OpenAI 的 ω≤9/4 矩阵乘法上界构造推广至任意特征域
Polynomial Separation of a Shared Tensor Leg and Matrix Multiplication in Arbitrary Characteristic
OpenAI 提出的 9/4 矩阵乘法上界原本依赖域特征，这篇论文用多项式插值重做构造，让它在任意域都成立。
一篇 arXiv 论文改进了 OpenAI 2026年10月2日预印本《An Upper Bound of 9/4 for the Matrix Multiplication Exponent》中 Proposition 3.1 的共享张量腿分离构造。当 M 个张量块共享第一个空间时，M+n-1 份拷贝即可退化到全直和，把源拷贝数从 5M 降到 2M-1。新构造只用多项式插值和截断局部替换，不依赖单位根或阶乘分母，在任何含足够多不同元素的域上都成立。论文还验证了原预印本剩余的谱与离散增长论证与特征无关，因而在代数闭包上每个特征都有 ω≤9/4。
Polynomial Separation of a Shared Tensor Leg and Matrix Multiplication in Arbitrary Characteristic
We improve the shared-leg separation construction in Proposition 3.1 of OpenAI, *An Upper Bound of $9/4$ for the Matrix Multiplication Exponent* (October 2, 2026). If $M$ tensor blocks have a common first space and independent matched sectors on their other two spaces, then $M+n-1$ copies of their matched sum degenerate to the full direct sum of the blocks, each tensored with an $n$-dimensional dot product. For $n=M$, this reduces the number of source copies from $5M$ to $2M-1$. The construction uses polynomial interpolation and truncated local substitution, with no roots of unity or factorial denominators. It works over every field containing enough distinct elements. We verify that the remaining spectral and discrete-growth arguments of the cited preprint are independent of characteristic. Over an algebraic closure, the replacement therefore gives $ω\leq9/4$ in every characteristic.
