55次加法足以实现秩为23的3×3矩阵乘法
55 Additions Suffice for 3x3 Matrix Multiplication at Rank 23
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中文总结 AI 辅助
该研究改进了3×3矩阵乘法的加法实现次数，得到55次加法的秩23线性电路，适用于所有结合环，提供了完整程序及校验。
中文摘要 AI 辅助
我们给出了一种对任意两个3×3矩阵进行秩为23的乘法的55次加法实现。该电路与其23个双线性乘积共同使用78次标量运算，这改进了Sun提出的此前56次加法的最新成果。该构造始于Perminov提出的三元张量的公开58次加法实现cr58_cn122；本研究的贡献在于，针对该张量的固定方向，得到了更短且可证明最优的线性电路：左侧输入使用13次加法，右侧输入使用14次加法，输出端使用28次加法。该最终电路通过转置一个14次加法的因子电路得到。由于系数字母表为{-1,0,1}且保留所有双线性乘积的顺序，该算法适用于所有结合环，无论是否可交换。我们提供了完整的直线程序和张量因子，以及四项精确计算校验，包括在整数环ℤ上对全部729个Brent恒等式的独立Python实现和this http URL实现。
英文摘要
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 $\mathbb Z$.
