AI 中文总结
本文通过结合Wang的自动化框架与D'Ambrosio的容量-轮廓策略，利用穷举计算证明$\mathbb{F}_2$上$3\times3$矩阵乘法双线性复杂度下界提升至21，并确定$\mathbb{F}_3$上$2\times3$乘$3\times3$矩阵乘法秩为15。
AI 中文摘要
我们证明了在$\mathbb{F}_2$上$3\times3$矩阵乘法的双线性复杂度至少为21，改进了下界20。对第一个输入的限制的下界约束了分解中每个子空间内第一因子的数量。加强的限制下界迫使所有矩阵秩至少为2的第一因子落入一个三维秩一子空间的同一个陪集中。穷举计算发现不存在包含20项的允许的第一因子轮廓。我们还证明了在$\mathbb{F}_3$上$2\times3$乘以$3\times3$矩阵乘法的秩恰好为15：三个轮廓在相应的计算中幸存，而简短的限制论证排除了它们。我们将Wang的用于张量秩下界的自动化框架与D'Ambrosio的容量与轮廓策略相结合。我们的计算贡献是秩一跨度搜索，它加强了子空间下界表，以及一个直接的、对称约简的轮廓枚举器，它同时强制执行所有子空间容量。
英文摘要
We prove that $3\times3$ matrix multiplication over $\mathbb{F}_2$ has bilinear complexity at least $21$, improving the previous lower bound of $20$. Lower bounds for restrictions of the first input constrain how many first factors of a decomposition can lie in each subspace. For a hypothetical $20$-term decomposition, the strengthened restriction bounds force all first factors of matrix rank at least two into a single coset of a three-dimensional rank-one subspace. An exhaustive computation finds no admissible first-factor profile with $20$ terms. We also prove that $2\times3$ by $3\times3$ matrix multiplication over $\mathbb{F}_3$ has rank exactly $15$: three profiles with $14$ terms survive the corresponding computation up to symmetry, and short restriction arguments exclude them. We combine Wang's automated framework for tensor-rank lower bounds with D'Ambrosio's capacity-and-profile strategy. Our computational contributions are rank-one-span searches that strengthen the subspace lower-bound table and a direct, symmetry-reduced profile enumerator that enforces all subspace capacities simultaneously.
