# P3118: Does the matrix-multiplication exponent equal two?

- ID: `P3118`
- Reference: `matrix-multiplication-exponent-two`
- Page: https://theoremdb.org/statements/P3118
- Record maturity: Reviewed problem with recorded work

## 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 \(\omega=2\)?

### Context

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.

### Problem setup

- **Definition (arithmetic operation).** A field addition, subtraction, multiplication, or division in the algebraic model.
- **Definition (exponent ω).** The infimum exponent for square matrix multiplication up to n^ε slack.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): The current located upper bound is ω<2.371339 and the trivial information-size lower bound is 2. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): 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](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): Closing any positive part of the gap between 2 and 2.371339 remains open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `matrix-multiplication-exponent-two`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://doi.org/10.1137/1.9781611978322.63
   - 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
   - journal_article; primary source; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/2404.16349
   - Source use: original_summary
   - Proves the current ω<2.371339 upper bound.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1137/1.9781611976465.32
   - journal_article; primary source; checked 2026-08-01
   - Source 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.