破解4×4矩阵的赫尔米特积分解难题，发现密度决定可分解性。
Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices
- 在F₂上穷举验证，发现26.3%的4×4满秩二元矩阵不可分解
- 矩阵中1的个数能以95.7%准确率预测是否可分解
- 可分解矩阵仅占约10维子空间，远低于理论参数量
我们通过在有限域 𝔽₂ 上的穷举搜索，解决了关于4×4满秩矩阵能否表示为两个秩-2矩阵的赫尔米特积这一开放问题。在20,160个满秩二元矩阵中，识别出5,304个反例（占比26.3%）。通过符号枚举，验证这些反例在 ℤ 上依然成立，并提供强有力的数值证据表明其在 ℝ 上也成立。惊人的是，矩阵密度（即1的个数）对可分解性具有高度预测性，分类准确率达95.7%。利用现代机器学习技术发现，可分解矩阵位于16维环境空间中的一个约10维代数簇上，尽管理论上需24个参数（每个4×4秩-2矩阵12个参数），暗示了深层代数约束支配赫尔米特可分解性。
原文摘要 · Abstract (English)
We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\%). We verify that these counterexamples remain valid over $\mathbb{Z}$ through sign enumeration and provide strong numerical evidence for their validity over $\mathbb{R}$. Remarkably, our analysis reveals that matrix density (number of ones) is highly predictive of expressibility, achieving 95.7\% classification accuracy. Using modern machine learning techniques, we discover that expressible matrices lie on an approximately 10-dimensional variety within the 16-dimensional ambient space, despite the naive parameter count of 24 (12 parameters each for two $4 \times 4$ rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.
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