Lower Bound of 22 for 3x3 Matrix Multiplication over the Integers

2026-10-01

AI Summary

This study addresses whether 3×3 recursive matrix multiplication with integer constants can surpass Strassen’s algorithm. Rather than relying solely on matrix rank, the proposed method directly encodes recursive algorithmic constraints to elevate the lower bound on multiplication count to 22. By integrating Wang’s decomposition with the Lean theorem prover, the authors formally verify 359 of 496 subproblems. This work rigorously establishes that 3×3 recursive matrix multiplication requires at least 22 multiplications, yielding a complexity lower bound of O(n^2.814) and definitively precluding any advantage over Strassen’s algorithm. Furthermore, the introduced formalized framework substantially reduces manual auditing overhead, offering a high-assurance verification paradigm for future research on matrix multiplication lower bounds.

Abstract

Strassen showed that two 2x2 matrices can be multiplied with 7 multiplications instead of 8. Applied recursively, his algorithm multiplies two nxn matrices with O(n^2.807) multiplications, beating the naive O(n^3). The best known 3x3 recursive matrix multiplication algorithm uses 23 multiplications O(n^2.854). The best published lower bound of 21 (on algorithms with integer constants) leaves room for an algorithm with O(n^2.771) multiplications, and thus does not rule out the possibility of an algorithm that would beat Strassen's. We prove a lower bound of 22 multiplications for any 3x3 recursive algorithm with integer constants, proving that no such algorithm can do better than O(n^2.814) multiplications, and eliminating the possibility of a 3x3 algorithm that beats Strassen's 2x2 method. The proof builds on a recent decomposition method from Wang, who approached the problem by turning it into 496 subproblems. We provide exact solutions for 359 of them. The proof is in Lean; verification requires auditing only a few short files. The Lean formalization directly encodes statements about the limitations of recursive algorithms for matrix multiplication, as opposed to just a statement about the rank of the problem.

Problem

• matrix multiplication
• lower bound
• recursive algorithm
• integer constants
• computational complexity

Innovation

• Matrix Multiplication
• Lower Bound
• Formal Verification
• Lean
• Recursive Algorithm