AI 中文总结
该研究证明每个3×3可嵌入行随机矩阵可表示为至多7个初等行随机矩阵的乘积，Frydman的例子说明该界是紧的，通过结合相关准则、定理及分析完成证明。
AI 中文摘要
我们证明，每个可嵌入的3×3行随机矩阵均可表示为至多7个初等行随机矩阵的乘积。Frydman的例子表明该界是紧的。证明结合了Frydman的六因子准则、螺旋定理，以及对三参数临界族的归约和沿其持续存在的成员资格证书的分析。
英文摘要
We prove that every embeddable $3\times3$ row-stochastic matrix is a product of at most seven elementary row-stochastic matrices. Frydman's example shows that this bound is sharp. The proof combines Frydman's six-factor criterion and spiral theorem with a reduction to a three-parameter critical family and an analysis of the membership certificates that persist along it.
