A 51-Addition Alternative-Basis Kernel for Rank-23 $3\times 3$ Matrix Multiplication

Abstract

We give a rank-23 algorithm for $3\times 3$ matrix multiplication using 51 additions and subtractions in alternative bases. The input and output conversions require five further additions, giving 56 additions in ordinary coordinates. The construction combines linear-program reduction with a search over sparse basis changes, and the resulting programs are distributed as a machine-checkable certificate. We verify correctness by exact coefficient expansion and distinguish the kernel cost from the cost of a complete multiplication.

arXiv comment7 pages. Certificate and verification code: https://github.com/Joshua-Stapleton/51-addition-alt-basis-3x3-rank23