AI 中文总结
本研究针对4×4矩阵乘法的48项分解，通过穷举支撑约简和核证书等方法，证明了秩半径至少12、强半径11、边界半径至少8的局部障碍，并揭示了相关判据的不可比性。
AI 中文摘要
我们研究固定双线性张量分解中的替换，计数对完整秩一加项（包括输出因子）的更改。缩短前沿记录固定大小子集的最大秩亏，并确定在给定更改预算内可达到的最小长度。对于有理数域上4×4矩阵乘法的48项Li–Wang–Hu分解D(2)，我们证明秩半径至少为12，强半径恰好为11，边界半径至少为8。因此，每个更短的复数分解必须更改至少十三个原始加项。一个精确的有理十二项替换达到了等长障碍。证明结合了穷举支撑约简、饱和投影核以及控制任意最小竞争者的零角补全论证。一个约化关联论证将核证书转移到张量空间邻域。一个Laurent正规形给出了十六项核心在任意非零复数参数下的强半径恰好为11。在实际参数曲线的非空Zariski开子集上，秩半径至少为12，强半径恰好为11，边界半径至少为8。我们还证明了完整的Kothari–Moitra–Wein充分判据与配备Sylvester的核判据的不可比性。这些结果描述了局部分解结构，而非完整矩阵乘法的新秩界。
英文摘要
We study replacements in fixed bilinear tensor decompositions, counting changes to complete rank-one summands, including output factors. The shortening frontier records the maximum rank defect of a fixed-size subset and determines the minimum length attainable within a change budget. For the rational 48-term Li--Wang--Hu decomposition \(D(2)\) of \(4\times4\) matrix multiplication over \(\mathbb{C}\), we prove rank radius at least 12, strong radius exactly 11, and border radius at least 8. Every shorter complex decomposition therefore changes at least thirteen original summands. An exact rational twelve-term replacement attains the equal-length barrier. The proofs combine exhaustive support reductions with saturated projected kernels and zero-corner completion arguments controlling arbitrary minimal competitors. A reduced-incidence argument transfers kernel certificates to tensor-space neighborhoods. A Laurent normal form gives strong radius exactly 11 for the sixteen-term core at every nonzero complex parameter. On a nonempty Zariski-open subset of the actual parameter curve, the rank radius is at least 12, the strong radius exactly 11, and the border radius at least 8. We also prove incomparability of the full Kothari--Moitra--Wein sufficient criterion and the Sylvester-equipped kernel criterion. These results describe local decomposition structure rather than a new rank bound for full matrix multiplication.
