we-solve (beta)

CS-126 · computer science · mathematics · Algorithms and data structures

Complex matrix multiplication at exponent nine-fourths

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Can square complex matrices be multiplied in $O_\varepsilon (n^{9/4+\varepsilon })$ arithmetic operations for every fixed $\varepsilon >0$, equivalently is the complex matrix-multiplication exponent at most 9/4?

Short English statement

Can square complex matrices be multiplied in $O_\varepsilon (n^{9/4+\varepsilon })$ arithmetic operations for every fixed $\varepsilon >0$, equivalently is the complex matrix-multiplication exponent at most 9/4?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-126-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-126

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 107; 2026-10-02).

Complex matrix multiplication at exponent nine-fourths

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 107; 2026-10-02), the asserted result is that the complex matrix-multiplication exponent is at most $9/4$.

Mathematical statement of the claimed result

Manuscript 1.
For every $\varepsilon>0$, the claim gives an exact complex-arithmetic algorithm multiplying two $n\times n$ matrices in \[ O_{\varepsilon}(n^{9/4+\varepsilon})\quad\text{operations},\qquad\omega\le9/4. \] Its associated tensor inequality uses $T=\sum_{i=1}^sT_i$ with every $T_i\ne0$, a shared first tensor factor and matched disjoint sectors on the other two. A tensor character $\lambda$ takes nonnegative real values and is additive, multiplicative and restriction-monotone, with $\lambda(1)=1$; define $p_X$ by $\lambda(x\sum_{j=1}^my_jz_j)=m^{p_X}$. For every probability vector $q$ the claimed inequality is \[ \lambda(T)\ge e^{p_XH(q)}\prod_i\lambda(T_i)^{q_i},\qquad H(q)=-\sum_iq_i\log q_i, \] with $0\log0=0$. These are exact arithmetic and tensor-character statements, without finite-precision assumptions.

Outline of the argument

For manuscript 1 (family 107; 2026-10-02), tensor degenerations separate blocks that already expose two independent variable legs. Character inequalities for polynomial multiplication, a determinant filtration, and discrete growth bound the detecting character values for matrix multiplication at the claimed exponent.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 107; 2026-10-02).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

Go to contribution →