over every characteristic-zero field that

$$\omega<2.258,\qquad \alpha>0.465,\qquad w(0.709)<2.092.$$

The square and specified rectangular inequalities also transfer to every field outside one finite set of positive characteristics. The manuscript does not compute that exceptional set. Its positive-characteristic transfer does not assert the same dual-exponent bound.

M3, also dated 24 September, claims

$$\omega_F<2.371054886006746$$

for every fixed field $F$, including every positive characteristic. Constants may depend on the field and the finite construction chosen for the asymptotic algorithm.

The results overlap without being interchangeable. M1 gives the strongest square exponent over $\mathbb C$; M2 supplies additional shapes and transfers; M3 has the broadest explicitly stated field coverage. Nothing in this digest extends the $9/4$ theorem to a setting that its statement does not name.

5. Why the advance matters

Importance: 5/5, provisional. Matrix multiplication is a central primitive connecting algebra, graph algorithms, optimization, and computational geometry. A reduction from an exponent around 2.37 to 2.25 would be a large change on the scale of recent progress, if the claim is correct.

Fast multiplication also supports fast elimination, determinant computation, and linear-system solving through established reductions. The precise computational model and any required zero tests still matter when transferring a result to another task.

The rectangular claims contribute separately. An algorithm that repeatedly multiplies a tall matrix by a wide one cannot always exploit a square algorithm efficiently. Controlling the inner dimension directly can improve the surrounding algorithm without changing the size of its final output.

The rating reflects this centrality and breadth. It is not a judgment about proof correctness, numerical stability, or immediate usefulness in scientific software.

6. The main idea

Separating calculations that share variables

Imagine several useful subcomputations inside a large algebraic expression. Two input groups reveal which subcomputation a term belongs to, but the third group is shared. The pieces are not yet independent, so one cannot simply add their benefits as though each had its own variables.

M1 supplies candidate labels to the shared group using a finite Fourier projection. It then assigns powers of an auxiliary parameter so that a correctly matched label has weight zero, while a mismatched label has positive weight. Taking the leading part removes mismatches.

The output consists of independent blocks, each carrying an additional dot-product factor. That extra factor is useful computational structure, and the construction accounts for the copies spent to produce it. The underlying aim is to separate the blocks without losing all the efficiency that made them valuable.

Measuring every construction consistently

The paper then uses numerical invariants called tensor characters. These add when independent tensors are placed side by side, multiply when constructions are combined, and cannot increase when a tensor is simplified by linear substitutions.

A character measures a matrix multiplication tensor by a power $n^{3t}$, for a parameter $t$. The argument compares its measurements of ordinary polynomial multiplication at different input lengths. Interpolation supplies an upper growth bound; the separation construction and a degree filtration supply lower growth constraints.

The claimed comparison forces $t\le3/4$. A spectral characterization linking these measurements to asymptotic rank then yields the exponent $3t\le9/4$. This is the conceptual route, rather than a list of explicit formulas for multiplying a moderately sized matrix.

What the companions add

M2 organizes repeated label separation and optimizes square and rectangular constructions with entropy calculations. Here entropy measures the number of ways prescribed proportions of tensor components can occur.

M3 staggers extraction stages across different tensor populations. A shortage of usable variables on one side at one stage can be offset by availability at another. It pools these capacities before taking the limiting bottleneck, recovers deleted components, and converts the resulting independent products into a field-uniform exponent bound.

7. Scope, limitations, and open questions

These are asymptotic arithmetic claims. They do not establish a practical crossover size, a floating-point error bound, or a corresponding bit-operation bound for arbitrary encoded inputs. A smaller exponent can coexist with enormous constants.

The strongest square claim leaves a substantial gap between 2 and 2.25. It also leaves open whether the same numerical bound holds in every positive characteristic. The rectangular results answer specific shape questions without determining the whole function $w(a)$.

Finally, a finite exceptional set of characteristics is different from an explicit list of exceptions. M2’s transfer theorem should not be used to certify a particular small prime unless its hypotheses establish that case. M3 provides a weaker numerical bound without this exclusion.

8. Sources and further reading

The historical papers linked above introduce the successive changes in viewpoint: recursive identities, interpolation, simultaneous products, and tensor extraction. For a first reading, the entry formula for matrix multiplication and the meaning of the exponent are enough to follow the question. The tensor language explains how the manuscripts connect finite algebraic identities with asymptotic algorithms.

Source manuscripts

• M1. OpenAI, An Upper Bound of 9/4 for the Matrix Multiplication Exponent (October 2, 2026).
• M2. OpenAI, Complex Matrix Multiplication Below 2.258 and Rectangular Bounds (September 24, 2026).
• M3. OpenAI, Staggered extraction for exact matrix multiplication over every field (September 24, 2026).

Sources consulted on 9 October 2026 at repository revision fd4aeeb. Download the manuscript citations.