Matrix multiplication with exponent at most 9/4

2 · Going further

The main problem is to bound the square matrix-multiplication exponent $\omega$. Over $\mathbb{C}$, Manuscript 1 establishes $\omega \le 9/4$: for every $\epsilon >0$, square matrices of size $n$ can be multiplied using $O_{\epsilon }(n^{9/4+\epsilon })$ arithmetic operations. This formulation allows the bound’s constant to depend on $\epsilon$; it does not assert an $O(n^{9/4})$ operation count.

Manuscript 2 supplies related square and rectangular bounds. Over every field of characteristic zero, it gives $\omega <2.258$, a dual exponent $\alpha >0.465$, and $\omega (1,0.709,1)<2.092$. The dual-exponent statement means that some inner dimension $n^a$, with $a>0.465$, permits multiplication of an $n\times n^a$ matrix by an $n^a\times n$ matrix in $n^{2+o(1)}$ arithmetic operations. The $0.709$ bound concerns the analogous rectangular shape with inner dimension $n^{0.709}$. Its strict square bound and this specific rectangular bound also hold outside a possible finite set of positive characteristics.

Manuscript 3 gives $\omega <2.371054886006746$ over every fixed field, including every positive characteristic. Its title identifies “staggered extraction,” but the supplied abstract does not explain that technique. More generally, none of the supplied abstracts describes the argument behind its bounds, so no mechanism can be reconstructed from this source. The companion manuscripts broaden the stated field coverage and add rectangular results; they are not presented here as steps in a supplied argument for the main bound.

Useful notions

• Matrix dimensions and matrix multiplication
• Growth rates and asymptotic notation
• Counting arithmetic operations
• Fields and their characteristic
• Formal verification in Lean

What was known before / what is new

Limits and assumptions

These are arithmetic-operation bounds, not stated measurements of running time. The $9/4$ result is stated over the complex numbers; it should not be extended to all fields from this source. Manuscript 2’s extension outside finitely many positive characteristics covers its strict square bound and its $0.709$ rectangular bound, not explicitly its dual-exponent bound. Manuscript 3 applies over every fixed field. All three manuscripts are labeled Lean-formalized, but the source supplies no finer account of the formalizations’ scope. It also supplies neither argument details nor a list of remaining open problems.

Mini glossary

Matrix-multiplication exponent

—

The exponent $\omega$ describes the asymptotic arithmetic-operation growth achievable for square matrix multiplication.

Arithmetic-operation model

—

This model measures computation by counting arithmetic operations rather than elapsed running time.

Inner dimension

—

For an $n\times m$ matrix multiplied by an $m\times n$ matrix, the shared dimension $m$ is the inner dimension.

Dual exponent

—

The dual exponent $\alpha$ describes the threshold for inner-dimension powers allowing $n^{2+o(1)}$ rectangular matrix multiplication.

Lean formalization

—

A Lean formalization expresses a mathematical result and its justification in a form checked by Lean.

Keywords : Matrix multiplication · Arithmetic exponent · Lean formalization

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family