[#P3118] Does the matrix-multiplication exponent equal two?

Matrix multiplication approaching quadratic complexity. — A structural automaton diagram of the statement's mathematical objects.

Problem. Let \(\omega\) be the infimum of the real numbers \(c\) such that two \(n\times n\) matrices over a field can be multiplied using \(O(n^{c+\varepsilon})\) arithmetic operations for every \(\varepsilon>0\). Is \(\omega=2\)?

1Context

Known frontier: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. Open boundary: Closing any positive part of the gap between 2 and 2.371339 remains open.

2Problem setup

Definition 1 (arithmetic operation). A field addition, subtraction, multiplication, or division in the algebraic model.

Definition 2 (exponent ω). The infimum exponent for square matrix multiplication up to n^ε slack.

Remark 1. Input and output size force ω≥2. Decades of tensor constructions have lowered the upper bound close to 2.37, while no superquadratic lower bound is known for unrestricted algebraic algorithms.

3What counts as a solution

Give algorithms proving ω≤2+ε for every ε>0.
Or prove a lower bound ω≥2+δ for some fixed δ>0.

1Status

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. Exact unresolved remainder: Closing any positive part of the gap between 2 and 2.371339 remains open.^[1][2]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. Exact unresolved remainder: Closing any positive part of the gap between 2 and 2.371339 remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Does the matrix-multiplication exponent equal two?. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. The exact unresolved remainder is: Closing any positive part of the gap between 2 and 2.371339 remains open. A complete resolution must meet the following acceptance conditions: - Give algorithms proving ω≤2+ε for every ε>0. - Or prove a lower bound ω≥2+δ for some fixed δ>0. Open full record →
	Result	Supported
claim · Claim 2 The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2.^[1][2] Relevance to this problem Locates the present research frontier immediately below Does the matrix-multiplication exponent equal two?. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. This leaves the following boundary unresolved: Closing any positive part of the gap between 2 and 2.371339 remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
	Trace	Supported
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. Unresolved remainder: Closing any positive part of the gap between 2 and 2.371339 remains open.^[1][2] Relevance to this problem Documents why Does the matrix-multiplication exponent equal two? was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - matrix multiplication exponent omega equals 2 open 2026 best bound - current best omega 2.371339 ADVXXZ 2025 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. Unresolved remainder: Closing any positive part of the gap between 2 and 2.371339 remains open. Related: Claim 1 Open full record →
	Trace	Reported
attempt · Route 2 Closing any positive part of the gap between 2 and 2.371339 remains open. Relevance to this problem Turns the remaining uncertainty in Does the matrix-multiplication exponent equal two? into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Closing any positive part of the gap between 2 and 2.371339 remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Give algorithms proving ω≤2+ε for every ε>0. - Or prove a lower bound ω≥2+δ for some fixed δ>0. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. Exact unresolved remainder: Closing any positive part of the gap between 2 and 2.371339 remains open.

Equivalent-formulation queries: matrix multiplication exponent omega equals 2 open 2026 best bound; current best omega 2.371339 ADVXXZ 2025
Strongest checked neighboring result: The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2.
Exact unresolved remainder: Closing any positive part of the gap between 2 and 2.371339 remains open.

How the 4 records connectTyped relations and evidence flow

ProblemDoes the matrix-multiplication exponent equal two?

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Is VP equal to VNP?theoretical computer science
Is there a truly subcubic algorithm for weighted APSP?theoretical computer science
Strong Exponential Time Hypothesistheoretical computer science

How to cite

TheoremDB contributors, “Does the matrix-multiplication exponent equal two?,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/matrix-multiplication-exponent-two

This page as plain text: matrix-multiplication-exponent-two.md

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1References

Packet source. J. Alman, R. Duan, V. Vassilevska Williams, Y. Xu, Z. Xu, and R. Zhou, “More Asymmetry Yields Faster Matrix Multiplication,” Proceedings of SODA 2025, 2005–2039. abstract and square-matrix exponent theorem. ↗ open copy ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the current ω<2.371339 upper bound.Also cited at J. Alman, R. Duan, V. Vassilevska Williams, Y. Xu, Z. Xu, and R. Zhou, “More Asymmetry Yields Faster Matrix Multiplication,” Proceedings of SODA 2025, 2005–2039. abstract and square-matrix exponent theorem.Source used to assess the problem's recorded status.For Does the matrix-multiplication exponent equal two?: This is the dated publication status for the canonical target Does the matrix-multiplication exponent equal two?.Source named by the research packet.
Josh Alman and Virginia Vassilevska Williams, “A Refined Laser Method and Faster Matrix Multiplication”. Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA) (2021), 522-539. DOI 10.1137/1.9781611976465.32. main exponent theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Develops the dominant modern approach and an earlier record bound.Source used to assess the problem's recorded status.For Does the matrix-multiplication exponent equal two?: Develops the dominant modern approach and an earlier record bound.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3118] Does the matrix-multiplication exponent equal two?

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance