Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes in the ternary field ($Z_T$), where coefficients are restricted to $\{-1, 0, 1\}$ to minimize naive additive complexity. The core of the method is a GPU-accelerated meta flip graph algorithm that maintains ternary safety through specialized arithmetic operations and sign symmetry breaking. Key results include new best ranks for the formats $4 \times 5 \times 12$, $5 \times 6 \times 10$, and $6 \times 7 \times 9$, the independent discovery of 32 schemes in $Z_T$ that match known optimal ranks (including 8 previously known only with rational coefficients), and 30 rank improvements in the binary field. The analysis of 164 known schemes shows that 92 can be implemented in $Z_T$, while 72 could not be found in the ternary field with current methods, defining the current boundaries of this approach. All software, results, and discovered schemes are provided as open-source.

翻译：矩阵乘法优化仍是计算数学中的一个基础性挑战。本文提出一种新方法，在三元域（$Z_T$）中探索矩阵乘法方案，其中系数限制为 $\{-1, 0, 1\}$，以最小化朴素加法复杂度。该方法的核心是一个GPU加速的元翻转图算法，通过专门的算术运算和符号对称性破缺来维持三元安全性。关键成果包括：针对格式 $4 \times 5 \times 12$、$5 \times 6 \times 10$ 和 $6 \times 7 \times 9$ 获得了新的最佳秩；独立发现了32个三元域中的方案，其匹配已知最优秩（包括8个先前仅知具有有理系数的方案）；以及在二元域中实现了30个秩的改进。对164个已知方案的分析表明，其中92个可在三元域中实现，而72个在当前方法下无法在三元域中找到，这界定了该方法的当前边界。所有软件、结果及发现的方案均已开源提供。