55 Additions Suffice for 3x3 Matrix Multiplication at Rank 23

Abstract

We give a 55-addition realization of rank-23 multiplication of two arbitrary $3\times3$ matrices. Together with its 23 bilinear products, the circuit uses 78 scalar operations. This improves the previous state of the art of 56 additions, due to Sun. The construction starts from Perminov's public 58-addition realization cr58_cn122 of a ternary tensor; the contribution is a shorter and, for this fixed orientation of that tensor, provably optimal linear circuit: 13 additions on the left input, 14 on the right input, and 28 at the output. The last circuit is obtained by transposing a 14-addition factor circuit. Because the coefficient alphabet is $\{-1,0,1\}$ and the order of every bilinear product is retained, the algorithm applies over every associative ring, commutative or not. We provide the full straight-line program and tensor factors together with four exact computational checks, including independent Python and Node.js implementations of all 729 Brent identities over $\mathbb Z$.